Projected particle methods for solving McKean Vlasov SDEs
Belomestny, Denis
We propose a novel projection-based particle method for solving the McKean-Vlasov stochastic differential equations. Our approach is based on a projection-type estimation of the marginal density of the solution in each time step. The projection-based particle method leads in many situation to a significant reduction of numerical complexity compared to the widely used kernel density estimation algorithms. We derive strong convergence rates and rates of density estimation. The convergence analysis in the case of linearly growing coefficients turns out to be rather challenging and requires some new type of averaging technique. This case is exemplified by explicit solutions to a class of McKean-Vlasov equations with affine drift. The performance of the proposed algorithm is illustrated by several numerical examples.
 So this is a joint work with John Shore Marcus who is also here from Weisstass Institute. I will start maybe with some introduction to this Marking-Vlasov type equation because I think not everybody here is familiar with this. Basically, Marking-Vlasov equation is connected related to nonlinear Fokker-Planck equation in the sense that if you have an XT random process which satisfies this equation, then the density of this process will satisfy this nonlinear Fokker-Planck equation. What is this equation you see here? This is the kind of diffusion equation which maybe everybody is familiar with, but with some important difference. The coefficients here, the coefficients of the corresponding differential equation depend on the distribution of X. So basically here mu s is the distribution, the measure corresponding to the distribution of X. So basically this is a kind of nonlinear diffusion equation. If mu is the density of mu dx and the density satisfies this nonlinear Fokker-Planck equation. So this is a deep relation between these two types of equations. Basically you can see this equation as a stochastic representation basically for Fokker-Planck equation as also in the case of standard diffusion where you have a linear Fokker-Planck equation which somehow is an equation for the density of the diffusion process. In my talk I will be concerned with some numerical methods of solving such type of nonlinear stochastic differential equations which are especially useful in the case where the coefficient of this equation somehow depends on X and Y and in some sense it cannot be represented in a simpler form because if you can, there are some special cases of this equation which can be even solved explicitly. I also talk about at the end of my talk on some explicitly solvable stochastic differential equation of Machin loss of type. But in the first part I will present the kind of numerical methods how to solve this type of equation in the case of so-called interacting kernels. So basically in the case we have real interaction between Xs and other Xs here because of this integration. So basically the very powerful numerical methods of solving this type of equation is of course particle methods which was developed in the 90s and this kind of method is consistent in considering a system of particles instead of one equation you run somehow many equations and replace the unknown measure mu which is the distribution of X you replace by the empirical distribution. So basically you put here instead of mu X which you don't know you replace it by mu Tn which is somehow depends on your particle or estimated using your particle and basically then you end up with a kind of n-dimensional system of standard SdE's. Okay n-dimensional of course in the case if X is one dimension and generally it is d times n-dimensional system which is now principle can be solved by the standard methods like various discretization methods you can use Euler scheme you can use different type of scheme the time discretization scheme to solve this and then these are famous and very nice result that there is a kind of convergence of the system we have here this particle system to the original system although this is not obvious because the particles are quite dependent and it is not somehow this is very amazing result that you have a kind of convergence also strong convergence and also convergence in the space of measures I will not go into details here much you can look at the papers of Sneedspann Millard which is somehow here but also many other papers. Okay the application there are also many applications of this type of equation I shortly mentioned only few of them this is a population dynamic also in our essence recently also financial mathematics for example which is what is done in the Weierstrass Institute is modeling of lithium batteries and also many other applications and for many of this application it is important to have an efficient numerical scheme to solve this type of equations in a reasonable time and as I said if you use a particle method the standard particle methods you need to consider this type of d-time and dimensional equation system of Sneeds and then you can discretize this for example by using this oiler scheme and this is already a kind of discretized version of this particle system and now you can try to solve it and if you count the number of operation needed to solve it you immediately see that you need usually n squared times L operations that L is the number of time steps to solve this is a kind of of course is a kind of rough estimate but important it was here that you have n squared so is the number of particle increases then the code somehow will be like n squared and this is it could be quite prohibitive and the aim of this project here or projectors aim of this talk is to present numerical scheme which can avoid this type of n square cost and of course not always but in some situations and if we look at this system we will immediately realize that this n square comes from these interactions because you need to compute all possible a function a and b to possible pairs for any i and j in general of course in some particular cases in the case of non-interacting kernels where you have a kind of separation of variables here you can avoid this but in general you need this type of this number of operations and the main challenge here difficulty is how to somehow reduce it and the idea is somehow simple the idea is to approximate this type of interacting system by more or less by a system of kind of non-interacting so what I mean exactly is you can think about having some because you have a density mut and if you can decompose this density into some basis function expansion this is a so-called projection estimator for mut so basically what is you see here you assume that you mut as a measure has a density which is also denoted by mut and this density has this representation with respect to some set of basis function phi k so basically this is a kind of representation which you you can always write somehow but important is of course is as a kind of infinite sound and if you cut this some at some point then you will get a kind of projection of this of measure on the first say capital K basis function projection on the linear space spanned by this first k basis function and and you see here if you consider this type of expansions and cut it somewhere and if you plug this type this mut into your equation you will see you will end up I will show it in details in a minute you will end up with a kind of sum of non-interacting kernels yes some linear combination of non-interacting kernels of course this number of kernels a number of summons in this sound will depend on the place where you cut your summation here so basically here is a coefficient gamma k this projection is coefficient of this projection basically simply scalar products of mut with the phi which also nothing else is the expectation of phi kxt and these coefficients in general can be computed from the particle system yes because if you have a sum approximation of xt you can approximate any expectation of course by using the empirical average yes okay and this is a somehow now we can think look at this type of the composition and this projection and plug this form of mut into our equation back and then we somehow end up with a as I said first with the infinite number of in non-interacting kernels with this coefficient basically we define this coefficients this is the coefficients which somehow also can be computed they are related to this scalar products of a with y and b with phi and also because then this type of the composition basically for your coefficients yes and now you see this is although this was somehow interacting here you see here some of infinite number of non-interacting kernels this is somehow main idea of our project to replace this interacting kernel by approximated by a sum of non-interacting ones of course as I said this sum is now internet sum but you can cut it somewhere and look at this approximation basically you will end up with this type of projected particle system where you have now is some from 0 to capital K so basically you project on the first care on the linear span of first K capital K basis function then you consider this type of equation and coefficients as I also mentioned can be computed by replacing the unknown expectation by empirical average okay what is the somehow advantage of considering this type of equation this system instead the original one so the main somehow reason is that now if you compute the numerical cost of computing this type of discretizing or solving this type of projected particle system we will see that this course will be proportional to the K times n and times the number of discretization step as written here so you see here this is a kind of discretized version of this projected particle system and you will see the total cost will be of order n times K times L and if you compare it to the cost of the original particle system and square L you will see this you will really have some cost reduction if K is a number of kind of a number basis function you are using for approximating is of order smaller than n because in the end of the day you of course like to have a kind of convergence so you will let K go to infinity independence on n so everything will go to infinity and yes so n will go to infinity to ensure convergence of the particle system K will go to infinity to ensure that your approximation error will be small and L go to infinity somehow to ensure that the time discretization error is small and but of course the question is how fast does K go to infinity and if K go to infinity somehow slower than n you will already have a kind of reduction in your in your cost and this is a kind of interesting question now what kind of convergence what are the convergence rates here and also another question is in which case you really can choose K of order smaller than n and still have a kind of convergence yes this is a kind of questions which I am going to answer here basically I mean if you go to convergence analysis of the particle system this convergence analysis usually done under some restrictions on the system and it always somehow heavily depends on this very simple observation that your particle system is a system of somehow identically distributed random variables but they are of course dependent and also there's a kind of symmetry in this particle system so although particle system is a kind of the high-dimensional system of dependent random variables it still has some nice properties like this which also are used in the proof and while all these somehow properties are the convergence somehow analysis of the particle system as a based on the idea that particle system of n increases they became become somehow less dependent and then you can use a kind of averaging if you average them to basically there's still a kind of low-flage numbers here and this is exactly how it works and we need a kind of assumption as well so we need assumptions on first on the basis functions this is for example we need that the basis function lipschitz continuous with some lipschitz constant Lk and also bounded by some sequence decay although here we allow for dependence of Lk and decay on k yes as they can increase one example of such basis function you can take orthonoma system such system like Laguerre system or Hermit system and something like this and you can always check what are these coefficients and then we need also some assumptions on this on the coefficients alpha k and beta k maybe I will go back and recall what is them what are these coefficients this is mainly coefficients here this is a particular case with this transformers of original coefficient and the scalar products of 5k and we need some conditions on alpha k and beta k somehow related to the behavior of this function on the for large x basically we allow for the linear growth of these coefficients and x but also we need a kind of assumption on the lipschitz continuity of this function in x and this is exactly the second type of assumption here this is a bound and also here this is the next one here you need to also kind of bound on the uniform