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consider an equity market with asset capitalizations @xmath0 at time @xmath1 , and with covariance and relative risk rates @xmath2 and @xmath3 , respectively . at any given time @xmath4 , these rates are _ nonanticipative functionals _ of past - and - present capitalizations @xmath5 , @xmath6 ; they are not specified with precision but are , rather , subject to `` knightian uncertainty . '' to wit , for a given collection @xmath7 of nonempty compact and convex subsets on @xmath8 , where @xmath9 is the space of real , symmetric , positive definite @xmath10 matrices , and @xmath11 is the origin in  @xmath12 , they are subject to the constraint @xmath13 in other words , the pair @xmath14 must take values at time @xmath4 inside the compact , convex set @xmath15 which is determined by the current location of the asset capitalization process ; but within this range , the actual value @xmath16 is allowed to depend on past capitalizations as well . [ to put it a little differently : the constraint ( [ e.3 ] ) is not necessarily `` markovian , '' as long as the sets in ( [ 1.b ] ) are not singletons . ]    under these circumstances , what is the highest return on investment relative to the market that can be achieved using nonanticipative investment rules , and with probability one under all possible market model configurations that satisfy the constraints of ( [ e.3 ] ) ? what are the weights in the various assets of an investment rule that accomplishes this ?    answers : subject to appropriate conditions , @xmath17 and @xmath18\\[-8pt ] \eqntext{i = 1 , \ldots , n , 0 \le t \le t,}\end{aligned}\ ] ] respectively . here the function @xmath19 $ ] is the smallest nonnegative solution , in the class @xmath20 , of the fully nonlinear parabolic partial differential inequality @xmath21 subject to the initial condition @xmath22 , with @xmath23 we use in ( [ fg ] ) , ( [ e.3.d ] ) and throughout this paper , the notation @xmath24 , @xmath25 , @xmath26 , @xmath27 and define @xmath28    we call the function @xmath29 the _ arbitrage function , _ as @xmath30 gives the smallest initial capital starting with which an investor , who uses nonanticipative investment rules , can match or outperform the market portfolio by time @xmath31 , if the initial configuration of asset capitalizations is @xmath32 at @xmath33 , and does so with probability one under any `` admissible '' market configuration that might materialize . it is perhaps worth noting that this function @xmath29 is characterized almost entirely in terms of the prevalent covariance structure @xmath34 . the relative risk @xmath35 enters only indirectly , namely , in determining the family of sets ( [ e.3.a ] ) which are admissible for the covariance structure . put a bit differently , the only role  @xmath35 plays is to ensure the asset capitalization process @xmath36 takes values in @xmath37 .    under additional regularity conditions , @xmath38 is the value of a stochastic control problem : the maximal probability that the diffusion process @xmath39 with initial configuration @xmath40 , values in the punctured nonnegative orthant @xmath41 of ( [ 1.b ] ) , infinitesimal generator @xmath42 and controlled through the choice of covariance function @xmath43 which satisfies @xmath44 for all @xmath45 , does not hit the boundary of the orthant @xmath46 by time @xmath47 . under appropriate conditions the function @xmath29 satisfies then , in the notation of ( [ e.3.d ] ) , the _ hamilton  jacobi  bellman _ ( _ hjb _ ) _ equation _ @xmath48 _ relation to extant work _ : stochastic control problems of the `` maximal probability of containment '' type were apparently pioneered by van mellaert and dorato ( @xcite ) ( see also @xcite , pages 157158 ) . the `` knightian uncertainty '' constraint imposed in ( [ e.3 ] ) is very similar to the formulation of stochastic control and stochastic game problems for one - dimensional diffusions pioneered by william sudderth , that sudderth and his collaborators developed in a series of articles that includes @xcite , heath et al . ( @xcite ) , @xcite , @xcite ; indeed , the developments in sections [ sec6][sec8 ] of our paper can be construed as a multidimensional extension of the sudderth approach . we rely strongly on krylov s ( @xcite , @xcite ) work , which studies solutions of stochastic differential equations with constraints on the drift and diffusion coefficients in terms of `` supermartingale problems '' and characterizes sets of stochastic integrals via appropriate supermartingales . the approach we adopt has a lot in common with the effort , started in the mid-1990s , to understand option pricing and hedging in the presence of uncertainty about the underlying volatility structure of assets . we have been influenced by this strand of work , particularly by the papers of @xcite , @xcite , @xcite , @xcite , @xcite ; other important papers include @xcite , @xcite , @xcite , @xcite , ekstrm and tysk ( @xcite ) , @xcite , @xcite , whereas the recent preprints by @xcite ( @xcite , @xcite ) contain very relevant results . similar in this spirit is the strand of work by shige peng and his collaborators , surveyed in @xcite , regarding the so - called `` @xmath49-brownian motion '' which exhibits volatility uncertainty [ see also @xcite , as well as @xcite for extensions to settings where the range of uncertainty is stochastic ] . whereas in both these strands the relevant fully nonlinear parabolic - type partial differential ( so - called `` black  scholes  barenblatt'')equation has a typically unique solution , here the main interest arises from _ lack of uniqueness _ on the part of the rather similar , fully nonlinear equation  ( [ e.3.c ] ) . let us mention that optimization problems in stochastic control , mathematical economics and finance that involve model uncertainty have also been treated by other authors , among them @xcite , @xcite , shied and wu ( @xcite ) , @xcite , fllmer and gundel ( @xcite ) , @xcite , @xcite , bayraktar and yao ( @xcite ) , bayraktar , karatzas and yao ( @xcite ) and @xcite [ see also the survey by @xcite ] .    _ preview _ : sections [ sec2 ] and [ sec3 ] set up the model for an equity market with knightian model uncertainty regarding its volatility and market - price - of - risk characteristics , and for investment rules in its context . section [ sec4 ] introduces the notion of optimal arbitrage in this context , whereas section [ sec5 ] discusses the relevance of the fully nonlinear parabolic partial differential inequality of hjb type ( [ e.3.in ] ) , ( [ e.3.d ] ) in characterizing the arbitrage function and in finding an investment rule that realizes the best outperformance of the market portfolio . section [ sec6 ] presents a verification - type result for this equation . sections [ sec7 ] and [ sec8 ] make the connection with the stochastic control problem of maximizing the probability of containment for an auxiliary it process , controlled in a nonanticipative way and in a manner that affects both its drift and dispersion characteristics . finally , section [ sec9 ] develops yet another characterization of the arbitrage function , this time as the min - max value of a zero - sum stochastic game ; the investment rule that realizes the best outperformance of ( optimal arbitrage with respect to ) the market , is now seen as the investor s best response to a `` least favorable '' market model configuration . we shall fix throughout a canonical , filtered measurable space @xmath50 , @xmath51 and assume that @xmath52 contains the space @xmath53 of all continuous functions @xmath54 . we shall specify this canonical space in more detail in section [ sec7 ] below , when such detail becomes necessary .    