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the credit valuation adjustment ( cva ) is , by definition , the difference between the risk - free portfolio value and the true portfolio value that takes into account default risk of the counterparty . in other words , cva is the market value of counterparty credit risk . after the financial crisis in 2007 - 2008 , it has been widely recognized that even major financial institutions may default . therefore , the market participants has become fully aware of counterparty credit risk . in order to reflect the counterparty credit risk in the price of over - the - counter ( otc ) derivative transactions , cva is widely used in the financial institutions today . although duffie - huang @xcite has already introduced the basic idea of cva in 1990 s , several people reconsidered the theory of cva related to collateralized derivatives ( cf . @xcite ) and also efficient numerical calculation methods appeared(cf . @xcite ) . there are two approaches to measuring cva : unilateral and bilateral ( cf . @xcite ) . under the unilateral approach , it is assumed that the bank that does the cva analysis is default - free . cva measured in this way is the current market value of future losses due to the counterparty s potential default . the problem with unilateral cva is that both the bank and the counterparty require a premium for the credit risk they are bearing and can never agree on the fair value of the trades in the portfolio . therefore , we have to consider not only the market value of the counterparty s default risk , but also the bank s own counterparty credit risk called debit value adjustment ( dva ) in order to calculate the correct fair value . bilateral cva ( it is calculated by netting unilateral cva and dva ) takes into account the possibility of both the counterparty default and the own default . it is thus symmetric between the own company and the counterparty , and results in an objective fair value calculation . mathematically , unilateral cva and dva are calculated in the same way , and bilateral cva is the difference of them . so we focus on the calculation of unilateral cva in this paper . cva is measured at the counterparty level and there are many assets in the portfolio generally . therefore , we have to be involved in the high dimensional numerical problem to obtain the value of cva . this is one of the reasons why cva calculation is difficult . on the other hand , each payoff usually depends only on a few assets . we will focus on this property and suggest an efficient calculation methods of cva in the present paper . let us consider the portfolio consist of the contracts on one counterparty . let @xmath0 be @xmath1-valued stochastic processes @xmath2 we think that @xmath3 is an underlying process . we consider the model that the macro factor is determined by @xmath4 and the payoff of each derivative at maturity @xmath5 is the form of @xmath6 let @xmath7 be the final maturity of all the contracts in the portfolio . let @xmath8 be the default time of the counterparty , @xmath9 be its hazard rate process , @xmath10 be the process of loss when the default takes place at time @xmath11 , and @xmath12 be the discount factor process from @xmath11 to @xmath13 . we assume that @xmath14 