bound on the lipids constant of these coefficients but it still somehow for us is important was it does not include the case of linear growing coefficients so a fine a fine micron loss of models because these are fine type of micron loss of models are widely used and also recently there are some application in the financial mathematics where you have exactly linear coefficients and B and this will result also in linear dependence on alpha and bet on the x so basically our set of assumptions is also include this type of linear models are fine models ok so then I will impose some rather standard assumption on the on the initial distribution and under these conditions there are also some known existence and uniqueness result for this type of equations proofed in the paper of Antonelli-Kohatsu-Igas basically we use this result and what we can prove we can prove basically this type of convergence for our projected particle system so you see here there's this strong error in LP no strong approximation error can be approximated bounded by these three terms basically this is the term which is a standard more or less which appears also in the standard particle system convergence analysis is simply due to central limit or low large numbers I mean result on the convergence of this and the second two terms are due to our approximation this projection approximation and what you see here is coefficients they already appeared in our assumptions but here gamma k where I recall it this is gamma k where defined here this is a basically this is a Fourier coefficients of this of the density of the underlying density which is of course not known but you if you know some properties of this density so you can prove this density has some properties for example a few density somehow is nl2 with respect to this to the corresponding l2 space weighted l2 space then and because phi k is an autonomous system you can prove that this converges to zero and if you need to kind of more precise information how fast it converges to zero you need to analyze also you can analyze the convergence of these coefficients to zero and they may depend for example on the smoothness of your density I will return to this problem to this somehow analysis of the coefficients later but for example if you take phi to be hermit polynomial then basically this condition this how fast this goes to zero it relates to the smoothness of your density basically your density is smooth that it will decrease fast anyway you have such a bound and this you can prove this of course this song goes to zero k goes to infinity so basically this go to zero under our assumption but of course the question is how fast and this will depend on first on the coefficients and second on on the behavior of your gamma k and but maybe to recall that coefficients here are related are not related somehow to the densities are only related to coefficients a and b yes if you look back what are because it maybe is instructive to look again and so because here if you look at this coefficients alpha and beta you will see that higher there somehow related to them only to the behavior of your coefficients a and b if you for example if again your functions five basis functional like hermit polynomials then you will have a kind of your coefficients it will somehow alpha k and beta k will decay to zero fast and this will depend on the smoothness on the function a and b and x and in you yes sorry and also and because alpha alpha k rk this coefficients are related to the somehow to alpha and to beta so basically if you will look again at this bound we'll see that we can make this convergence faster we can expect faster convergence here all these two terms to zero if either your densities move or your coefficient is a smooth so of course there's some relation between density and coefficient but it's not so clear and also what is nice here even if your density is not smooth even if you don't know that your density is smooth because maybe it's difficult to analyze but you observe your coefficients a and b and you know that this is a smooth enough and this a k this will go to zero fast enough so this is what this theorem says and what does it imply for complexity analysis so basically it implies that if say coefficient decay polynomially then you will get something like this polynomial decay of this sum I mean this second two terms they are all sums and I simply didn't write here anything but it means simply that the sum of second two terms and but you can also have an exponential decay and let us look at these two cases and consider then the complexity analysis basically what we are interested in and the choice of k how to choose k in such a way is that we have still convergent we have some convergence rate but the k should not increase fast so we can take for example here in the first k k is k equal to n to the power one over two eta and then the second case we can take even k logarithmically and then of course we will see there's a ratio of costs of the part projected particle system and the cost of the standard system this kind of non-projected will be like this and you see there's a real you have really gain in the cost especially in the second case there's almost one over n this is a this gain and here's of course depends on your atom but anyway you will under assumptions basically the message is like this and the assumption of smoothness of your coefficients you can you may have some real cost reduction for the projected particle system okay now it was what we have done before it was for kind of a non discretized system if we include discretization this is nothing particularly happens we have simply we have to add this term square root of h if we use the order scheme if we use other type of schemes it will be maybe higher order but if we use the kind of order scheme we simply have to add to our previous bound you need to add this square root of h this is the kind of and then we have it then overall error of approximation will consist of basically of four terms yes this is now the question is okay we have proved the kind of strong convergence of our system to the original process x to the this is the kind of strong convergence but another question is which is also interesting what is about density approximating the density mu because in many application we are not really interested in x as itself we are interested in the density because for example if I like to solve this nonlinear focal Planck equation we are interested in the density mu t and basically here is the approximation result for density mu so if you have your particle system you simulated it you can take now this consider now this estimate using somehow your base function is a kind of empirical estimate for your density projection estimate which is known as statistics it usually called projection estimate projection estimator for density and if you look at this projection estimator you will you can analyze that L2 known this weighted L2 known with respect to some weight function we use this estimate and you will somehow end up with very simple bound what you need now to look at it these two terms but basically the second term again as also before what is related to the convergence of your coefficients gamma k to zero which is in turn related more to the smoothness of your density but the first term can be analyzed using our previous result on the strong type convergence of your particle system because the gamma kn is based on particle system xt ik is our particle system which is which we simulated yes so basically it goes like this first we somehow can simulate our particle system using this projection type estimator and then in the next step you can estimate your density and then the analysis of the second first term can be done by employing this type of Liebschitz continuity function phi k basis function and then you somehow relate this error to the strong approximation error of your particle system and also you have some variance term which will not harm and then you end up with this type of approximation where again here you cover one square root convergence basically and here's some term which may depend on k additionally here you have also term which was already appeared is already appeared in our main theorem and I also already commented on this and also this is clear yes so basically we have this additional term here and all other terms are basically can be analyzed but of course the main here in order to derive precise convergence rate in terms of k and n we need to know how these coefficients lk and dk behave so basically this bounds on the on the Liebschitz constant of the basis function and on the absolute value of this basis function are important they play here important role but you can derive under various assumptions on your basis functions and assumptions on your somehow your coefficients you can derive explicit convergence rate here for the density now okay this is about this approximation maybe again or the message to take out here this is that you can really achieve cost reduction if under some additional assumptions on your coefficients of your i's on the coefficient of your sd for example assuming that your coefficient is smooth or on the assumption if you that your density underlying density in the t is smooth under this assumption you can achieve cost reduction with respect as compared to the original particle methods now let us look at this type a fine type of microglasm equation which somehow have some very interesting properties and this are fine type of equation also can be used to model to consider generalization of some affine models in finance because I find models in financial must be very quite popular for many reasons I have some tractable structure also the option pricing can be done in this model but here you can generalize this as a type of affine models with coefficients which may depend for example on this expectation so basically on the distribution of xd in this also makes sense in some applications to consider the generalization and here for example this is a kind of general affine model with diffusion coefficient depending only on this expectation and for this type of models you can even derive the explicit solution of this and this is quite amazing it's also somehow maybe it's not so amazing because in a standard affine models you can also derive something but here this type of models are different structure because they are non-linear they depend also on the distribution here in form of this expectation so basically if you introduce these Gauss transforms you can derive an explicit solution by solving this system of standard ODEs so basically you somehow transform the solution of this problem to the solution or that kind of ODEs system if you maybe know in the case of standard affine models you also in order to find for example the characteristic function of the affine model you also need to solve a kind of Riccati equation it's also Riccati equation also type of non-linear as an ODEs system it's ODEs equation and here you end up this this type of equation which in some cases can be solved explicitly for example you can prove such a nice result that if you have a marking velocity equation of this type with some bound equations then this is equivalent to the following equation which is now a kind of standard SDE non-autonomous standard SDE so basically standard SDE with coefficients which depend on the time non-autonomous here so basically and this type of equation has explicit solution you can look up at the book of a platinum and this is a kind of a standard formula for this type because it's also fine non-autonomous equation has this solution but basically now you see you have a kind of explicit solution in terms of this function h a 0 h a 1 and this function they solve the system of ODEs basically this is nice and you can also consider some particular examples to see even more explicit solution you can look for this also a fine type of models where all coefficients also are fine in u yes because here you can of course study this in general but you can also look at the equation where your function a and not a 1 also are fine and then you will end up with a some explicit form of this gauss transform and you also will and can solve this you have this equation riccati type equation for function at and this riccati type equation can be solved explicitly and if you solve that explicitly you will end up with this exact form of the solution of this equation and you see that that x t is nothing other than a gaussian kind of gaussian process with a mean variance function which can be computed explicitly so you can you can look at different cases here this is interesting that for this type of equations you can study also the problems of existence and uniqueness because and also the problem of explosion related problem of explosion because for some type of coefficients you cannot simply apply the known results on the existence and convergence for example on the existence of uniqueness from the paper of kohatsu higa and antonyli but you can study this in separate way using this for