on this space , we shall consider a vector of continuous , adapted processes @xmath55 with values in @xmath37 ; its components will represent stock capitalizations in an equity market with @xmath56 assets , and thus the total market capitalization will be the sum @xmath57 we shall also fix throughout a collection @xmath58 of nonempty , compact and convex subsets of @xmath8 as in ( [ 1.b ] ) . we shall consider @xmath59-valued functionals @xmath60 and @xmath61-valued functionals @xmath62 , all of them defined on @xmath63 and progressively measurable [ see @xcite , definition  3.5.15 ] . we shall assume that , for every continuous function @xmath64 and @xmath65 , these functionals satisfy the constraint and integrability conditions , respectively , @xmath66\\[-8pt ] \int_0^t \bigl ( \| \vartheta(t , \mathfrak{w } ) \|^2 + \operatorname{tr } ( \alpha(t , \mathfrak{w } ) ) \bigr ) \,\mathrm{d } t & < & \infty.\nonumber\end{aligned}\ ] ] we shall also consider @xmath67-matrix - valued functionals @xmath68 where @xmath69      for a given collection of sets @xmath70 as in ( [ 1.b ] ) and a  fixed initial configuration @xmath71 of asset capitalizations , we shall call _ admissible system _ a collection @xmath72 consisting of the underlying filtered space @xmath73 , @xmath74 , of a probability measure @xmath75 on it , and of a pair of processes @xmath76 , with @xmath77 an @xmath78-dimensional @xmath79-brownian motion and @xmath80 taking values in @xmath37 . these processes have the dynamics @xmath81 , \qquad   x_i(0)=x_i>0\ ] ] for some progressively measurable functionals @xmath82 and @xmath83 which satisfy  ( [ 1.2 ] ) and ( [ alphasigma ] ) above . we shall think of this admissible system @xmath72 as a _ model _ subject to `` knightian '' uncertainty ; this is expressed by the requirement @xmath84 in ( [ 1.2 ] ) , ( [ 1.b ] ) about the market price of risk and the covariance structure of the asset capitalization vector process @xmath85 . in order not to lose sight of the underlying probability space , we shall denote by @xmath86 ( resp . , @xmath87 ) the probability measure ( resp . , the corresponding expectation operator ) on this space . finally , @xmath88 will denote the collection of all such admissible systems or `` models '' with initial configuration @xmath89 . we shall think of @xmath90 as a _ meta - model _ , a collection of admissible models , and of the collection @xmath91 as a `` family of meta - models . '' the interpretation is that the components of the driving brownian motion  @xmath92 represent the independent factors of the resulting model ; the entries of the matrix @xmath93 are the local volatility rates of the asset capitalization vector process @xmath85 at time @xmath94 ; the entries of the matrix @xmath95 as in  ( [ alphasigma ] ) represent the local covariance rates ; whereas the components of the vector  @xmath96 are the market price of risk ( also called _ relative risk _ ) rates prevalent at time @xmath4 . in particular , @xmath97 is , in the notation of ( [ alphasigma ] ) , the vector of mean rates of return for the various assets at time @xmath4 , when the equations of ( [ 1.1 ] ) are cast in the more familiar form @xmath98    the integrability condition of ( [ 1.2 ] ) guarantees that the process @xmath85 takes values in @xmath37 , @xmath99-a.s . ; it implies also that the exponential process @xmath100 is well defined and a strictly positive local martingale , thus also a supermartingale . this process @xmath101 plays the role of a state - price - density or `` deflator '' in the present context . just as in our earlier works , fernholz and karatzas ( @xcite , @xcite ) as well as @xcite  mostly in a markovian context , and without model uncertainty  an important feature of this subject is that @xmath101 has to be allowed to be a _ strict _ local martingale , that is , that @xmath102 be allowed to hold for some , if not all , @xmath103 .      constraint ( [ e.3 ] ) brings us in the realm of the krylov ( @xcite , @xcite ) approach , which studies stochastic differential equations with constraints on the drift and diffusion coefficients in terms of `` supermartingale problems . '' in particular , theorem 2.2 of @xcite shows that , under a suitable regularity condition on the family of sets  @xmath70 in  ( [ 1.b ] ) , solving stochastic equation  ( [ 1.1 ] ) subject to the requirements of  ( [ 1.2 ] ) can be cast as a _ supermartingale problem , _ as follows . consider the nonlinear partial differential operator associated with ( [ b1 ] ) , ( [ b ] ) , namely @xmath104 \\[10pt ] \eqntext{(q , p , \mathbf { z } ) \in \mathbb{s}^n \times\mathbb{r}^n \times(0 , \infty ) ^n .}\end{aligned}\ ] ] the supermartingale problem is to find a probability measure @xmath105 on the filtered measurable space @xmath50 , @xmath51 , under which @xmath85 takes values in @xmath37 a.s . , and the process @xmath106 is a local supermartingale for every @xmath107 of class @xmath20 with compact support . the regularity condition on the family @xmath70 in ( [ 1.b ] ) that we alluded to earlier , mandates that the function @xmath108 if , in addition , the family of sets @xmath70 in ( [ 1.b ] ) satisfies the linear growth condition @xmath109 and the upper - semicontinuity condition @xmath110 for every @xmath111 and some real constant @xmath112 , then theorem 3.2 in @xcite shows that the family @xmath113 is convex and sequentially compact in the topology of vague convergence of probability measures . we shall also consider the subcollection @xmath114 of _ markovian _ admissible systems , for which the functionals @xmath115 and @xmath116 as in ( [ 1.2])([1.1 ] ) are given as @xmath117 with measurable functions @xmath118 , that satisfy @xmath119 under condition ( [ lingro ] ) it follows then from so - called markovian selection results [ @xcite ; @xcite , chapter 12 ; @xcite , section 4.5 ] that the state process @xmath120 of ( [ 1.1 ] ) can be assumed to be ( strongly ) markovian under @xmath121 , @xmath122 . we shall make this selection whenever admissible systems in @xmath123 are invoked . consider now an investor who is `` small '' in the sense that his actions do not affect market prices . he starts with initial fortune @xmath124 and uses a rule that invests a proportion @xmath125 of current wealth in the @xmath126th asset of the equity market , for any given time @xmath127 and all @xmath128 ; the remaining proportion @xmath129 is held in cash ( equivalently , in a zero - interest money market ) . here @xmath130 is a progressively measurable functional assumed to satisfy the requirement @xmath131\\[-8pt ] \eqntext{\forall t \in(0 , \infty)}\end{aligned}\ ] ] for every continuous function @xmath132 . ( thus , the requirement ( [ 1.3 ] ) will be in force under all admissible systems . ) we shall denote throughout by @xmath133 the collection of all such ( nonanticipative ) _ investment rules _ , and by @xmath134 the sub - collection of all _ markovian _ investment rules , that is , those that can be expressed as @xmath135 , @xmath136 for some measurable function @xmath137 . an investment rule is called _ bounded _ , if the functional @xmath138 is bounded uniformly on @xmath139 ; for a bounded investment rule , requirement ( [ 1.3 ] ) is satisfied automatically , thanks to ( [ 1.2 ] ) . an investment rule is called _ portfolio _ if the functional @xmath140 satisfies @xmath141 on @xmath142 , and a portfolio is called _ long - only _ if @xmath143 also hold on this domain . a long - only portfolio is clearly bounded . given an initial wealth @xmath144 , an investment rule @xmath145 and an admissible model @xmath146 , the resulting wealth process @xmath147 satisfies the dynamics @xmath148\ ] ] and the initial condition @xmath149 . in conjunction with ( [ expo ] ) in the differential form @xmath150 and the product rule of the stochastic calculus , this gives @xmath151\\[-8pt ] \eqntext{0 \le t < \infty.