is the function of @xmath15 and that @xmath16 and @xmath17 are the function of @xmath18    let @xmath19 be total value of all contracts in the portfolio at time @xmath11 under the assumption that counterparty is default free . then @xmath19 is given by @xmath20,\ ] ] where @xmath21 denotes the expectation with respect to the risk neutral measure . then unilateral cva on this portfolio is the restructuring cost when the counterparty defaults . so unilateral cva is given by @xmath22 \nonumber\\ & = e[\int_0^t l(t)\exp(-\int_0^t \lambda(s)ds ) \lambda(t)d(0,t ) ( \tilde{v}_0(t ) \vee 0 )   dt ] \nonumber\\ & = e[\int_0^t l(t)\exp(-\int_0^t \lambda(s)ds ) \lambda(t ) ( v_0(t ) \vee 0 )   dt ] \label{defcva},\end{aligned}\ ] ] where @xmath23,\ ] ] and @xmath24 is a function such as @xmath25 since @xmath26 is a function of @xmath27 we denote it by @xmath28 then cva is given by the following form . @xmath29 \vee 0 )   dt ] \label{defcva2}.\end{aligned}\ ] ]    now we prepare the mathematical setting . let @xmath30 be fixed , @xmath31 , @xmath32 @xmath33 and @xmath34 . + let @xmath35 @xmath36 @xmath37 be the borel algebra over @xmath35 and @xmath38 be the wiener measure on @xmath39 let @xmath40 @xmath41 be given by @xmath42 @xmath43 then @xmath44 is a @xmath45-dimensional brownian motion . let @xmath46 @xmath47    let @xmath48 @xmath49 @xmath50 here @xmath51 denotes the space of @xmath52-valued smooth functions defined in @xmath53 whose derivatives of any order are bounded . we regard elements in @xmath54 as vector fields on @xmath55    now let us consider the following stratonovich stochastic differential equations . @xmath57 where @xmath58 + let @xmath59 and @xmath60 be @xmath61 then we have @xmath62 there is a unique solution @xmath63 to this equation . then @xmath64 also satisfies the solution to the following stratonovich stochastic differential equation . @xmath65 where @xmath66 is @xmath67    we assume that vector fields @xmath68 satisfy condition ( ufg ) stated in the section [ vec ] . let @xmath69 be defined by ( [ defe ] ) in section [ vec ] . by @xcite , if @xmath70 the distribution law of @xmath71 under @xmath38 has a smooth density function @xmath72 for @xmath73 + let @xmath74 we assume that the underlying asset process is @xmath75 . we also assume that @xmath76 let @xmath77 denotes the space of functions on @xmath52 given by @xmath78 where @xmath79 . + @xmath80 denotes the space of lipschitz continuous functions on @xmath81 and we define a semi - norm @xmath82 on @xmath83 by @xmath84 let @xmath85 be the linear subspace of @xmath80 spanned by @xmath86 +   + we define linear operators @xmath87 by @xmath88 , \quad f\in { lip}({\bf r}^n),\ ] ] and @xmath89 by @xmath90 , \quad f \in { lip}({\bf r}^{\tilde{n}_m}).\ ] ] we remind that @xmath91 is represented by @xmath92 we assume that @xmath93 \times { \bf r}^n \to [ 0,\infty)$ ] satisfies the following two conditions . + ( 1 ) @xmath94 is differentiable in @xmath11 and there is an integer @xmath95 and a constant @xmath96 such that @xmath97}| \frac{\partial}{\partial t}g(t , x)|   \leqq c_1(1+|x|^{n_1 } ) , \quad x \in { \bf r}^n.