example for this type of test examples these are fine you can look at different possibilities and you also might have here even even a kind of explosion i will maybe go to this okay so this is about somehow you have in most cases you can derive an exact form of solution and also this is for example this type of model which is known as a shimitsu yamada model which is widely used in physics for this type of models you also can derive explicit solution and for this type of models more generally you have this coefficient a of this form the situation becomes more difficult because now usually if you look at the paper of antonyli kohatsu higa as i mentioned they derive uniqueness convergence only under some additional assumption for example this type of assumption of linear growth but here you have at the look you have x time u so you already have you may not have linear growth because you have the product here so basically if you allow that your coefficients are like this you are you cannot apply the results of existence and uniqueness and you you immediately realize if you look at this that you also may have also a kind of explosion here at least for some coefficients and if you look at this you have a again the solution which is explicitly you have equation which is explicitly solvable and you can also and it depends also on explosion depends on what happens here it's a denominator and it depends on the sign of this ratio if the sign of this ratio can be positive or negative and then you may have explosion or you may not have explosion it's some time point and this is the explosion issue somehow very delicate here because this type of model even even for standard affine models in the finance this is interesting question how there are many papers devoid to studying the explosion issue in the affine models and here the issue this problem is even more delicate because here you have two types of first you have dependence on x and your coefficient and you have dependence on u and this is of course this interaction between x and u how they somehow interact may lead to explosion or may lead to some problem I mean this is the interesting analysis which is has not been done in the literature at all and here is only kind of first step the kind of examples and for this how to study this type of explosion or not explosion in this type of if you are interested in this I can refer to the paper for affine models there are several important papers Martin Keller rest so there are other people they studied for us study the explosion issues in the affine model here then is a completely open question what kind of explosion may happen here and finally I would like to consider one numerical example so this interacting kernels where you have this interacting kernel cube is it exponential a normal one basically so you have here it depends on the difference but it's still an interacting kernel and in this particular case you can compute your you can employ use this hermit polynomials and compute your coefficient explicitly so this is somehow is a kind of toy example but somehow shows also the behavior of coefficients you are like to analyze for example here this it follows from this formula that your coefficient alpha for example has this structure so it decays in n it decays very fast yes so basically and then you can compute basically this density mu t estimate density as described both as described and then you can look at this l2 arrow of the density and those two pictures somehow I intended to indicate that the choice of k test is important yeah because what was k test k test was a number of basis function which is somehow used in the second second step of estimation so maybe I should stress that this k is was the number of basis function used to approximate particle system and k test was the number of basis function yes basis function used to in the second step to approximate density they must not coincide and even though even more here you see that you need to choose k test in a proper way otherwise you will hear you have some value which minimizes this arrow and then if you go further it may happen that your result will be worse and this density which is somehow estimated is this one such as density using I think five basis function here meet this estimating like this this is in fact this is not normal density this is some kind of Gaussian density no not even gaussian this is a kind of diff is because this structure is not affine so you you may not you don't know really what kind of density explicit form I think it would be difficult to derive explicit form but this is estimate of the density and this is of course a kind of simple example but it shows that you can somehow choose your basis function for your problem yes because okay here as a here meet polynomials are nice but in other cases you can use Laguerre polynomials you can add that type of autonomous system to approximate your density like this is my all so thank you very much for your attention okay thank you very much Dennis for the talk is there any question so thank you for your talk and this is more a curiosity than than else in the very first slide you mentioned the an application of making blasts of modeling is lithium batteries yeah could you tell me in which sense and how do you use making blasts of to to model maybe John can say more about this the new making continuous mechanics and so on and basically these particles there are particles are ions which are polluting through some solution in an electrochemical situation and which have interaction but unfortunately I'm not a physicist to explain this this kind of application but because we are working on this application we have also some collaboration but it's started shortly and so I know he's playing I know about general how it works but I think this is a type of a special type of equations but they're all interacting equation yes interacting I think this is always this is important for us it is an interacting system which somehow still if you like to solve it like this it will require squared operations and so we can apply this method but I am more familiar with financial applications so you have for example this type of affine models and so on I thank you for a talk so in the original equation which you showed if I add jumps as well yeah can we again do something similar project a particle system or you mean for a fine model so for this general general making also yeah I think so but we have not yet studied this type of I think there are some recent results on the Michael Russell equation with jumps and of course it would be interesting to look also at this type of equation but we have not yet done this yes thanks do you have any idea how to extend this method to a higher dimension or even slightly higher dimension like the basis would inevitably extension extension actually exponentially go away you mean no this is a system as a kind of multi-dimensional yes we have d principle we have general d here so this is the d-dimensional Markov-Lasov equation but what do you mean maybe infinite dimensional what do you mean No I mean in a higher dimension of density you inevitably need to use higher order basis for how do you pick the higher order basis do you have any insight in picking basis for a density function in a higher order situation so that you don't need like free to power n for your I mean the how to choose basis function in the high dimensional situation like yes there's a very interesting question and of course it depends on the structure of your problem I think this is a general this is a kind of difficult question and how to approximate a high dimensional density using some expansion but any at work example there's a recent results on the tensor type of approximations for your density where you use tensor on special type of basis for high dimensional problems so this is the kind of this is not directly related to somehow to this this is a somehow general problem how to approximate density on the high dimensional space this is a general difficult problem but here if you have some information on the structure for example it might happen that your density maybe you have some structure or maybe your coefficients have structure in maybe they are not something like some general function of x maybe they depend on the difference or maybe they depend on the sum of something like this if you have some structure yes and your coefficient then you may try to choose basis also using this information the structure for example this kind of additive structure or this tensor structure of your coefficient may imply this tensor structure of your density and also may lead to some proper choice of your basis function yes you need some assumptions otherwise you will of course all you will have this curse of dimensionality in any case yes but if you have some structure on your model you may be able to avoid it okay I think it's time to thank Denise again
 So this is a joint work with John Shore Marcus who is also here from Weisstass Institute.  I will start maybe with some introduction to this Marking-Vlasov type equation because I think  not everybody here is familiar with this. Basically, Marking-Vlasov equation is  connected related to nonlinear Fokker-Planck equation in the sense that if you have an  XT random process which satisfies this equation, then the density of this process will satisfy  this nonlinear Fokker-Planck equation. What is this equation you see here? This is the kind of  diffusion equation which maybe everybody is familiar with, but with some important difference.  The coefficients here, the coefficients of the corresponding differential equation depend on  the distribution of X. So basically here mu s is the distribution, the measure corresponding to  the distribution of X. So basically this is a kind of nonlinear diffusion equation. If mu is  the density of mu dx and the density satisfies this nonlinear Fokker-Planck equation. So this  is a deep relation between these two types of equations. Basically you can see this equation  as a stochastic representation basically for Fokker-Planck equation as also in the case of  standard diffusion where you have a linear Fokker-Planck equation which somehow is an  equation for the density of the diffusion process. In my talk I will be concerned with some numerical  methods of solving such type of nonlinear stochastic differential equations which are  especially useful in the case where the coefficient of this equation somehow depends on X and Y and  in some sense it cannot be represented in a simpler form because if you can, there are some  special cases of this equation which can be even solved explicitly. I also talk about at the end  of my talk on some explicitly solvable stochastic differential equation of Machin loss of type.  But in the first part I will present the kind of numerical methods how to solve this type of  equation in the case of so-called interacting kernels. So basically in the case we have real  interaction between Xs and other Xs here because of this integration. So basically the very powerful  numerical methods of solving this type of equation is of course particle methods which was developed  in the 90s and this kind of method is consistent in considering a system of particles instead of  one equation you run somehow many equations and replace the unknown measure mu which is the  distribution of X you replace by the empirical distribution. So basically you put here instead  of mu X which you don't know you replace it by mu Tn which is somehow depends on your particle  or estimated using your particle and basically then you end up with a kind of n-dimensional system  of standard SdE's. Okay n-dimensional of course in the case if X is one dimension and generally it  is d times n-dimensional system which is now principle can be solved by the standard methods  like various discretization methods you can use Euler scheme you can use different type of  scheme the time discretization scheme to solve this and then these are famous and very nice result  that there is a kind of convergence of the system we have here this particle system to the original  system although this is not obvious because the particles are quite dependent and it is not somehow  this is very amazing result that you have a kind of convergence also strong convergence and also  convergence in the space of measures I will not go into details here much you can look at the papers  of Sneedspann Millard which is somehow here but also many other papers. Okay the application  there are also many applications of this type of equation I shortly mentioned only few of them  this is a population dynamic also in our essence recently also financial mathematics for example  which is what is done in the Weierstrass Institute is modeling of lithium batteries and also many  other applications and for many of this application it is important to have an efficient numerical  scheme to solve this type of equations in a reasonable time and as I said if you use a  particle method the standard particle methods you need to consider this type of d-time and  dimensional equation