}\end{aligned}\ ] ] for any initial configuration @xmath152 , initial wealth @xmath153 , investment rule @xmath154 and admissible model @xmath155 , the product @xmath156 is therefore under @xmath157 a continuous , positive local martingale , thus also a supermartingale . once again , it is important that this process be allowed to be a strict local martingale . the choice of markovian investment rule @xmath158 given by @xmath159 leads to the long - only _ market portfolio _ @xmath160\\[-8pt ] \eqntext{\mbox{with weights } \mu_i ( t ) = x_i ( t ) / x(t ) , \ i=1 , \ldots , n , \ 0 \le t < \infty}\end{aligned}\ ] ] in the notation of ( [ 1.a ] ) . it follows from ( [ 1.5 ] ) that investing according to this portfolio amounts to owning the entire market , in proportion to the initial wealth , @xmath161 . reading ( [ 1.6 ] ) for the market portfolio of ( [ 1.7.a ] ) , and recalling ( [ 1.a ] ) , leads to @xmath162\\[-8pt ] \eqntext{0 \le t < \infty}\end{aligned}\ ] ] or equivalently @xmath163 , where @xmath164\\[-8pt ] \eqntext{\displaystyle \mbox{satisfies } \int_0^t \| \widetilde{\vartheta } ( t , \mathfrak{w } ) \|^2 \,\mathrm{d } t < \infty}\end{aligned}\ ] ] for all @xmath165 , thanks to ( [ 1.2 ] ) . with this notation , it follows from ( [ 1.c ] ) that @xmath166 on the strength of the integrability condition in ( [ 1.d ] ) , the dambis  dubins  schwartz representation [ e.g. , @xcite , page 174 ] of the @xmath167-local martingale @xmath168 gives @xmath169 where @xmath170 is one - dimensional , standard brownian motion under @xmath171.whereas the equations of ( [ 1.1 ] ) can be written as @xmath172 for @xmath128 , with @xmath173 we then have the representation @xmath174\\[-8pt ] & \qquad\hspace*{3pt}= & \exp\biggl\ { \int_0^\cdot ( \widetilde{\vartheta } ( t , \mathfrak { x } ) ) ' \,\mathrm{d } \widetilde { w}(t ) - { 1 \over2 } \int_0^\cdot \| \widetilde{\vartheta } ( t , \mathfrak { x } ) \|^2 \,\mathrm{d } t \biggr\}\nonumber\end{aligned}\ ] ] for the normalized reciprocal of the deflated total market capitalization , and @xmath175 for the dynamics market weights in ( [ 1.7.a ] ) ; here @xmath176 is the @xmath126th unit vector in  @xmath177 . let us consider now the smallest proportion @xmath178\\[-8pt ] & & \qquad\hspace*{198pt}\forall\mathcal{m } \in\mathfrak{m } ( \mathbf{x } ) \bigr\}\nonumber\end{aligned}\ ] ] of the initial total market capitalization @xmath179 which allows the small investor , starting with initial capital @xmath180 and through judicious choice of investment rule in the class @xmath181 , to match or exceed the performance of the market portfolio over the time - horizon @xmath182 $ ] , and to do this with @xmath167-probability one , under any model @xmath183 that might materialize . we shall refer to @xmath184 of ( [ 1.7 ] ) as the _ arbitrage function _ for the family of meta - models @xmath185 , and think of it as a version of the arbitrage function studied in fernholz and karatzas ( @xcite ) which is `` robust '' with respect to @xmath186 . the quantity of ( [ 1.7 ] ) is strictly positive ; see proposition [ proposition_1 ] below and the discussion following it . on the other hand , the set of ( [ 1.7 ] ) contains the number @xmath187 , so clearly @xmath188    if @xmath189 , then for every @xmath190and even for @xmath191 when the infimum in ( [ 1.7 ] ) is attained , as indeed it is in the context of theorem 1 below there exists an investment rule @xmath192 such that @xmath193 holds for every @xmath194 . in other words , the investment rule @xmath195 leads then to _ strong arbitrage relative to the market _ portfolio in the terminology of @xcite  here with the extra feature that such arbitrage is now _ robust _ , that is , holds under any possible admissible system or `` model '' that might materialize . if , on the other hand , @xmath196 , then such outperformance of ( equivalently , strong arbitrage relative to ) the market is just not possible over all meta - models @xmath197 . in either case , the highest return on investment relative to the market @xmath198 achievable using ( nonanticipative ) investment rules , is given as @xmath199 . [ remark_1 ] instances of @xmath189 occur , when there exists a constant @xmath200 such that either @xmath201 or @xmath202 holds for every @xmath203 . see the survey paper @xcite , examples 11.1 and 11.2 [ as well as @xcite , @xcite for additional examples ] . [ proposition_1 ] the quantity of ( [ 1.7 ] ) satisfies @xmath204\\[-8pt ] \eqntext{\displaystyle \mbox{where } \phi(t,\mathbf{x}):= \sup_{\mathcal{m } \in \mathfrak{m } ( \mathbf{x } ) } \biggl ( \frac { \mathbb{e}^ { \mathbb { p}^\mathcal{m } } [ l(t ) x(t ) ] } { x_1 + \cdots+ x_n } \biggr ) .}\end{aligned}\ ] ] furthermore , under conditions ( [ measur])([uppsemicont ] ) , there exists an admissible system @xmath205 such that @xmath206 } { x_1 + \cdots+ x_n } .\ ] ]    take an arbitrary element @xmath207 of the set on the right - hand side of ( [ 1.7 ] ) and an arbitrary admissible system @xmath155 . there exists then an investment rule @xmath208 with the inequality @xmath209 valid @xmath210-a.s . on the strength of ( [ expo ] ) and ( [ 1.6 ] ) , the process @xmath211 is a  @xmath210-supermartingale ; thus ( [ repr ] ) and ( [ 1.d ] ) lead to @xmath212 \nonumber\\[-8pt]\\[-8pt ] & \ge&\mathbb{e}^ { \mathbb{p}^\mathcal{m } } [ l ( t ) x(t ) ] > 0 .\nonumber\end{aligned}\ ] ] the inequality @xmath213 in ( [ 1.8 ] ) follows now from the arbitrariness of @xmath207 and @xmath214 . the existence of an admissible system @xmath215 that satisfies ( [ 1.88 ] ) follows from theorem 3.4 in @xcite , in conjunction with the dynamics of ( [ 1.1 ] ) and ( [ expotoo ] ) . although strong arbitrage relative to the market may exist within the framework of the models @xmath155 studied here ( cf . remark [ remark_1 ] ) , the existence of a strictly positive supermartingale deflator process @xmath216 as in ( [ expo ] ) proscribes scalable arbitrage opportunities , also known as _ unbounded profits with bounded risk _ ( _ upbr _ ) ; this is reflected in the inequality @xmath217 of  ( [ 1.8 ] ) . we refer the reader to @xcite for the origin of the resulting _ nupbr _ concept , and to @xcite for an elaboration of this point in a different context , namely , the existence and properties of the numraire portfolio . finally , let us write ( [ 1.8 ] ) as @xmath218 } { x_1 + \cdots+ x_n } .\hspace*{-10pt}\ ] ] we have for this quantity the interpretation @xmath219 as the `` arbitrage function for the model @xmath220 , '' at least when the matrix @xmath221 in invertible for every @xmath222 and when @xmath223-martingales can be represented as stochastic integrals with respect to the brownian motion @xmath92 in ( [ 1.1 ] ) . consider now a continuous function @xmath224 with @xmath225 which is of class @xmath20 on @xmath226 and satisfies on this domain the fully nonlinear partial differential inequality ( pdi ) @xmath227\\[-8pt ] \eqntext{\forall a \in\mathcal{a } ( \mathbf { z } ) .