\ ] ] ( 2 ) @xmath94 is @xmath98-times coninuously differentiable in @xmath99 and there is an integer @xmath100 and a constant @xmath101 such that @xmath97 } |\frac{\partial^{\alpha}}{\partial x^{\alpha}}g(t , x)| \leqq c_2(1+|x|^{n_2}),\quad x \in { \bf r}^n\ ] ] for any multi index @xmath102 + we assume that a discounted payoff functions @xmath103 in equation ( [ defcva2 ] ) belong to @xmath104 under the assumptions above , cva @xmath105 is given by @xmath106 \vee0\}dt ] .\end{aligned}\ ] ] we will introduce numerical calculation methods by monte carlo simulation for @xmath105 . +   + let @xmath107 be a probability space , and @xmath108 @xmath109 be continuous stochastic processes such that each probability law on @xmath110 of @xmath111 under @xmath112 is the same as that of @xmath113 for all @xmath114 @xmath115 and that @xmath116 @xmath109 are independent . + let us define projections @xmath117 by @xmath118 and define @xmath119 by @xmath120 we define random linear operators ( stochastic mesh operators ) @xmath121 on @xmath122 by @xmath123    let @xmath124 denotes the set of partitions @xmath125 such that @xmath126 and that @xmath127 let @xmath128 we define estimators @xmath129 in the following . @xmath130 . \label{app2}\end{aligned}\ ] ] our main results are following . [ main2 ] let @xmath131 let @xmath132 be a sequence and suppose that there is a constant @xmath133 such that @xmath134 then there exists a constant @xmath135 such that @xmath136 \leqq c_1 ( l^{-\frac{1}{1+\alpha_0}}+|\delta|)\ ] ] for any @xmath137 and @xmath138    [ main1 ] let @xmath139 and let @xmath132 be a sequence such that there is a constant @xmath140 such that @xmath141 suppose that there are constants @xmath142 $ ] and @xmath143 such that @xmath144 for all @xmath145.$ ] + then then there exists a constant @xmath146 and @xmath147 @xmath148 such that @xmath149 and @xmath150 for any @xmath151 and @xmath138    [ actual ] let @xmath152 be @xmath153 then by theorem [ main2 ] , we see that @xmath154 and @xmath155 theorem [ main1 ] shows that the estimation of @xmath156 may be better than @xmath157 . we can compute the estimators @xmath158 practically in the following way . + first , we generate a family of independent paths @xmath159 next , by using @xmath160 , we compute @xmath161 then our estimator @xmath162 is @xmath163 we used the same paths for monte carlo simulation and construction of stochastic mesh operator .    for @xmath164 , we generate another independent family of independent paths @xmath165 and we compute @xmath166    in the above computations of @xmath162 and @xmath164 , we use the values of @xmath167 , only . + as for the computation of @xmath168 we do not use @xmath169 explicitly . we use @xmath169 only to judge whether @xmath170 or not . so the approximation has no error when the signs of @xmath171 and @xmath172 are the same , even if there are large differences between them . let @xmath173 and @xmath174 for @xmath175 , let @xmath176 if @xmath177 @xmath178 @xmath179 and let @xmath180 @xmath181 also , for each @xmath182 @xmath183    we define vector fields @xmath184},$ ] @xmath185 inductively by @xmath186 } = v_i ,   \quad i = 0,1,\ldots , d,\ ] ] @xmath187 } = [ v_{[\alpha ] } , v_i ] ,   \qquad i = 0,1,\ldots , d.