system of Sneeds and then you can discretize this for example by using this  oiler scheme and this is already a kind of discretized version of this particle system and  now you can try to solve it and if you count the number of operation needed to solve it you  immediately see that you need usually n squared times L operations that L is the number of time  steps to solve this is a kind of of course is a kind of rough estimate but important it  was here that you have n squared so is the number of particle increases then the code somehow  will be like n squared and this is it could be quite prohibitive and the aim of this project  here or projectors aim of this talk is to present numerical scheme which can avoid this type of n  square cost and of course not always but in some situations and if we look at this system we will  immediately realize that this n square comes from these interactions because you need to compute  all possible a function a and b to possible pairs for any i and j in general of course in some  particular cases in the case of non-interacting kernels where you have a kind of separation  of variables here you can avoid this but in general you need this type of this number of  operations and the main challenge here difficulty is how to somehow reduce it and the idea is somehow  simple the idea is to approximate this type of interacting system by more or less by a  system of kind of non-interacting so what I mean exactly is you can think about having some  because you have a density mut and if you can decompose this density into some basis function  expansion this is a so-called projection estimator for mut so basically what is you see here you  assume that you mut as a measure has a density which is also denoted by mut and this density has  this representation with respect to some set of basis function phi k so basically this is a kind  of representation which you you can always write somehow but important is of course is  as a kind of infinite sound and if you cut this some at some point then you will get a kind of  projection of this of measure on the first say capital K basis function projection on the linear  space spanned by this first k basis function and and you see here if you consider this type of  expansions and cut it somewhere and if you plug this type this mut into your equation you will see  you will end up I will show it in details in a minute you will end up with a kind of sum of  non-interacting kernels yes some linear combination of non-interacting kernels of  course this number of kernels a number of summons in this sound will depend on the place  where you cut your summation here so basically here is a coefficient gamma k this projection  is coefficient of this projection basically simply scalar products of mut with the phi  which also nothing else is the expectation of phi kxt and these coefficients in general can be  computed from the particle system yes because if you have a sum approximation of xt you can  approximate any expectation of course by using the empirical average yes okay and this is a somehow  now we can think look at this type of the composition and this projection and plug this  form of mut into our equation back and then we somehow end up with a as I said first with the  infinite number of in non-interacting kernels with this coefficient basically we define this  coefficients this is the coefficients which somehow also can be computed they are related to  this scalar products of a with y and b with phi and also because then this type of the  composition basically for your coefficients yes and now you see this is although this was somehow  interacting here you see here some of infinite number of non-interacting kernels this is somehow  main idea of our project to replace this interacting kernel by approximated by a sum of  non-interacting ones of course as I said this sum is now internet sum but you can cut it somewhere  and look at this approximation basically you will end up with this type of projected particle  system where you have now is some from 0 to capital K so basically you project on the first  care on the linear span of first K capital K basis function then you consider this type of  equation and coefficients as I also mentioned can be computed by replacing the unknown expectation  by empirical average okay what is the somehow advantage of considering this type of equation  this system instead the original one so the main somehow reason is that now if you compute the  numerical cost of computing this type of discretizing or solving this type of projected  particle system we will see that this course will be proportional to the K times n and times the  number of discretization step as written here so you see here this is a kind of discretized  version of this projected particle system and you will see the total cost will be of order n times  K times L and if you compare it to the cost of the original particle system and square L you  will see this you will really have some cost reduction if K is a number of kind of a number  basis function you are using for approximating is of order smaller than n because in the end  of the day you of course like to have a kind of convergence so you will let K go to infinity  independence on n so everything will go to infinity and yes so n will go to infinity to  ensure convergence of the particle system K will go to infinity to ensure that your  approximation error will be small and L go to infinity somehow to ensure that the time  discretization error is small and but of course the question is how fast does K go to infinity  and if K go to infinity somehow slower than n you will already have a kind of reduction in your  in your cost and this is a kind of interesting question now what kind of convergence what  are the convergence rates here and also another question is in which case you really can choose  K of order smaller than n and still have a kind of convergence yes this is a kind of questions  which I am going to answer here basically I mean if you go to convergence analysis of the particle  system this convergence analysis usually done under some restrictions on the system and it  always somehow heavily depends on this very simple observation that your particle system is a system  of somehow identically distributed random variables but they are of course dependent and also there's  a kind of symmetry in this particle system so although particle system is a kind of the  high-dimensional system of dependent random variables it still has some nice properties  like this which also are used in the proof and while all these somehow properties are the  convergence somehow analysis of the particle system as a based on the idea that particle system  of n increases they became become somehow less dependent and then you can use a kind of averaging  if you average them to basically there's still a kind of low-flage numbers here and this is  exactly how it works and we need a kind of assumption as well so we need assumptions on  first on the basis functions this is for example we need that the basis function  lipschitz continuous with some lipschitz constant Lk and also bounded by some sequence decay although  here we allow for dependence of Lk and decay on k yes as they can increase one example of such  basis function you can take orthonoma system such system like Laguerre system or Hermit  system and something like this and you can always check what are these coefficients and then we need  also some assumptions on this on the coefficients alpha k and beta k maybe I will go back and recall  what is them what are these coefficients this is mainly coefficients here this is a particular  case with this transformers of original coefficient and the scalar products of 5k and we need some  conditions on alpha k and beta k somehow related to the behavior of this function on the for large  x basically we allow for the linear growth of these coefficients and x but also we need a kind of  assumption on the lipschitz continuity of this function in x and this is exactly the second  type of assumption here this is a bound and also here this is the next one here you need to also  kind of bound on the uniform bound on the lipids constant of these coefficients but it still  somehow for us is important was it does not include the case of linear growing coefficients so  a fine a fine micron loss of models because these are fine type of micron loss of models are widely  used and also recently there are some application in the financial mathematics where you have exactly  linear coefficients and B and this will result also in linear dependence on alpha and bet on the  x so basically our set of assumptions is also include this type of linear models are fine models  ok so then I will impose some rather standard assumption on the on the initial distribution and  under these conditions there are also some known existence and uniqueness result for this type of  equations proofed in the paper of Antonelli-Kohatsu-Igas basically we use this result and what we can  prove we can prove basically this type of convergence for our projected particle system so you see here  there's this strong error in LP no strong approximation error can be approximated bounded  by these three terms basically this is the term which is a standard more or less which appears  also in the standard particle system convergence analysis is simply due to central limit or  low large numbers I mean result on the convergence of this and the second two terms are due to our  approximation this projection approximation and what you see here is coefficients they already  appeared in our assumptions but here gamma k where I recall it this is gamma k where defined here  this is a basically this is a Fourier coefficients of this of the density of the underlying density  which is of course not known but you if you know some properties of this density so you can prove  this density has some properties for example a few density somehow is nl2 with respect to this  to the corresponding l2 space weighted l2 space then and because phi k is an autonomous system you  can prove that this converges to zero and if you need to kind of more precise information how fast  it converges to zero you need to analyze also you can analyze the convergence of these coefficients  to zero and they may depend for example on the smoothness of your density I will return to this  problem to this somehow analysis of the coefficients later but for example if you take phi to be  hermit polynomial then basically this condition this how fast this goes to zero it relates to  the smoothness of your density basically your density is smooth that it will decrease fast  anyway you have such a bound and this you can prove this of course this song goes to zero k goes  to infinity so basically this go to zero under our assumption but of course the question is how  fast and this will depend on first on the coefficients and second on on the behavior  of your gamma k and but maybe to recall that coefficients here are related are not related  somehow to the densities are only related to coefficients a and b yes if you look back what  are because it maybe is instructive to look again and so because here if you look at this  coefficients alpha and beta you will see that higher there somehow related to them only to the  behavior of your coefficients a and b if you for example if again your functions five basis  functional like hermit polynomials then you will have a kind of your coefficients it will somehow  alpha k and beta k will decay to zero fast and this will depend on the smoothness on the function  a and b and x and in you yes sorry and also and because alpha alpha k rk this coefficients are  related to the somehow to alpha and to beta so basically if you will look again at this bound  we'll see that we can make this convergence faster we can expect faster convergence here  all these two terms to zero if either your densities move or your coefficient is a smooth  so of course there's some relation between density and coefficient but it's not so clear  and also what is nice here even if your density is not smooth even if you don't know that your  density is smooth because maybe it's difficult to analyze but you observe your coefficients a and  b and you know that this is a smooth enough and this a k this will go to zero fast enough so this  is what this theorem says and what does it imply for complexity analysis so basically it implies  that if say coefficient decay polynomially then you will get something like this polynomial  decay of this sum I mean this second two terms they are all sums and I simply didn't write here  anything but it means simply that the sum of second two terms and but you can also have an  exponential decay and let us look at these two cases and consider then the complexity analysis  basically what we are interested in and the choice of k how to choose k in such a way is that we  have still convergent we have some convergence rate but the k should not increase fast so we  can take for example here in the first k k is k equal to n to the power one over two eta and then  the second case