}\end{aligned}\ ] ] we shall denote by @xmath228 the collection of all such continuous functions @xmath229 which are of class @xmath20 on @xmath230 and satisfy ( [ 2.1 ] ) and ( [ 2.2 ] ) . this collection @xmath228 is nonempty , since we can take @xmath231 ; however , @xmath228 need not contain only one element . let us fix an initial configuration @xmath232 and consider any admissible system @xmath233 . applying it s rule to the process @xmath234 in conjunction with ( [ 1.c ] ) and ( [ 1.1 ] ) , we obtain its @xmath210-semimartingale decomposition as @xmath235\\[-8pt ] & & { } + \sum_{\nu=1}^n \bigl [ r_\nu(t , \mathfrak{x } ) - u \bigl(t - t , \mathfrak{x}(t ) \bigr ) \widetilde{\vartheta}_\nu(t , \mathfrak{x } ) \bigr ] \,\mathrm{d}w_\nu(t ) .\nonumber\end{aligned}\ ] ] here we have used the notation of ( [ 1.d ] ) , and have set @xmath236    from the inequality of ( [ 2.2 ] ) , coupled with the fact that @xmath237 holds for all @xmath238 , this last expression is clearly not positive . as a  result , the positive process @xmath239 of ( [ 2.4 ] ) is a @xmath210-supermartingale , namely , @xmath240 \nonumber\\[-8pt]\\[-8pt ] & = & \mathbb{e}^ { \mathbb{p}^\mathcal{m } } [ l(t)x(t ) | \mathcal{f}(t ) ] \nonumber\end{aligned}\ ] ] holds @xmath167-a.s . , @xmath241 and @xmath242 ; in particular , @xmath243 \\ & = & \mathbb{e}^ { \mathbb{p}^\mathcal{m } } [ l(t ) x(t ) ] \qquad   \forall\mathcal{m } \in\mathfrak{m } ( \mathbf{x } ) .\end{aligned}\ ] ] with the notation of ( [ 1.8 ] ) , we obtain in this manner the following analog of the inequality in proposition [ proposition_1 ] : @xmath244 digging in this same spot , just a bit deeper , leads to our next result ; this is very much in the spirit of theorem 5 in @xcite and of section ii.2 in lions ( @xcite ) . [ proposition_2 ] for horizon @xmath103 , initial configuration @xmath245 and function @xmath229 in the collection @xmath246 , we have the inequality @xmath247 furthermore , the markovian investment rule @xmath248 generated by this function @xmath249 through @xmath250\\[-8pt ] & & { } + \frac{z_i } { z_1 + \cdots+ z_n } , \qquad   ( t,\mathbf{z } ) \in[0,t ] \times(0 , \infty)^n\nonumber\end{aligned}\ ] ] for each @xmath251 , satisfies for every admissible system @xmath252 the inequality @xmath253    for a fixed initial configuration @xmath254 , an arbitrary admissible model @xmath155 and any function @xmath255 , let us recall the notation of  ( [ 2.4 ] ) and re - cast the dynamics of  ( [ 2.5 ] ) as @xmath256 here by virtue of ( [ 2.5 ] ) , ( [ 2.6 ] ) and ( [ 1.d ] ) we have written @xmath257\\[-8pt ] & & { } - \vartheta_\nu(t , \mathfrak{x})\nonumber\end{aligned}\ ] ] for @xmath258 and have introduced in the notation of ( [ 2.7 ] ) the continuous , increasing process @xmath259    the expression of ( [ 2.10 ] ) suggests considering the markovian investment rule @xmath260 as in ( [ 2.11.a ] ) ; then we cast the expression of ( [ 2.10 ] ) as @xmath261 on the strength of ( [ 1.6 ] ) , the value process generated by this investment rule @xmath262 starting with initial wealth @xmath263 , satisfies the equation @xmath264 juxtaposing this to ( [ 2.9 ] ) , and using the positivity of @xmath265 along with the nonnegativity and nondecrease of @xmath266 , we obtain the @xmath167-a.s . comparison @xmath267 , thus @xmath268 with @xmath269 this leads to ( [ 2.11.b ] ) , in conjunction with ( [ 2.1 ] ) . we conclude from  ( [ 2.11.b ] ) that the number @xmath270 belongs to the set on the right - hand side of ( [ 1.7 ] ) , and the first comparison in ( [ 2.8 ] ) follows ; the second is just a  restatement of ( [ 1.8 ] ) . suppose that the function @xmath271 of ( [ 1.8 ] ) belongs to the collection @xmath272 . then @xmath273 is the smallest element of @xmath274 ; the infimum in ( [ 1.7 ] ) is attained ; we can take @xmath275 in ( [ 2.11.b ] ) , ( [ 2.11.a ] ) ; and the inequality in ( [ 2.8 ] ) holds as equality , that is , @xmath276 coincides with the arbitrage function @xmath277    _ interpretation : _ imagine that the small investor is a manager who invests for a pension fund and tries to track or exceed the performance of an index ( the market portfolio ) over a finite time - horizon . he has to do this in the face of uncertainty about the characteristics of the market , including its covariance and price - of - risk structure , so he acts with extreme prudence and tries to protect his clients against the most adverse market configurations imaginable [ the range of such configurations is captured by the constraints  ( [ e.3 ] ) , ( [ 1.b ] ) ] . if such adverse circumstances do not materialize his strategy generates a surplus , captured here by the increasing process @xmath278 of ( [ 2.11 ] ) with @xmath279 , which can then be returned to the ( participants in the ) fund . we are borrowing and adapting this interpretation from @xcite . similarly , the markovian investment rule @xmath260 generated by the function @xmath279 in ( [ 2.11.a ] ) , ( [ fg ] ) implements the best possible outperformance of the market portfolio , as in ( [ 2.11.b ] ) . for the purposes of this section we shall impose the following _ growth _ condition on the family @xmath280 of subsets of  @xmath281 in ( [ e.3.a ] ) , ( [ 1.b ] ) : there exists a constant @xmath282 , such that for all @xmath283 we have @xmath284 we shall also need the following _ strong ellipticity _ condition , which mandates that for every nonempty , compact subset @xmath285 of @xmath37 , there exists a real constant @xmath286 such that @xmath287    [ assa ] there exist a continuous function @xmath288 , a @xmath289-function @xmath290 , and a continuous square root @xmath291 of @xmath292 , namely @xmath293 such that , with the vector - valued function @xmath294 defined by @xmath295 condition ( [ d.22 ] ) is satisfied , whereas the system of stochastic differential equations @xmath296 , \nonumber\\[-8pt]\\[-8pt ] \eqntext{x_i(0)=x_i>0 , i=1,\ldots , n}\end{aligned}\ ] ] has a solution in which the state process @xmath120 takes values in @xmath37 . a bit more precisely , this assumption posits the existence of a markovian admissible system @xmath297 consisting of a filtered probability space @xmath298 , @xmath74 and of two continuous , adapted process @xmath85 and  @xmath92 on it , such that under the probability measure  @xmath299 the process  @xmath92 is @xmath300-dimensional brownian motion , the process @xmath85 takes values in @xmath37 a.s . and ( [ 1.1 ] ) holds with @xmath301 as in ( [ d.2 ] ) , and with @xmath302 , @xmath303 ( @xmath304 ) . the system of equations ( [ d.1 ] ) can be cast equivalently as @xmath305\\[-8pt ] & & \hphantom{x_i(t ) \biggl [ } { } + \biggl ( \sum_{j=1}^n \mathbf{a}_{i j } ( \mathfrak { x } ( t ) ) x_j ( t ) d_j h ( \mathfrak { x } ( t ) ) \biggr)\ , \mathrm{d } t \biggr].\nonumber\end{aligned}\ ] ]    [ assb ] in the notation of the previous paragraph and under the condition @xmath306 we define on @xmath37 the continuous functions @xmath307 and @xmath308 $ ] and assume that the function @xmath309 , \qquad   ( \tau , \mathbf{x } ) \in[0 , \infty)^n\ ] ] is continuous on @xmath310 and of class @xmath311 on @xmath312    sufficient conditions for assumptions [ assa ] , [ assb ] to hold are provided in @xcite , sections 8 and 9 . it is also shown there , that we have the @xmath313-martingale property @xmath314 = x(t ) l(t ) \cdot\gamma\bigl(t - t , x ( t ) \bigr ) , \qquad   0 \le t \le t\ ] ] for the function @xmath315 this function is of class @xmath20 on @xmath316 and satisfies the initial condition @xmath317 on @xmath37 as well as the _ linear _ second - order parabolic equation @xmath318\\[-8pt ] \eqntext{(\tau , \mathbf { z } ) \in(0 , \infty ) \times(0 , \infty)^n .