\ ] ] here @xmath188 @xmath189 for @xmath177 @xmath178 and @xmath190    we assume that a system of vector fields @xmath191 satisfies the following condition ( ufg ) . ( ufg ) there is an integer @xmath192 and there are functions @xmath193 @xmath194 @xmath195 satisfying the following . @xmath196 }   = \sum_{\beta \in { \cal a}^{*}_{\leqq \ell_0 } } \varphi_{\alpha,\beta}v_{[\beta ] } } , \qquad   \alpha \in { \cal a}^{*}.\ ] ]    a system of vector fields @xmath197 also satisfies the @xmath198 condition .    _ proof . _ we prove following by induction on @xmath199 @xmath200 } ( f \circ \pi_m ) = ( \tilde{v}^{(m)}_{[\alpha ] } f)\circ \pi_m , \quad f \in c_b^{\infty}({\bf r}^{\tilde{n}_m } ) ,   \end{aligned}\ ] ] for any @xmath201 and @xmath202 + it is trivial in the case of @xmath203 by the assumption for induction , @xmath204 } ( f \circ \pi_m ) = ( { v}_{[\alpha ] } v_i- v_i { v}_{[\alpha ] } ) ( f \circ \pi_m)\ ] ] @xmath205 } ( ( \tilde{v}^{(m)}_{i } f)\circ \pi_m ) - v_i ( ( \tilde{v}^{(m)}_{[\alpha]}f )   \circ \pi_m)\ ] ] @xmath206}(\tilde{v}^{(m)}_{i } f ) ) \circ \pi_m - ( \tilde{v}^{(m)}_{i } ( \tilde{v}^{(m)}_{[\alpha]}f ) )   \circ \pi_m.\ ] ] so we have ( [ inductive ] ) . from ( ufg ) condition , we have @xmath207 } ( f \circ \pi_m)=   \sum _ { \beta \in { \cal a}^{*}_{\leqq \ell_0 } } \varphi _ { \alpha , \beta }   { v}_{[\beta ] } ( f \circ \pi_m)\ ] ] @xmath208}^{(m)}f ) \circ \pi_m.\ ] ] let @xmath209 be @xmath210 then @xmath211}f=(v_{[\alpha]}^{(m)}f)\circ \pi_m \circ j_m\ ] ] @xmath212}^{(m)}f ) \circ \pi_m ) \circ j_m = \sum _ { \beta \in { \cal a}^{*}_{\leqq \ell_0 } } ( \varphi _ { \alpha , \beta } \circ j_m ) \tilde{v}_{[\beta]}^{(m)}f.\ ] ] so we have our assertion . +   + let @xmath213 @xmath214 @xmath215 be a @xmath216 symmetric matrix given by @xmath217}(\tilde{x}_m)\tilde{v}^j_{m,[\alpha ] } ( \tilde{x}_m ) , \qquad i , j=1,\ldots , \tilde{n}_m.\ ] ] let @xmath218 and @xmath219 by @xcite , we see that if @xmath220 the distribution law of @xmath71 under @xmath38 has a smooth density function @xmath221 for @xmath222 moreover , by @xcite we see that @xmath223 @xmath224 we have @xmath225 by @xcite . in this section , we use the notation in @xcite . we have the following lemma similarly to the proof of @xcite lemma 8 ( 3 ) .    [ rev ] for any @xmath226 let @xmath227 and @xmath228 then @xmath229}f ) ( x(t , x ) ) ] = t^{-\|\alpha\| / 2}e^{\mu}[\phi_{\alpha}(t , x ) f(x(t , x))],\ ] ] and @xmath97 , x \in { \bf r}^n , p\in ( 1 , \infty ) } e[|\phi_{\alpha}(t , x)|^p ] < \infty.\ ] ] +    let @xmath230 be a smooth function such that @xmath231 let @xmath232 and @xmath233 be @xmath234 then for any @xmath235 @xmath236    [ rev2 ] if @xmath237 then @xmath238    _ proof . _ let @xmath239 then for any @xmath235 @xmath240 and @xmath241 we have @xmath242 then @xmath243 is a cauchy sequence in @xmath244 because @xmath245 \}}\|d\phi(t , x)\|_h , \quad n\geqq m\geqq 1.\end{aligned}\ ] ] because @xmath246 is a closed operator , we have @xmath247 for any @xmath248 . so we have the assertion . + let us denote @xmath249    [ disclem2 ] let @xmath250 . then there exists a @xmath251 such that @xmath252\\ & \leqq c \|\nabla f\|_{\infty } \int_{s}^t(r^{-1/2}+(t - r)^{-1/2})dr , \end{aligned}\ ] ] for any @xmath253 and any @xmath254 . _ let @xmath255 be @xmath256=(p_{t - t}f)(x(t , x^*)).\ ] ] @xmath255 is a @xmath257-martingale . let @xmath258    let @xmath259 by it formula , @xmath260 note that , @xmath261 @xmath262 @xmath263 and @xmath264 so we have @xmath265\ ] ] @xmath266 dr\ ] ] @xmath267dr+ \sum_{j=1}^d \int_s^t e^{\mu}[|(v_jg)(v_j ( p_{t - r}f))(x(r , x^*))|]dr.