we can take even k logarithmically and then of course we will see there's a ratio  of costs of the part projected particle system and the cost of the standard system this kind of  non-projected will be like this and you see there's a real you have really gain in the cost  especially in the second case there's almost one over n this is a this gain and here's of  course depends on your atom but anyway you will under assumptions basically the message is like  this and the assumption of smoothness of your coefficients you can you may have some real  cost reduction for the projected particle system okay now it was what we have done before it was  for kind of a non discretized system if we include discretization this is nothing particularly  happens we have simply we have to add this term square root of h if we use the order scheme if  we use other type of schemes it will be maybe higher order but if we use the kind of order scheme  we simply have to add to our previous bound you need to add this square root of h this is the  kind of and then we have it then overall error of approximation will consist of basically of  four terms yes this is now the question is okay we have proved the kind of strong convergence of  our system to the original process x to the this is the kind of strong convergence but another  question is which is also interesting what is about density approximating the density mu because  in many application we are not really interested in x as itself we are interested in the density  because for example if I like to solve this nonlinear focal Planck equation we are interested  in the density mu t and basically here is the approximation result for density mu so if you  have your particle system you simulated it you can take now this consider now this estimate  using somehow your base function is a kind of empirical estimate for your density projection  estimate which is known as statistics it usually called projection estimate projection  estimator for density and if you look at this projection estimator you will you can analyze  that L2 known this weighted L2 known with respect to some weight function we use this  estimate and you will somehow end up with very simple bound what you need now to look at it  these two terms but basically the second term again as also before what is related to the  convergence of your coefficients gamma k to zero which is in turn related more to the  smoothness of your density but the first term can be analyzed using our previous result on the  strong type convergence of your particle system because the gamma kn is based on particle system  xt ik is our particle system which is which we simulated yes so basically it goes like this  first we somehow can simulate our particle system using this projection type estimator  and then in the next step you can estimate your density and then the analysis of the second  first term can be done by employing this type of Liebschitz continuity function phi k basis  function and then you somehow relate this error to the strong approximation error of your particle  system and also you have some variance term which will not harm and then you end up with this  type of approximation where again here you cover one square root convergence basically and here's  some term which may depend on k additionally here you have also term which was already appeared  is already appeared in our main theorem and I also already commented on this and also this  is clear yes so basically we have this additional term here and all other terms are basically  can be analyzed but of course the main here in order to derive precise convergence rate in terms  of k and n we need to know how these coefficients lk and dk behave so basically this bounds on the  on the Liebschitz constant of the basis function and on the absolute value of this basis function  are important they play here important role but you can derive under various assumptions on your  basis functions and assumptions on your somehow your coefficients you can derive explicit convergence  rate here for the density now okay this is about this approximation maybe again or the  message to take out here this is that you can really achieve cost reduction if under some  additional assumptions on your coefficients of your i's on the coefficient of your sd for example  assuming that your coefficient is smooth or on the assumption if you that your density underlying  density in the t is smooth under this assumption you can achieve cost reduction with respect as  compared to the original particle methods now let us look at this type a fine type of  microglasm equation which somehow have some very interesting properties and this are fine type  of equation also can be used to model to consider generalization of some affine models in finance  because I find models in financial must be very quite popular for many reasons I have some  tractable structure also the option pricing can be done in this model but here you can  generalize this as a type of affine models with coefficients which may depend for example on  this expectation so basically on the distribution of xd in this also makes sense in some applications  to consider the generalization and here for example this is a kind of general affine model  with diffusion coefficient depending only on this expectation and for this type of models you can  even derive the explicit solution of this and this is quite amazing it's also somehow maybe  it's not so amazing because in a standard affine models you can also derive something but here  this type of models are different structure because they are non-linear they depend also on  the distribution here in form of this expectation so basically if you introduce these Gauss transforms  you can derive an explicit solution by solving this system of standard ODEs so basically you  somehow transform the solution of this problem to the solution or that kind of ODEs system if you  maybe know in the case of standard affine models you also in order to find for example the  characteristic function of the affine model you also need to solve a kind of Riccati equation  it's also Riccati equation also type of non-linear as an ODEs system it's ODEs equation and here you  end up this this type of equation which in some cases can be solved explicitly for example you  can prove such a nice result that if you have a marking velocity equation of this type with some  bound equations then this is equivalent to the following equation which is now a kind of standard  SDE non-autonomous standard SDE so basically standard SDE with coefficients which depend on the time  non-autonomous here so basically and this type of equation has explicit solution you can look up  at the book of a platinum and this is a kind of a standard formula for this type because it's also  fine non-autonomous equation has this solution but basically now you see you have a kind of explicit  solution in terms of this function h a 0 h a 1 and this function they solve the system of ODEs  basically this is nice and you can also consider some particular examples to see  even more explicit solution you can look for this also a fine type of models where all coefficients  also are fine in u yes because here you can of course study this in general but you can also  look at the equation where your function a and not a 1 also are fine and then you will end up with  a some explicit form of this gauss transform and you also will and can solve this you have this  equation riccati type equation for function at and this riccati type equation can be solved explicitly  and if you solve that explicitly you will end up with this exact form of the solution of this  equation and you see that that x t is nothing other than a gaussian kind of gaussian process  with a mean variance function which can be computed explicitly  so you can you can look at different cases here this is interesting that for this type of equations  you can study also the problems of existence and uniqueness because and also the problem of  explosion related problem of explosion because for some type of coefficients you cannot simply  apply the known results on the existence and convergence for example on the existence of  uniqueness from the paper of kohatsu higa and antonyli but you can study this in separate  way using this for example for this type of test examples these are fine you can look at different  possibilities and you also might have here even even a kind of explosion i will maybe go to this  okay so this is about somehow you have in most cases you can derive  an exact form of solution and also this is for example this type of model which is known as  a shimitsu yamada model which is widely used in physics for this type of models you also can derive  explicit solution and for this type of models more generally you have this coefficient a of this  form the situation becomes more difficult because now usually if you look at the paper of antonyli  kohatsu higa as i mentioned they derive uniqueness convergence only under some additional assumption  for example this type of assumption of linear growth but here you have at the look you have x time  u so you already have you may not have linear growth because you have the product here so basically  if you allow that your coefficients are like this you are you cannot apply the results of  existence and uniqueness and you you immediately realize if you look at this that you also may have  also a kind of explosion here at least for some coefficients and if you look at this you have  a again the solution which is explicitly you have equation which is explicitly solvable and you can  also and it depends also on explosion depends on what happens here it's a denominator and it depends  on the sign of this ratio if the sign of this ratio can be positive or negative and then you may  have explosion or you may not have explosion it's some time point and this is the explosion issue  somehow very delicate here because this type of model even even for standard affine models in the  finance this is interesting question how there are many papers devoid to studying the explosion issue  in the affine models and here the issue this problem is even more delicate because here you have  two types of first you have dependence on x and your coefficient and you have dependence on u  and this is of course this interaction between x and u how they somehow interact  may lead to explosion or may lead to some problem I mean this is the interesting analysis which is  has not been done in the literature at all and here is only kind of first step the kind of examples  and for this how to study this type of explosion or not explosion in this type of if you are interested  in this I can refer to the paper for affine models there are several important papers  Martin Keller rest so there are other people they studied for us study the explosion issues  in the affine model here then is a completely open question what kind of explosion may happen here  and finally I would like to consider one numerical example so this interacting kernels where you  have this interacting kernel cube is it exponential a normal one basically so you have here it depends  on the difference but it's still an interacting kernel and in this particular case you can compute  your you can employ use this hermit polynomials and compute your coefficient explicitly so  this is somehow is a kind of toy example but somehow shows also the behavior of coefficients you are  like to analyze for example here this it follows from this formula that your coefficient alpha  for example has this structure so it decays in n it decays very fast yes so basically  and then you can compute basically this density mu t estimate density as described  both as described and then you can look at this l2 arrow of the density and those two pictures  somehow I intended to indicate that the choice of k test is important yeah because what was k test  k test was a number of basis function which is somehow used in the second second step of estimation  so maybe I should stress that this k is was the number of basis function used to approximate  particle system and k test was the number of basis function yes basis function used to in the second  step to approximate density they must not coincide and even though even more here you see that you  need to choose k test in a proper way otherwise you will hear you have some value which minimizes  this arrow and then if you go further it may happen that your result will be worse and  this density which is somehow estimated is this one such as density using I think five basis function  here meet this estimating like this this is in fact this is not normal density this is some kind  of Gaussian density no not even gaussian this is a kind of diff is because this structure is not  affine so you you may not you don't know really what kind of density explicit form I think it  would be difficult to derive explicit form but this is estimate of the density and this is of  course a kind of simple example but it shows that you can somehow choose your basis function for your  problem yes because okay here as a