}\end{aligned}\ ] ]    [ proposition_3 ] under the assumptions [ assa ] , [ assb ] and the conditions ( [ 3.2.gg ] ) , ( [ 3.2 ] ) , suppose that the functions @xmath319 and @xmath320 satisfy the inequality @xmath321\\[-8pt ] \qquad & & \qquad \ge\sum_{i=1}^n \sum_{j=1}^n z_i z_j a_{ij } \biggl ( \frac{1 } { 2 } d^2_{ij } \gamma(\tau , \mathbf{z})+ \frac{d_{i } \gamma(\tau , \mathbf{z})}{z_1 + \cdots+z_n } \biggr ) \qquad   \forall a \in\mathcal{a } ( \mathbf { z})\nonumber\end{aligned}\ ] ] for every @xmath322 . then , in the notation of ( [ 1.7])([1.9 ] ) , we have : @xmath323 for the markovian admissible system @xmath324 posited in assumption  [ assa ] ; the conclusions of proposition [ proposition_2 ] and its corollary for @xmath325 ; as well as the @xmath167-a.s . comparison @xmath326,\qquad 0\le t \le t , \ ] ] which holds for every @xmath194 and as equality for @xmath327 .    under condition ( [ d.11 ] ) the function @xmath328 belongs to the collection  @xmath228 , as ( [ 2.2 ] ) is satisfied with @xmath329 on the strength of ( [ d.10 ] ) and ( [ d.11 ] ) ; thus , we deduce @xmath330 from ( [ 2.7.b ] ) . on the other hand , equality  ( [ d.8 ] ) with @xmath33 , and the definition of @xmath331 in ( [ 1.8 ] ) , give @xmath332 } { x_1 + \cdots+ x_n } = \mathfrak{u}_{\mathcal{m}_o } ( t,\mathbf{x } ) \le\phi(t,\mathbf{x } ) , \ ] ] so the equality @xmath333 follows . in other words , we identify @xmath334 as a markovian admissible system that satisfies ( [ 1.88 ] ) and attains the supremum in ( [ 1.8 ] ) . the remaining claims come from proposition [ proposition_2 ] and its corollary [ in particular , from reading ( [ 2.8 ] ) with @xmath335 and from ( [ 2.7.7 ] ) , ( [ d.8 ] ) . [ remark_2 ] proposition [ proposition_3 ] holds under conditions weaker than those imposed in assumptions [ assa ] and [ assb ] above , at the `` expense '' of a certain localization . more precisely , one posits the existence of locally bounded and locally lipschitz functions @xmath336 and @xmath337 ( @xmath304 ) for which ( [ d.22 ] ) , ( [ 3.2.gg ] ) , ( [ 3.2 ] ) and ( [ d.4 ] ) are satisfied with @xmath338 , and for which there exists a markovian admissible system @xmath339 whose state process @xmath340 in ( [ d.1 ] ) is , under @xmath341 , a strong markov process with values in @xmath342 a.s . using results from the theory of stochastic flows [ @xcite , @xcite ] and from parabolic partial differential equations [ @xcite , ekstrm and tysk ( @xcite ) ] , theorem  2 in ruf ( @xcite ) shows that the function @xmath343 is then of class @xmath311 locally on @xmath344 , and solves there equation ( [ d.10 ] ) . we have now gone as far as we could without having to specify the nature of our filtered measurable space @xmath345 , @xmath346 . to proceed further , we shall need to choose this space carefully . we shall take as our sample space the set @xmath52 of right - continuous paths @xmath347 . here @xmath348 is an additional `` absorbing point '' ; paths stay at @xmath348 once they get there , that is , after @xmath349 , and are continuous on @xmath350 ; we are employing here , and throughout this work , the usual convention @xmath351 . we also select @xmath352 and @xmath353 , where @xmath354 is the zero matrix . with @xmath355 , the filtration @xmath356 is a _ standard system _ in the terminology of @xcite . this means that each @xmath357 is isomorphic to the borel @xmath358-algebra on some polish space , and that for any decreasing sequence @xmath359 where each @xmath360 is an atom of the corresponding @xmath361 , for some increasing sequence @xmath362 , we have @xmath363 [ see the appendix in fllmer ( @xcite ) , as well as @xcite and fllmer ( @xcite ) ] .    with all this in place we take @xmath364 as our filtered measurable space , where @xmath365    an admissible system @xmath366 , @xmath367 defined as in section [ sec2 ] consists of this filtered measurable space @xmath368 , of a probability measure @xmath86 on it , of an @xmath300-dimensional brownian motion @xmath92 on the resulting probability space and of the coordinate mapping process @xmath369 which is assumed to satisfy ( [ 1.1 ] ) , ( [ 1.2 ] ) and to take values in @xmath37 , @xmath370-a.s . we shall take @xmath371 in the notation of ( [ lambda ] ) , ( [ expo ] ) and ( [ 1.a ] ) , and note @xmath372 .      with this setup , there exists a probability measure @xmath373 on @xmath374 such that @xmath375 we express this property ( [ bigtriangleleft ] ) by writing @xmath376 . under the measure  @xmath377 , the process @xmath378 of ( [ 1.1.b ] ) is brownian motion ; whereas the processes @xmath379 and @xmath380 of ( [ mu ] ) , ( [ lambda ] ) in section [ subsec3.1 ] are nonnegative @xmath381-martingales .    the `` absorbing state '' @xmath348 acts here as a proxy for @xmath210-null sets to which the new measure @xmath377 may assign positive mass ; the possible existence of such sets makes it important that the filtration @xmath382 be `` pure , '' that is , _ not _ completed by @xmath167-null sets . this probability measure @xmath383 satisfies @xmath384 } { x_1 + \cdots + x_n } & = & \mathbb{e}^{\mathbb{p}^\mathcal{m } } \bigl [ ( 1 / \lambda(t ) ) \cdot\mathbf { 1}_{\ { \mathcal{t } > t\ } } \bigr ] \nonumber\\[-8pt]\\[-8pt ] & = & \mathbb{q } ( \mathcal { t } > t ) \qquad \forall t \in[0 , \infty)\nonumber\end{aligned}\ ] ] and @xmath385    we also have @xmath386-a.e . on @xmath387 @xmath388 and @xmath389 whereas , @xmath390-a.e . on @xmath391 , we have @xmath392    we deduce from ( [ foell ] ) that the arbitrage function of ( [ 1.9 ] ) for the model @xmath194 is given by the _ probability of _ `` _ _ containment _ _ '' _ under the measure _  @xmath383 , namely , the @xmath393-probability that the process @xmath394 , started at @xmath395 , stays in @xmath37 throughout the time - horizon @xmath396 $ ] .    at this point we shall impose the following requirements on @xmath397 , the family of compact , convex subsets of @xmath398 in ( [ 1.b ] ) , ( [ e.3 ] ) : there exists a constant @xmath399 , such that for all @xmath400 we have the strengthening @xmath401 of the growth condition in ( [ 3.2.gg ] ) , as well as the `` shear '' condition @xmath402 \le c .\ ] ] then the following identity holds @xmath390-a.s . : @xmath403 for justification of the claims made in this subsection , we refer to section  7 in fernholz and karatzas ( @xcite ) , as well as @xcite , @xcite and @xcite  in addition , of course , to the seminal work by fllmer ( @xcite , @xcite ) . the special structure of the filtered measurable space @xmath404 that we selected in this section is indispensable for this construction and for the representation ( [ foell ] ) ; whereas the inequality @xmath405 from condition ( [ thetatrace ] ) is important for establishing the representation of ( [ ttaui ] ) . let us fix then an initial configuration @xmath406 and denote by @xmath407 the collection of stochastic systems @xmath408 that consist of the filtered measurable space @xmath409 , of a probability measure @xmath410 , of an @xmath59-valued brownian motion @xmath411 under @xmath377 , and of the cordinate mapping process @xmath412 which satisfies @xmath377-a.s . the system of the stochastic equations in ( [ 1.1.a ] ) @xmath413 here the elements @xmath414 , @xmath415 of the matrix @xmath416 are progressively measurable functionals that satisfy , in the notation of ( [ e.3.a ] ) , @xmath417    as in section [ subsec2.3 ] , we shall denote by @xmath418 the subcollection of @xmath419 that consists of _ markovian _ auxiliary admissible systems , namely , those for which the equations of ( [ aux ] ) are satisfied with @xmath420 and @xmath421 , @xmath422 and with measurable functions @xmath423 and @xmath424 that satisfy the condition @xmath425 , @xmath426 . we invoke the same markovian selection results as in section  [ subsec2.3 ] , to ensure that the process @xmath427 is strongly markovian under any given @xmath428 , @xmath429 .    