\ ] ] now by the definition of @xmath268 and @xmath269 , we have , @xmath270dr \leqq \int_s^t   e^{\mu}[|m(r)|^2]^{1/2}e^{\mu}[|(l_t g)(x(r , x^*))|^2]^{1/2}dr\ ] ] @xmath271}e^{\mu}[|(l_t g)(x(r , x^*))|^2]^{1/2 } \int_s^t   e^{\mu}[|m(r)|^2]^{1/2}dr.\ ] ] by burkholder s inequality , @xmath272^{1/2}dr \leqq e^{\mu}[\langle m \rangle_t]^{1/2 } ( t - s )   \leqq \sup_{r\in ( s , t ) } \|v_j ( p_{t - r}f)\|_{\infty}(t - s).\ ] ] on the other hand , we have , @xmath273dr\ ] ] @xmath274dr\ ] ] @xmath275 for any @xmath253 and any @xmath254 . + on the other hand @xmath276 @xmath277 @xmath278 notice that @xmath279 we have @xmath280\ ] ] @xmath281\ ] ] @xmath282 where @xmath283,\ ] ] @xmath284.\ ] ] let @xmath285 then by lemma [ rev2 ] , @xmath286 let @xmath287 be defined by the formula of lemma [ rev ] . then we have @xmath288,\ ] ] and @xmath289 , x\in { \bf r}^n } e^{\mu}[|\phi_{g , i}(t , x)|^p ] < \infty.\ ] ] then there exists a constant @xmath251 such that @xmath290 also we have @xmath291\|v_j^2 ( p_{t - r}f)\|_{\infty},\ ] ] for any @xmath253 and any @xmath292 . +   + let vector field @xmath293 be represented by @xmath294 then we have @xmath295 where @xmath296 and @xmath297.\ ] ] moreover , we have @xmath298 then by corollary 9 of @xcite , since @xmath299 and there is a constant @xmath251 such that @xmath300 and @xmath301 for any @xmath302 and any @xmath292 . +   + so we have @xmath303\ ] ] @xmath304 letting @xmath305 we have our assertion . let @xmath250 . there exists a constant @xmath306 such that @xmath307\ ] ] @xmath308 for any @xmath309 and any @xmath254 . _ for @xmath310 , there exists @xmath311 such that @xmath312 and @xmath313 , for any @xmath314 . so we obtain the result from lemma [ disclem2 ] . +    [ disclem1 ] let @xmath315 and @xmath316 there exists a constant @xmath306 such that @xmath317\nonumber \\ \leqq & c(\|\nabla h\|_{\infty}+ \|\nabla^2 h\|_{\infty } ) ( t - t),\label{disc1}\\ \text{and } \nonumber\\ & e^{\mu } [ |g(t , x(t , x))(p^{(m)}_{t - t}(h\vee 0))(\tilde{x}^{(m)}(t,\tilde{x}_m^*))-(h\vee 0)(\tilde{x}^{(m)}(t,\tilde{x}_m^*))| ] \nonumber \\ \leqq & c(\|\nabla h\|_{\infty}+ \|\nabla^2 h\|_{\infty } ) ( t - t ) \label{disc2}.\end{aligned}\ ] ] for any @xmath318    _ proof . _ ( [ disc1 ] ) follows from it s formula . so we show ( [ disc2 ] ) . + let @xmath319 are as defined in ( [ defphi ] ) . let @xmath320 by it s formula @xmath321 @xmath322 @xmath323 @xmath324    so we have @xmath325-\bar{\varphi}_k(h(\tilde{x}^{(m)}(t,\tilde{x}_m^*))\ ] ] @xmath326 ds\ ] ] @xmath327 ds\ ] ] notice that @xmath328 then we have @xmath329-\bar{\varphi}_k(h(\tilde{x}^{(m)}(t,\tilde{x}_m^*))|]\ ] ] @xmath330\|\tilde{l}_m h\|_{\infty}(t - t)\ ] ] @xmath331 ds.\ ] ] on the other hand , we have @xmath332 @xmath333 @xmath334 + let @xmath335 and @xmath336 be defined by the formula of lemma [ rev ] . then it follows that @xmath337\|_{\infty}\ ] ] @xmath338 \|(\varphi_k \circ h ) \tilde{v}^{(m)}_{i } h \|_{\infty } \leqq c s^{-1/2 } \| \tilde{v}^{(m)}_{i } h \|_{\infty},\ ] ] and @xmath339 \|_{\infty }   \leqq c\| ( \tilde{v}^{(m)}_{i})^2 h \|_{\infty}.