here meet polynomials are nice but in other cases you can  use Laguerre polynomials you can add that type of autonomous system to approximate your density  like this is my all so thank you very much for your attention  okay thank you very much Dennis for the talk is there any question  so thank you for your talk and this is more a curiosity than than else in the very first  slide you mentioned the an application of making blasts of modeling is lithium batteries yeah could  you tell me in which sense and how do you use making blasts of to to model maybe John can say more about this  the new making continuous mechanics and so on and basically these  particles there are particles are ions which are polluting through some solution in an  electrochemical situation and which have interaction but unfortunately I'm not a physicist to explain  this this kind of application but because we are working on this application we have also some  collaboration but it's started shortly and so I know he's playing I know about general how it  works but I think this is a type of a special type of equations but they're all interacting  equation yes interacting I think this is always this is important for us it is an interacting  system which somehow still if you like to solve it like this it will require squared  operations and so we can apply this method but I am more familiar with financial applications  so you have for example this type of affine models and so on  I thank you for a talk so in the original equation which you showed if I add jumps  as well yeah can we again do something similar project a particle system or  you mean for a fine model so for this general general making also yeah I think so but we have  not yet studied this type of I think there are some recent results on the Michael Russell equation  with jumps and of course it would be interesting to look also at this type of equation but we have  not yet done this yes thanks do you have any idea how to extend this method to a higher dimension  or even slightly higher dimension like the basis would inevitably extension extension  actually exponentially go away you mean no this is a system as a kind of multi-dimensional  yes we have d principle we have general d here so this is the d-dimensional  Markov-Lasov equation but what do you mean maybe infinite dimensional what do you mean  No I mean in a higher dimension of density you inevitably need to use higher order basis for  how do you pick the higher order basis do you have any insight in picking  basis for a density function in a higher order situation so that you don't need  like free to power n for your I mean the how to choose basis function in the high dimensional  situation like yes there's a very interesting question and of course it depends on the structure  of your problem I think this is a general this is a kind of difficult question and  how to approximate a high dimensional density using some expansion but any at work example  there's a recent results on the tensor type of approximations for your density where you use  tensor on special type of basis for high dimensional problems so this is the kind of this is not directly  related to somehow to this this is a somehow general problem how to approximate density on the  high dimensional space this is a general difficult problem but here if you have some information  on the structure for example it might happen that your density maybe you have some structure  or maybe your coefficients have structure in maybe they are not something like some general  function of x maybe they depend on the difference or maybe they depend on the sum of something like  this if you have some structure yes and your coefficient then you may try to choose basis  also using this information the structure for example this kind of additive structure  or this tensor structure of your coefficient may imply this tensor structure of your density and  also may lead to some proper choice of your basis function yes you need some assumptions otherwise  you will of course all you will have this curse of dimensionality in any case yes but if you have  some structure on your model you may be able to avoid it okay I think it's time to thank Denise again
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So this is a joint work with John Shore Marcus who is also here from Weisstass Institute. I will start maybe with some introduction to this Marking-Vlasov type equation because I think not everybody here is familiar with this. Basically, Marking-Vlasov equation is connected related to nonlinear Fokker-Planck equation in the sense that if you have an XT random process which satisfies this equation, then the density of this process will satisfy this nonlinear Fokker-Planck equation. What is this equation you see here? This is the kind of diffusion equation which maybe everybody is familiar with, but with some important difference. The coefficients here, the coefficients of the corresponding differential equation depend on the distribution of X. So basically here mu s is the distribution, the measure corresponding to the distribution of X. So basically this is a kind of nonlinear diffusion equation. If mu is the density of mu dx and the density satisfies this nonlinear Fokker-Planck equation. So this is a deep relation between these two types of equations. Basically you can see this equation as a stochastic representation basically for Fokker-Planck equation as also in the case of standard diffusion where you have a linear Fokker-Planck equation which somehow is an equation for the density of the diffusion process. In my talk I will be concerned with some numerical methods of solving such type of nonlinear stochastic differential equations which are especially useful in the case where the coefficient of this equation somehow depends on X and Y and in some sense it cannot be represented in a simpler form because if you can, there are some special cases of this equation which can be even solved explicitly. I also talk about at the end of my talk on some explicitly solvable stochastic differential equation of Machin loss of type. But in the first part I will present the kind of numerical methods how to solve this type of equation in the case of so-called interacting kernels. So basically in the case we have real interaction between Xs and other Xs here because of this integration. So basically the very powerful numerical methods of solving this type of equation is of course particle methods which was developed in the 90s and this kind of method is consistent in considering a system of particles instead of one equation you run somehow many equations and replace the unknown measure mu which is the distribution of X you replace by the empirical distribution. So basically you put here instead of mu X which you don't know you replace it by mu Tn which is somehow depends on your particle or estimated using your particle and basically then you end up with a kind of n-dimensional system of standard SdE's. Okay n-dimensional of course in the case if X is one dimension and generally it is d times n-dimensional system which is now principle can be solved by the standard methods like various discretization methods you can use Euler scheme you can use different type of scheme the time discretization scheme to solve this and then these are famous and very nice result that there is a kind of convergence of the system we have here this particle system to the original system although this is not obvious because the particles are quite dependent and it is not somehow this is very amazing result that you have a kind of convergence also strong convergence and also convergence in the space of measures I will not go into details here much you can look at the papers of Sneedspann Millard which is somehow here but also many other papers. Okay the application there are also many applications of this type of equation I shortly mentioned only few of them this is a population dynamic also in our essence recently also financial mathematics for example which is what is done in the Weierstrass Institute is modeling of lithium batteries and also many other applications and for many of this application it is important to have an efficient numerical scheme to solve this type of equations in a reasonable time and as I said if you use a particle method the standard particle methods you need to consider this type of d-time and dimensional equation system of Sneeds and then you can discretize this for example by using this oiler scheme and this is already a kind of discretized version of this particle system and now you can try to solve it and if you count the number of operation needed to solve it you immediately see that you need usually n squared times L operations that L is the number of time steps to solve this is a kind of of course is a kind of rough estimate but important it was here that you have n squared so is the number of particle increases then the code somehow will be like n squared and this is it could be quite prohibitive and the aim of this project here or projectors aim of this talk is to present numerical scheme which can avoid this type of n square cost and of course not always but in some situations and if we look at this system we will immediately realize that this n square comes from these interactions because you need to compute all possible a function a and b to possible pairs for any i and j in general of course in some particular cases in the case of non-interacting kernels where you have a kind of separation of variables here you can avoid this but in general you need this type of this number of operations and the main challenge here difficulty is how to somehow reduce it and the idea is somehow simple the idea is to approximate this type of interacting system by more or less by a system of kind of non-interacting so what I mean exactly is you can think about having some because you have a density mut and if you can decompose this density into some basis function expansion this is a so-called projection estimator for mut so basically what is you see here you assume that you mut as a measure has a density which is also denoted by mut and this density has this representation with respect to some set of basis function phi k so basically this is a kind of representation which you you can always write somehow but important is of course is as a kind of infinite sound and if you cut this some at some point then you will get a kind of projection of this of measure on the first say capital K basis function projection on the linear space spanned by this first k basis function and and you see here if you consider this type of expansions and cut it somewhere and if you plug this type this mut into your equation you will see you will end up I will show it in details in a minute you will end up with a kind of sum of non-interacting kernels yes some linear combination of non-interacting kernels of course this number of kernels a number of summons in this sound will depend on the place where you cut your summation here so basically here is a coefficient gamma k this projection is coefficient of this projection basically simply scalar products of mut with the phi which also nothing else is the expectation of phi kxt and these coefficients in general can be computed from the particle system yes because if you have a sum approximation of xt you can approximate any expectation of course by using the empirical average yes okay and this is a somehow now we can think look at this type of the composition and this projection and plug this form of mut into our equation back and then we somehow end up with a as I said first with the infinite number of in non-interacting kernels with this coefficient basically we define this coefficients this is the coefficients which somehow also can be computed they are related to this scalar products of a with y and b with phi and also because then this type of the composition basically for your coefficients yes and now you see this is although this was somehow interacting here you see here some of infinite number of non-interacting kernels this is somehow main idea of our project to replace this interacting kernel by approximated by a sum of non-interacting ones of course as I said this sum is now internet sum but you can cut it somewhere and look at this approximation basically you will end up with this type of projected particle system where you have now is some from 0 to capital K so basically you project on the first care on the linear span of first K capital K basis function then you consider this type of equation and coefficients as I also mentioned can be computed by replacing the unknown expectation by empirical average okay what is the somehow advantage of considering this type of equation this system instead the original one so the main somehow reason is that now if you compute the numerical cost of computing this type of discretizing or solving this type of projected particle system we will see that this course will be proportional to the K times n and times the number of discretization step as written here so you see here this is a kind of discretized version of