by analogy with ( [ ttaui ] ) , we consider @xmath430 then for every @xmath431 we have @xmath432 whereas @xmath433 , @xmath434 holds for every @xmath435 . [ remark_7 ] as in section [ subsec2.2 ] , solving the stochastic equation ( [ aux ] ) subject to condition ( [ const1 ] ) amounts to requiring that the process @xmath436 be a local supermartingale , for every continuous @xmath437 which is of class @xmath311 on @xmath438 and has compact support ; here @xmath439 is the nonlinear second - order partial differential operator in ( [ e.3.d ] ) . [ remark_3 ] the total capitalization process @xmath440 satisfies , by virtue of ( [ aux ] ) , the equation @xmath441 , \\ \widetilde{n}(\cdot ) & : = & \sum_{\nu=1}^n \int_0^\cdot\biggl ( \sum_{i=1}^n ( x_i ( t ) / x(t ) ) \sigma_{i \nu } ( t , \mathfrak { x } ) \biggr)\ , \mathrm{d } \widetilde{w}_\nu(t ) .\end{aligned}\ ] ] under the measure @xmath442 , the process @xmath443 is a continuous local martingale with quadratic variation @xmath444 from ( [ 3.2.g ] ) , so the total capitalization process @xmath445 takes values in @xmath446 , @xmath447-a.e . ; here @xmath448 is a one - dimensional @xmath447-brownian motion . this is in accordance with our selection of the punctured nonnegative orthant @xmath449 in ( [ 1.b ] ) as the state - space for the process @xmath120 under @xmath377 .    under this measure , the relative weights @xmath450 , @xmath128 are nonnegative local martingales and supermartingales , in accordance with ( [ mu ] ) , and since @xmath451 these processes are bounded , so they are actually martingales . once any one of the processes @xmath452 [ i.e. , any one of the processes @xmath453 becomes zero , it stays at zero forever ; of course , not all of them can vanish at the same time . in section [ subsec7.1 ] we started with an arbitrary admissible system @xmath454 and produced an `` auxiliary '' admissible system @xmath455 , for which the property ( [ foell ] ) holds . thus , for every @xmath456 we deduce @xmath457\\[-8pt ] & \hspace*{3pt}\ge & \sup_{\mathcal { m } \in\mathfrak { m } ( \mathbf { x } ) } \biggl ( \frac { \mathbb{e}^ { \mathbb{p}^\mathcal{m } } [ l(t ) x(t ) ] } { x_1 + \cdots+ x_n } \biggr)= \phi(t , \mathbf { x } ) .\nonumber\end{aligned}\ ] ]      we suppose from now onwards that , for every progressively measurable functional @xmath458 which satisfies @xmath459 we can select a progressively measurable functional @xmath460 with @xmath461 [ see the `` measurable selection '' results in chapter 7 of @xcite ] . we introduce now the functional @xmath462 as in ( [ 1.d ] ) and also , by analogy with ( [ ttau.a ] ) , the stopping rule @xmath463 [ cf . @xcite , where stopping rules of this type also play very important roles in the study of arbitrage ] . we recall now ( [ const2 ] ) ; on the strength of the requirement ( [ thetatrace ] ) , this gives @xmath464 for every @xmath465 , and @xmath466 , @xmath467 for every @xmath468 . in conjunction with ( [ 3.2.g ] ) , we obtain from ( [ ttau111 ] ) that @xmath469 holds for every @xmath465 , and that @xmath470 , @xmath471 holds for every @xmath472 . we deduce for the stopping rules of ( [ ttau1 ] ) and ( [ ttaui1 ] ) the identification @xmath473 . let us fix now a stochastic system @xmath474 as in section [ subsec7.2 ] , pick a progressively measurable functional @xmath475 with @xmath476 for all @xmath477 and select a progressively measurable functional @xmath478 as in ( [ thetaalpha ] ) . for _ this _ @xmath115 and _ this _ @xmath479 , we define @xmath480 by ( [ ttau101 ] ) as well as @xmath481\\[-8pt ] \eqntext{\mbox{for } 0 \le t < \mathcal{t}}\end{aligned}\ ] ] as in ( [ lambda ] ) , and set @xmath482 in the notation of ( [ ttau1 ] ) . the resulting process @xmath380 is a local martingale and a supermartingale under @xmath377 , and we have @xmath483 , @xmath381-a.e .    we introduce also the sequence of @xmath79-stopping rules @xmath484 which satisfy @xmath485 and @xmath486 @xmath487 , for every @xmath488 . from novikov s theorem [ e.g. , @xcite , page 198 ] , @xmath489 is a uniformly integrable @xmath381-martingale ; in particular , @xmath490 holds for every @xmath491 . thus , the recipe @xmath492 , \qquad   a \in\mathcal{f}(s_n)\ ] ] defines a consistent sequence , or `` tower , '' of probability measures @xmath493 on @xmath494 . appealing to the results in @xcite , pages 140143 [ see also the appendix of fllmer ( @xcite ) ] , we deduce the existence of a  probability measure @xmath495 on @xmath496 such that @xmath497 \quad \mbox{holds for every } a \in \mathcal{f}(s_n ) , n \in\mathbb{n}.\hspace*{-25pt}\ ] ] ( here again , the special structure imposed in this section on the filtered measurable space @xmath498 is indispensable . ) therefore , for every @xmath103 we have @xmath499 = \mathbb{e}^\mathbb{q } \bigl [ \lambda ( t ) \cdot \mathbf { 1}_{\{s_n > t\ } } \bigr]\ ] ] by optional sampling , whereas monotone convergence leads to @xmath500 .\ ] ] the following result echoes similar themes in @xcite . [ lemma_1 ] the process @xmath380 of ( [ lamda ] ) , ( [ lamda_too ] ) is a @xmath393-martingale , if and only if we have @xmath501 [ i.e. , if and only if the process @xmath394 never hits the boundary of the orthant @xmath502 , @xmath503-a.s . ] . if @xmath504 holds , the nonnegativity of @xmath380 and ( [ wald ] ) give @xmath505 \le\mathbb{e}^\mathbb{q } [ \lambda ( t ) ] \qquad   \forall t \in(0 , \infty ) .\vadjust{\goodbreak}\ ] ] but @xmath380 is a @xmath393-supermartingale , so the reverse inequality @xmath506 \le \lambda ( 0 ) = 1 $ ] also holds . we conclude that @xmath507=1 $ ] holds for all @xmath103 , so @xmath380 is a @xmath393-martingale . if , on the other hand , @xmath380 is a @xmath508-martingale , then @xmath506=1 $ ] and  ( [ wald ] ) give @xmath509 for every @xmath510 , from optional sampling and the fact that @xmath511 holds @xmath373-a.e . on @xmath512 . we conclude @xmath513 ; in conjunction with the identification @xmath514 and ( [ ttaui1 ] ) , this means that the cordinate mapping process @xmath515 never reaches the boundary of ( i.e. , takes values in ) the strictly positive orthant @xmath516 , @xmath503-a.e . when the conditions of lemma [ lemma_1 ] prevail , the process @xmath517 is brownian motion under the probability measure @xmath518 introducedin  ( [ tower ] ) . this measure satisfies the equations of ( [ foell ] ) , whereas the process  @xmath394 solves @xmath167-a.s . the system @xmath519 , \qquad   x_i(0)=x_i>0\ ] ] for @xmath520 , as in ( [ 1.1 ] ) . it is then not hard to check that @xmath101 defined by ( [ expo ] ) satisfies @xmath167-a.s . the identity @xmath521 in accordance with ( [ lambda ] ) . we formalize these considerations as follows . [ assc ] suppose that the collection of sets @xmath522 in ( [ 1.b ] ) satisfies ( [ 3.2.g ] ) , ( [ thetatrace ] ) and that for any given progressively measurable functional @xmath523 which satisfies ( [ alpha ] ) and @xmath524 there exists a progressively measurable functional @xmath460 that satisfies the condition ( [ thetaalpha ] ) , thus also by virtue of ( [ thetatrace ] ) @xmath525    the analysis of this subsection shows that , under assumption [ assc ] and starting with any initial configuration @xmath406 and with an arbitrary `` auxiliary '' admissible stochastic system @xmath526 , @xmath527 in @xmath528 as in section [ subsec7.2 ] , the process @xmath380 of ( [ lamda_too ] ) is a @xmath508-martingale , and we can construct a `` primal '' admissible system @xmath529 as in section [ subsec2.1 ] [ i.e. , with the canonical process @xmath427 taking values in @xmath37 , @xmath167-a.s . ] , for which ( [ foell ] ) holds , and we have @xmath530 as in ( [ bigtriangleleft ] ) . we deduce @xmath531 } { x_1 + \cdots+ x_n } \biggr)= \phi(t , \mathbf { x } ) .