this projected particle system and you will see the total cost will be of order n times K times L and if you compare it to the cost of the original particle system and square L you will see this you will really have some cost reduction if K is a number of kind of a number basis function you are using for approximating is of order smaller than n because in the end of the day you of course like to have a kind of convergence so you will let K go to infinity independence on n so everything will go to infinity and yes so n will go to infinity to ensure convergence of the particle system K will go to infinity to ensure that your approximation error will be small and L go to infinity somehow to ensure that the time discretization error is small and but of course the question is how fast does K go to infinity and if K go to infinity somehow slower than n you will already have a kind of reduction in your in your cost and this is a kind of interesting question now what kind of convergence what are the convergence rates here and also another question is in which case you really can choose K of order smaller than n and still have a kind of convergence yes this is a kind of questions which I am going to answer here basically I mean if you go to convergence analysis of the particle system this convergence analysis usually done under some restrictions on the system and it always somehow heavily depends on this very simple observation that your particle system is a system of somehow identically distributed random variables but they are of course dependent and also there's a kind of symmetry in this particle system so although particle system is a kind of the high-dimensional system of dependent random variables it still has some nice properties like this which also are used in the proof and while all these somehow properties are the convergence somehow analysis of the particle system as a based on the idea that particle system of n increases they became become somehow less dependent and then you can use a kind of averaging if you average them to basically there's still a kind of low-flage numbers here and this is exactly how it works and we need a kind of assumption as well so we need assumptions on first on the basis functions this is for example we need that the basis function lipschitz continuous with some lipschitz constant Lk and also bounded by some sequence decay although here we allow for dependence of Lk and decay on k yes as they can increase one example of such basis function you can take orthonoma system such system like Laguerre system or Hermit system and something like this and you can always check what are these coefficients and then we need also some assumptions on this on the coefficients alpha k and beta k maybe I will go back and recall what is them what are these coefficients this is mainly coefficients here this is a particular case with this transformers of original coefficient and the scalar products of 5k and we need some conditions on alpha k and beta k somehow related to the behavior of this function on the for large x basically we allow for the linear growth of these coefficients and x but also we need a kind of assumption on the lipschitz continuity of this function in x and this is exactly the second type of assumption here this is a bound and also here this is the next one here you need to also kind of bound on the uniform bound on the lipids constant of these coefficients but it still somehow for us is important was it does not include the case of linear growing coefficients so a fine a fine micron loss of models because these are fine type of micron loss of models are widely used and also recently there are some application in the financial mathematics where you have exactly linear coefficients and B and this will result also in linear dependence on alpha and bet on the x so basically our set of assumptions is also include this type of linear models are fine models ok so then I will impose some rather standard assumption on the on the initial distribution and under these conditions there are also some known existence and uniqueness result for this type of equations proofed in the paper of Antonelli-Kohatsu-Igas basically we use this result and what we can prove we can prove basically this type of convergence for our projected particle system so you see here there's this strong error in LP no strong approximation error can be approximated bounded by these three terms basically this is the term which is a standard more or less which appears also in the standard particle system convergence analysis is simply due to central limit or low large numbers I mean result on the convergence of this and the second two terms are due to our approximation this projection approximation and what you see here is coefficients they already appeared in our assumptions but here gamma k where I recall it this is gamma k where defined here this is a basically this is a Fourier coefficients of this of the density of the underlying density which is of course not known but you if you know some properties of this density so you can prove this density has some properties for example a few density somehow is nl2 with respect to this to the corresponding l2 space weighted l2 space then and because phi k is an autonomous system you can prove that this converges to zero and if you need to kind of more precise information how fast it converges to zero you need to analyze also you can analyze the convergence of these coefficients to zero and they may depend for example on the smoothness of your density I will return to this problem to this somehow analysis of the coefficients later but for example if you take phi to be hermit polynomial then basically this condition this how fast this goes to zero it relates to the smoothness of your density basically your density is smooth that it will decrease fast anyway you have such a bound and this you can prove this of course this song goes to zero k goes to infinity so basically this go to zero under our assumption but of course the question is how fast and this will depend on first on the coefficients and second on on the behavior of your gamma k and but maybe to recall that coefficients here are related are not related somehow to the densities are only related to coefficients a and b yes if you look back what are because it maybe is instructive to look again and so because here if you look at this coefficients alpha and beta you will see that higher there somehow related to them only to the behavior of your coefficients a and b if you for example if again your functions five basis functional like hermit polynomials then you will have a kind of your coefficients it will somehow alpha k and beta k will decay to zero fast and this will depend on the smoothness on the function a and b and x and in you yes sorry and also and because alpha alpha k rk this coefficients are related to the somehow to alpha and to beta so basically if you will look again at this bound we'll see that we can make this convergence faster we can expect faster convergence here all these two terms to zero if either your densities move or your coefficient is a smooth so of course there's some relation between density and coefficient but it's not so clear and also what is nice here even if your density is not smooth even if you don't know that your density is smooth because maybe it's difficult to analyze but you observe your coefficients a and b and you know that this is a smooth enough and this a k this will go to zero fast enough so this is what this theorem says and what does it imply for complexity analysis so basically it implies that if say coefficient decay polynomially then you will get something like this polynomial decay of this sum I mean this second two terms they are all sums and I simply didn't write here anything but it means simply that the sum of second two terms and but you can also have an exponential decay and let us look at these two cases and consider then the complexity analysis basically what we are interested in and the choice of k how to choose k in such a way is that we have still convergent we have some convergence rate but the k should not increase fast so we can take for example here in the first k k is k equal to n to the power one over two eta and then the second case we can take even k logarithmically and then of course we will see there's a ratio of costs of the part projected particle system and the cost of the standard system this kind of non-projected will be like this and you see there's a real you have really gain in the cost especially in the second case there's almost one over n this is a this gain and here's of course depends on your atom but anyway you will under assumptions basically the message is like this and the assumption of smoothness of your coefficients you can you may have some real cost reduction for the projected particle system okay now it was what we have done before it was for kind of a non discretized system if we include discretization this is nothing particularly happens we have simply we have to add this term square root of h if we use the order scheme if we use other type of schemes it will be maybe higher order but if we use the kind of order scheme we simply have to add to our previous bound you need to add this square root of h this is the kind of and then we have it then overall error of approximation will consist of basically of four terms yes this is now the question is okay we have proved the kind of strong convergence of our system to the original process x to the this is the kind of strong convergence but another question is which is also interesting what is about density approximating the density mu because in many application we are not really interested in x as itself we are interested in the density because for example if I like to solve this nonlinear focal Planck equation we are interested in the density mu t and basically here is the approximation result for density mu so if you have your particle system you simulated it you can take now this consider now this estimate using somehow your base function is a kind of empirical estimate for your density projection estimate which is known as statistics it usually called projection estimate projection estimator for density and if you look at this projection estimator you will you can analyze that L2 known this weighted L2 known with respect to some weight function we use this estimate and you will somehow end up with very simple bound what you need now to look at it these two terms but basically the second term again as also before what is related to the convergence of your coefficients gamma k to zero which is in turn related more to the smoothness of your density but the first term can be analyzed using our previous result on the strong type convergence of your particle system because the gamma kn is based on particle system xt ik is our particle system which is which we simulated yes so basically it goes like this first we somehow can simulate our particle system using this projection type estimator and then in the next step you can estimate your density and then the analysis of the second first term can be done by employing this type of Liebschitz continuity function phi k basis function and then you somehow relate this error to the strong approximation error of your particle system and also you have some variance term which will not harm and then you end up with this type of approximation where again here you cover one square root convergence basically and here's some term which may depend on k additionally here you have also term which was already appeared is already appeared in our main theorem and I also already commented on this and also this is clear yes so basically we have this additional term here and all other terms are basically can be analyzed but of course the main here in order to derive precise convergence rate in terms of k and n we need to know how these coefficients lk and dk behave so basically this bounds on the on the Liebschitz constant of the basis function and on the absolute value of this basis function are important they play here important role but you can derive under various assumptions on your basis functions and assumptions on your somehow your coefficients you can derive explicit convergence rate here for the density now okay this is about this approximation maybe again or the message to take out here this is that you can really achieve cost reduction if under some additional assumptions on your coefficients of your i's on the coefficient of your sd for example assuming that your coefficient is smooth or on the assumption if you that your density underlying density in the t is smooth under this assumption you can achieve cost reduction with respect as compared to the original particle methods now let us look at this type a fine type of microglasm equation which somehow have some very interesting properties and this are fine type of equation also can be used to model to consider generalization of some affine models in finance because I find models in financial must be very quite popular for many reasons I have some tractable structure also