\ ] ] the reverse inequality @xmath532 was established in ( [ 3.7.f ] ) . this way , for every function @xmath533 in the collection @xmath246 , we can strengthen ( [ 2.8 ] ) to @xmath534\\[-8pt ] \eqntext{\forall ( t,\mathbf{x } ) \in(0,\infty ) \times(0,\infty)^n .}\end{aligned}\ ] ] we have established the following result .    [ proposition_4 ] recall the functions @xmath535 , @xmath536 and @xmath537 defined on @xmath538 by ( [ 1.7 ] ) , ( [ 1.8 ] ) and ( [ 3.7.f ] ) , respectively , and impose assumption  [ assc ] . then ( [ 3.8 ] ) holds for every function @xmath539 . [ remark_4 ] here is a situation where assumption [ assc ] prevails : suppose that ( [ 3.2.g ] ) holds and that , for every @xmath540 and @xmath541 , we have @xmath542 for @xmath543 given by @xmath544 , @xmath545 and @xmath546 . then for any progressively measurable @xmath547 that satisfies ( [ aa1 ] ) we select the progressively measurable functional @xmath548 via @xmath549 , @xmath550 . this choice induces @xmath551 which obeys @xmath552 as in ( [ aa2 ] ) for all @xmath553 ; the process @xmath380 of ( [ lamda ] ) and ( [ lamda_too ] ) is a @xmath393-martingale , whereas ( [ 1.1 ] ) becomes @xmath554 , \quad   x_i(0)=x_i>0\ ] ] for @xmath128 . the condition ( [ aa1 ] ) guarantees now that @xmath555 takes values in @xmath342 , @xmath556-a.s . in the resulting primal admissible system @xmath529 . the quantity @xmath557 defined in ( [ 3.7.f ] ) is the value of a stochastic control problem : namely , the _ maximal _ `` _ _ containment _ _ '' _ probability _ , over all measures @xmath558 with @xmath559 , that the process @xmath427 with dynamics ( [ aux ] ) , initial configuration @xmath560 , and controlled through the choice of progressively measurable functional @xmath561 as in ( [ const1 ] ) , ( [ const2 ] ) , does _ not _ hit the boundary of the positive orthant by time @xmath562 .    let us suppose that the resulting function @xmath537 is continuous on @xmath438 . then it can be checked [ as in lions ( @xcite ) , lemma ii.1 and @xcite , theorem ii.4 ] that it satisfies as well the following dynamic programming principle : _ for every initial configuration @xmath563 , the process _ @xmath564\\[-8pt ] \eqntext{0\le t \le t \mbox { \textit{is a } } \mathbb{q}^ { \mathcal{n}}\mbox{-\textit{supermartingale } , } \forall\mathcal{n } \in\mathfrak { n}(\mathbf { x } ) .}\end{aligned}\ ] ] [ see @xcite , @xcite , @xcite and @xcite for results in a similar vein . ] equivalently , the process @xmath565 , @xmath242 is a @xmath566-supermartingale , for every @xmath567 . consider now an arbitrary continuous function @xmath568 which satisfies @xmath569 on @xmath516 , and is such that , for every @xmath570 and @xmath571 , the process @xmath572 is a  @xmath573-supermartingale . we shall denote by @xmath574 the collection of all such functions and note that @xmath575 and @xmath576 . from optional sampling we have then for every @xmath577 the comparisons @xmath578 = \mathbb{q}^ { \mathcal{n } } ( \mathcal{t } > t ) , \ ] ] thus also @xmath579 , @xmath580 .    in other words , _ the function @xmath581 defined in _ ( [ 3.7.f ] ) _ is the smallest element of the collection _ @xmath582 . it is also clear from this line of reasoning that @xmath583 attains the supremum @xmath584 in ( [ 3.7.f ] ) , if and only if the process @xmath585 is a @xmath586-martingale . [ theorem_1 ] suppose that assumption [ assc ] and conditions ( [ measur])([uppsemicont ] ) hold and that the function @xmath537 of ( [ 3.7.f ] ) is continuous on @xmath438 . then the infimum in ( [ 1.7 ] ) is attained , and @xmath587    consider an arbitrary function @xmath588 in the collection @xmath589 just defined and fix an arbitrary pair @xmath590 ; then for every @xmath591 , consider a mollification @xmath592 of the function @xmath593 with @xmath594 . proposition [ proposition_2 ] gives then @xmath595 . since @xmath596 is arbitrary , this shows that @xmath597 is dominated by @xmath598 , the infimum of @xmath599 over all functions @xmath600 . but the reverse inequality @xmath601 holds on the strength of ( [ 3.8 ] ) , so ( [ 3.8.a ] ) follows .      under the conditions of theorem [ theorem_1 ] , the arbitrage function @xmath602 is equal to the function @xmath603 of ( [ 3.7.f ] ) and is continuous on @xmath604 . thanks to the dynamic programming principle of  ( [ dp ] ) , it is also a viscosity solution of the _ hamilton  jacobi  bellman _ ( _ hjb _ ) _ equation _ @xmath605 on @xmath438 [ cf . lions ( @xcite ) , theorem iii.1 or @xcite , theorem  i.1 ] . if in addition to being continuous , as we assumed in theorem [ theorem_1 ] , the function @xmath606 of ( [ 3.7.f ] ) is of class @xmath20 locally on @xmath607 , then the arbitrage function @xmath602 is not only a viscosity solution but actually a  classical solution of the hjb equation ( [ 3.9 ] ) . this is the case , for instance , under the combined conditions of theorem [ theorem_1 ] and proposition [ proposition_3 ] ; then the arbitrage function @xmath608 also satisfies on the domain @xmath438 the _ linear _ parabolic equation @xmath609 with @xmath288 as in assumption [ assa ] or remark [ remark_2 ] , in addition to the initial condition @xmath610 in particular , the arbitrage function @xmath602 satisfies , in this case , the requirement ( [ d.11 ] ) and belongs to the class @xmath246 of section [ sec5 ] . recalling propositions [ proposition_1][proposition_3 ] and theorem [ theorem_1 ] , we summarize the above discussion as follows . [ theorem_2 ] suppose that conditions ( [ measur])([uppsemicont ] ) , ( [ 3.2.gg ] ) , ( [ 3.2 ] ) and assumptions [ assa ] , [ assb ] and [ assc ] are in force . then the arbitrage function @xmath602 is the smallest element of the class @xmath246 , as well as a classical solution of both the hjb equation ( [ 3.9 ] ) and of the linear parabolic equation ( [ 7.25 ] ) , subject to ( [ 3.9.a ] ) . furthermore ( [ 3.8.a ] ) holds , the infimum in ( [ 1.7 ] ) is attained , and the markovian investment rule @xmath611 in ( [ 2.11.a ] ) with @xmath612 satisfies ( [ 2.11.b ] ) for every admissible system @xmath183 . [ remark_5 ] we note that theorem [ theorem_2 ] is in agreement with general regularity theory for fully nonlinear parabolic equations , as in lions ( @xcite ) , theorem ii.4 ( see also @xcite , section 6.5 ; krylov ( @xcite ) ; wang ( @xcite , @xcite , @xcite ) , theorems ii.3.2 and iii.2 ; or @xcite , chapter xiv ) . as we mentioned already , assumptions [ assa ] and [ assb ] can be replaced in theorem [ theorem_2 ] by the conditions of remark [ remark_2 ] . we conjecture that the conclusions of theorem [ theorem_2 ] should hold under even weaker assumptions but leave this issue for future research . we also remark that the function @xmath613 , @xmath614 satisfies an hjb - type equation simpler than ( [ 3.9 ] ) , namely , the _ pucci maximal equation _ , @xmath615 along with the initial condition @xmath616 . in the setting of theorem  [ theorem_2 ] , equation ( [ 3.10 ] ) reduces to @xmath617      let us go back to the volatility - stabilized model introduced in @xcite , but now with some `` knightian '' uncertainty regarding its volatility structure @xmath618 for some given @xmath619 . the case @xmath620 corresponds to the variance structure of the model studied in fernholz and karatzas ( @xcite , @xcite ) . more specifically let us assume that , for any given @xmath621 , the compact , convex subset @xmath622 of @xmath9 in ( [ e.3.a ] ) consists of all matrices @xmath623 with @xmath624 for @xmath625 and @xmath626 the sets of ( [ 1.b ] ) are given as @xmath627 \bigr\}\ ] ] for some given constants @xmath628 $ ] , @xmath629 ; these choices satisfy ( [ 3.2.g ] ) , ( [ thetatrace ] ) .    