the option pricing can be done in this model but here you can generalize this as a type of affine models with coefficients which may depend for example on this expectation so basically on the distribution of xd in this also makes sense in some applications to consider the generalization and here for example this is a kind of general affine model with diffusion coefficient depending only on this expectation and for this type of models you can even derive the explicit solution of this and this is quite amazing it's also somehow maybe it's not so amazing because in a standard affine models you can also derive something but here this type of models are different structure because they are non-linear they depend also on the distribution here in form of this expectation so basically if you introduce these Gauss transforms you can derive an explicit solution by solving this system of standard ODEs so basically you somehow transform the solution of this problem to the solution or that kind of ODEs system if you maybe know in the case of standard affine models you also in order to find for example the characteristic function of the affine model you also need to solve a kind of Riccati equation it's also Riccati equation also type of non-linear as an ODEs system it's ODEs equation and here you end up this this type of equation which in some cases can be solved explicitly for example you can prove such a nice result that if you have a marking velocity equation of this type with some bound equations then this is equivalent to the following equation which is now a kind of standard SDE non-autonomous standard SDE so basically standard SDE with coefficients which depend on the time non-autonomous here so basically and this type of equation has explicit solution you can look up at the book of a platinum and this is a kind of a standard formula for this type because it's also fine non-autonomous equation has this solution but basically now you see you have a kind of explicit solution in terms of this function h a 0 h a 1 and this function they solve the system of ODEs basically this is nice and you can also consider some particular examples to see even more explicit solution you can look for this also a fine type of models where all coefficients also are fine in u yes because here you can of course study this in general but you can also look at the equation where your function a and not a 1 also are fine and then you will end up with a some explicit form of this gauss transform and you also will and can solve this you have this equation riccati type equation for function at and this riccati type equation can be solved explicitly and if you solve that explicitly you will end up with this exact form of the solution of this equation and you see that that x t is nothing other than a gaussian kind of gaussian process with a mean variance function which can be computed explicitly so you can you can look at different cases here this is interesting that for this type of equations you can study also the problems of existence and uniqueness because and also the problem of explosion related problem of explosion because for some type of coefficients you cannot simply apply the known results on the existence and convergence for example on the existence of uniqueness from the paper of kohatsu higa and antonyli but you can study this in separate way using this for example for this type of test examples these are fine you can look at different possibilities and you also might have here even even a kind of explosion i will maybe go to this okay so this is about somehow you have in most cases you can derive an exact form of solution and also this is for example this type of model which is known as a shimitsu yamada model which is widely used in physics for this type of models you also can derive explicit solution and for this type of models more generally you have this coefficient a of this form the situation becomes more difficult because now usually if you look at the paper of antonyli kohatsu higa as i mentioned they derive uniqueness convergence only under some additional assumption for example this type of assumption of linear growth but here you have at the look you have x time u so you already have you may not have linear growth because you have the product here so basically if you allow that your coefficients are like this you are you cannot apply the results of existence and uniqueness and you you immediately realize if you look at this that you also may have also a kind of explosion here at least for some coefficients and if you look at this you have a again the solution which is explicitly you have equation which is explicitly solvable and you can also and it depends also on explosion depends on what happens here it's a denominator and it depends on the sign of this ratio if the sign of this ratio can be positive or negative and then you may have explosion or you may not have explosion it's some time point and this is the explosion issue somehow very delicate here because this type of model even even for standard affine models in the finance this is interesting question how there are many papers devoid to studying the explosion issue in the affine models and here the issue this problem is even more delicate because here you have two types of first you have dependence on x and your coefficient and you have dependence on u and this is of course this interaction between x and u how they somehow interact may lead to explosion or may lead to some problem I mean this is the interesting analysis which is has not been done in the literature at all and here is only kind of first step the kind of examples and for this how to study this type of explosion or not explosion in this type of if you are interested in this I can refer to the paper for affine models there are several important papers Martin Keller rest so there are other people they studied for us study the explosion issues in the affine model here then is a completely open question what kind of explosion may happen here and finally I would like to consider one numerical example so this interacting kernels where you have this interacting kernel cube is it exponential a normal one basically so you have here it depends on the difference but it's still an interacting kernel and in this particular case you can compute your you can employ use this hermit polynomials and compute your coefficient explicitly so this is somehow is a kind of toy example but somehow shows also the behavior of coefficients you are like to analyze for example here this it follows from this formula that your coefficient alpha for example has this structure so it decays in n it decays very fast yes so basically and then you can compute basically this density mu t estimate density as described both as described and then you can look at this l2 arrow of the density and those two pictures somehow I intended to indicate that the choice of k test is important yeah because what was k test k test was a number of basis function which is somehow used in the second second step of estimation so maybe I should stress that this k is was the number of basis function used to approximate particle system and k test was the number of basis function yes basis function used to in the second step to approximate density they must not coincide and even though even more here you see that you need to choose k test in a proper way otherwise you will hear you have some value which minimizes this arrow and then if you go further it may happen that your result will be worse and this density which is somehow estimated is this one such as density using I think five basis function here meet this estimating like this this is in fact this is not normal density this is some kind of Gaussian density no not even gaussian this is a kind of diff is because this structure is not affine so you you may not you don't know really what kind of density explicit form I think it would be difficult to derive explicit form but this is estimate of the density and this is of course a kind of simple example but it shows that you can somehow choose your basis function for your problem yes because okay here as a here meet polynomials are nice but in other cases you can use Laguerre polynomials you can add that type of autonomous system to approximate your density like this is my all so thank you very much for your attention okay thank you very much Dennis for the talk is there any question so thank you for your talk and this is more a curiosity than than else in the very first slide you mentioned the an application of making blasts of modeling is lithium batteries yeah could you tell me in which sense and how do you use making blasts of to to model maybe John can say more about this the new making continuous mechanics and so on and basically these particles there are particles are ions which are polluting through some solution in an electrochemical situation and which have interaction but unfortunately I'm not a physicist to explain this this kind of application but because we are working on this application we have also some collaboration but it's started shortly and so I know he's playing I know about general how it works but I think this is a type of a special type of equations but they're all interacting equation yes interacting I think this is always this is important for us it is an interacting system which somehow still if you like to solve it like this it will require squared operations and so we can apply this method but I am more familiar with financial applications so you have for example this type of affine models and so on I thank you for a talk so in the original equation which you showed if I add jumps as well yeah can we again do something similar project a particle system or you mean for a fine model so for this general general making also yeah I think so but we have not yet studied this type of I think there are some recent results on the Michael Russell equation with jumps and of course it would be interesting to look also at this type of equation but we have not yet done this yes thanks do you have any idea how to extend this method to a higher dimension or even slightly higher dimension like the basis would inevitably extension extension actually exponentially go away you mean no this is a system as a kind of multi-dimensional yes we have d principle we have general d here so this is the d-dimensional Markov-Lasov equation but what do you mean maybe infinite dimensional what do you mean No I mean in a higher dimension of density you inevitably need to use higher order basis for how do you pick the higher order basis do you have any insight in picking basis for a density function in a higher order situation so that you don't need like free to power n for your I mean the how to choose basis function in the high dimensional situation like yes there's a very interesting question and of course it depends on the structure of your problem I think this is a general this is a kind of difficult question and how to approximate a high dimensional density using some expansion but any at work example there's a recent results on the tensor type of approximations for your density where you use tensor on special type of basis for high dimensional problems so this is the kind of this is not directly related to somehow to this this is a somehow general problem how to approximate density on the high dimensional space this is a general difficult problem but here if you have some information on the structure for example it might happen that your density maybe you have some structure or maybe your coefficients have structure in maybe they are not something like some general function of x maybe they depend on the difference or maybe they depend on the sum of something like this if you have some structure yes and your coefficient then you may try to choose basis also using this information the structure for example this kind of additive structure or this tensor structure of your coefficient may imply this tensor structure of your density and also may lead to some proper choice of your basis function yes you need some assumptions otherwise you will of course all you will have this curse of dimensionality in any case yes but if you have some structure on your model you may be able to avoid it okay I think it's time to thank Denise again
numerical methods of solving such type of nonlinear stochastic differential equations which said if you use a particle method the standard course in some particular cases in the case of to replace this interacting kernel by of convergence what are the convergence rates of the particle system as a based on the idea use a kind of averaging if you average them to linear growing coefficients so a fine a fine standard particle system convergence analysis is of the density of the underlying density which is complexity analysis basically what we are strong convergence of our system to the original the density mu because in many application we are respect as compared to the original particle generalize this as a type of affine models with even derive the explicit solution of this and model which is widely used in physics for this