condition ( [ 3.2 ] ) is satisfied in this case automatically ( in fact , with @xmath630 ) , as are ( [ 3.2.gg ] ) and ( [ d.4 ] ) : it suffices to take @xmath631 and @xmath632 , which induces @xmath633 in ( [ d.2 ] ) . these functions are all locally bounded and locally lipschitz continuous on @xmath502 . the hjb equation ( [ 3.9 ] ) satisfied by the arbitrage function @xmath634 becomes @xmath635 , \ ] ] and reduces to the _ linear _ parabolic equation @xmath636 of ( [ 7.25 ] ) for the choice of variances in ( [ volstab ] ) . the reason for this reduction is that the expression on the left - hand side of ( [ 3.11 ] ) is negative , so we have considerable simplification in this case .    in this example , the arbitrage function @xmath608 can be represented as @xmath637\ ] ] in terms of the components of the @xmath342-valued capitalization process @xmath638 . these are now time - changed versions @xmath639 , @xmath128 of the independent squared - bessel processes @xmath640 run with a time change @xmath641 common for all components , and with @xmath642 independent standard brownian motions [ see fernholz and karatzas ( @xcite , @xcite ) , @xcite and pall ( @xcite ) for more details ] . for any given investment rule @xmath145 and admissible system @xmath183 , let us consider the quantity @xmath643 this measures , as a proportion of the initial total market capitalization , the smallest initial capital that an investor who uses the rule @xmath138 and operates within the market model @xmath72 , needs to set aside at time @xmath33 in order for his wealth to be able to `` catch up with the market portfolio '' by time @xmath269 , with @xmath210-probability one . our next result exhibits the arbitrage function @xmath644 of ( [ 1.7 ] ) as the min  max value of a zero - sum stochastic game between two players : the investor , who tries to select the rule @xmath645 so as to make the quantity of ( [ 9.1 ] ) as small as possible and `` nature , '' or the goddess tyche herself , who tries to thwart him by choosing the admissible system or `` model '' @xmath646 to his detriment . [ theorem_3 ] under the conditions of theorem [ theorem_1 ] , we have @xmath647    for the quantities of ( [ 9.1 ] ) and ( [ 1.9 ] ) we claim @xmath648 indeed , if the set on the right - hand side of ( [ 9.1 ] ) is empty , we have@xmath649 and nothing to prove ; if , on the other hand , this set is not empty , then for any of its elements @xmath207 the process @xmath650 is a @xmath651-supermartingale , and therefore ( [ 1.89 ] ) , that is , @xmath652 , still holds and  ( [ 9.3 ] ) follows again .    taking the infimum with respect to @xmath145 on the left - hand side of ( [ 9.3 ] ) , then the supremum of both sides with respect to @xmath183 , we obtain @xmath653 from ( [ 1.9 ] ) . the quantity @xmath654 is the lower value of the stochastic game under consideration .    in order to complete the proof of ( [ 9.2 ] ) it suffices , on the strength of theorem [ theorem_1 ] , to show that the upper value @xmath655 of this game satisfies @xmath656 to see this , we introduce for each given investment rule @xmath657 the quantity @xmath658 that is , the smallest proportion @xmath207 of the initial market capitalization that allows an investor using the rule @xmath138 to be able to `` catch up with the market portfolio '' by time @xmath269 with @xmath659-probability one , no matter which admissible system ( model ) @xmath660 might materialize . we have clearly @xmath661 which leads to @xmath662 and proves ( [ 9.5 ] ) . let us place ourselves now in the context of theorem [ theorem_2 ] and observe that proposition [ proposition_1 ] , along with proposition [ proposition_2 ] and its corollary , yields @xmath663 by virtue of ( [ 9.3 ] ) for @xmath664 and of ( [ 2.11.b ] ) for @xmath665 . here @xmath334 is the `` least favorable admissible system '' that attains the supremum over @xmath666 in ( [ 1.8 ] ) , and @xmath667 denotes the investment rule of ( [ 2.11.a ] ) with @xmath668 .    in this setting , the investment rule @xmath669 attains the infimum@xmath670 in ( [ 9.8 ] ) , and we obtain then @xmath671\\[-8pt ] \eqntext{\forall \mathcal{m } \in\mathfrak{m } ( \mathbf { x})}\end{aligned}\ ] ] on the strength of ( [ 9.7 ] ) . putting ( [ 9.9 ] ) and ( [ 9.10 ] ) together we deduce @xmath672 the saddle property of the pair @xmath673 . in particular , the investment rule @xmath674 of ( [ 2.11.a ] ) with @xmath665 is seen to be the investor s best response to the least favorable admissible system @xmath675 of proposition [ proposition_1 ] , and vice - versa . in this sense the investor , once he has figured out a least favorable admissible system @xmath334 , can allow himself the luxury to `` forget '' about model uncertainty and concentrate on finding an investment rule @xmath669 that satisfies @xmath676 , @xmath677 as in ( [ 9.9 ] ) , that is , on outperforming the market portfolio with the least initial capital within the context of the least favorable model @xmath678 . we are grateful to professors jaka cvitani , nicolai krylov , mete soner , nizar touzi , hans fllmer , erhan bayraktar , johan tysk , and to the two reviewers , for helpful advice on the subject matter of this paper and for bringing relevant literature to our attention . numerous helpful discussions with drs . robert fernholz , adrian banner , vasileios papathanakos , phi - long nguyen - thanh , and especially with johannes ruf , are gratefully acknowledged .
in an equity market model with `` knightian '' uncertainty regarding the relative risk and covariance structure of its assets , we characterize in several ways the highest return relative to the market that can be achieved using nonanticipative investment rules over a given time horizon , and under any admissible configuration of model parameters that might materialize . 
 one characterization is in terms of the smallest positive supersolution to a fully nonlinear parabolic partial differential equation of the hamilton 
 jacobi 
 bellman type . under appropriate conditions , 
 this smallest supersolution is the value function of an associated stochastic control problem , namely , the maximal probability with which an auxiliary multidimensional diffusion process , controlled in a manner which affects both its drift and covariance structures , stays in the interior of the positive orthant through the end of the time - horizon . 
 this value function is also characterized in terms of a stochastic game , and can be used to generate an investment rule that realizes such best possible outperformance of the market .    and    .    .