\ ] ] so we have @xmath340 ds\ ] ] @xmath341 @xmath342 letting @xmath343 we have the assertion . let @xmath344 and @xmath345 . there exists a constant @xmath251 such that @xmath346   \leqq c(t - t),\end{aligned}\ ] ] for any @xmath347    _ proof . _ notice that @xmath348 and lemma [ disclem1 ] is valid for @xmath349 . on the other hand , for @xmath350 , we have the expression that @xmath351 @xmath352 so our assertion follows from lemma [ disclem1 ] . let @xmath353 be given by @xmath354.\ ] ] let @xmath355 be such that @xmath356 then we have @xmath357 is as follows . @xmath358\ ] ]    let @xmath359 @xmath360 be sub @xmath361-algebra of @xmath37 given by @xmath362 , \ \ell = 1,2,\ldots \ } .\ ] ] +    [ time disc ] there exists a constant @xmath306 such that @xmath363 . _ let @xmath364 then by lemma [ disclem2 ] , there is a constant @xmath251 such that + @xmath365 \\ - & e^{\mu } [ g(t_i , x(t_i , x^*))(p_{t_{k}-t_i } \tilde{f}_{k})(x(t_i , x^ * ) ) \vee 0]| dt\\ \leqq    & c\sum_{k=1}^k \sum_{i = i_{(k-1)}}^{i_{(k)}-1 } \int_{t_i}^{t_{i+1 } } dt \int_{t_i}^t ( r^{-1/2}+(t_k - r)^{-1/2})dr\\ \leqq & c|\delta|   \sum_{k=1}^k \sum_{i = i_{(k-1)}}^{i_{(k)}-1}\int_{t_i}^{t_{i+1 } } ( r^{-1/2}+(t_k - r)^{-1/2})dr . \end{aligned}\ ] ] so we have @xmath366 so the assertion follows . to estimate the stochastic mesh operator , we use the following estimation of transition kernel @xmath367 obtained by proposition 8 of @xcite . [ heatkernel ] let @xmath368 be given by @xmath369 then for any @xmath370 and @xmath315 there is a @xmath251 such that @xmath371 , \ x , y \in e_m,\ ] ] and @xmath372 , \ x , y \in e_m.\ ] ] in particular , for any @xmath373 and @xmath374 there is a @xmath251 such that @xmath375 , \ x , y \in e_m.\ ] ]    let @xmath376 from proposition 13 , 21 and proposition 15 ( 1 ) of @xcite , we have the followings . [ basic ] let @xmath377 and @xmath378 then we have @xmath379   = ( p^{(m)}_{s , t}f)(x ) , \qquad \nu_s^{(m)}-a.e.x \in e_m.\ ] ] and @xmath380 \leqq   \frac{1}{l } \int_{e_m } \frac{p^{(m)}(t - s , x , y)^2 |f(y)|^2}{q_{s , t}^{(m , l,\omega)}(y)}\ ; dy .\ ] ]    [ key ] let @xmath381 then there exists a @xmath382 such that @xmath383\right)^{1/2 } \\ & \leqq   c   l^{-(1-\delta)/2}(t_k - t)^{-(1+\delta)(\tilde{n}+1)\ell_0/4 } ( \int_{e_m } f(y)^2 ( 1+|y|^2)^{-\tilde{n}_m}dy)^{1/2}.\end{aligned}\ ] ] for any @xmath384 any @xmath315 and any @xmath385    [ exset ] let @xmath386 @xmath387 } z_l^{(m , k)}(s;\delta).\end{aligned}\ ] ] then we have the followings . + @xmath388 for any @xmath381 , and @xmath248 , there is a @xmath389 such that @xmath390^{1/p}\leqq c_{p,\delta}\varepsilon^{-5\ell_0}l^{-p\delta^2/2 + 1/p}\ ] ] for any @xmath391 , k=1,\ldots , k,$ ] and @xmath392 + @xmath393 let @xmath394 and @xmath395 if @xmath396 , and @xmath397 , then @xmath398 for any @xmath399 , \quad k=1,\ldots , k.$ ] and @xmath392    now we introduce the following sets and functions . let @xmath400 @xmath401 be given by @xmath402 and @xmath403 let @xmath404 \times e \times \omega \to [ 0 , \infty ) , i=1,2,3,$ ] be the measurable functions given by @xmath405 let @xmath406 be the transition kernel of @xmath407    [ d12 ] let @xmath381 . then there exists a constant @xmath382 such that @xmath408|g(t , x)| p(t , x^ * , dx ) \leqq    c l^{-(1-\delta)^3 } 1 _ { [ 0 , t_k- \varepsilon ) } ( t ) , \label{d1 } \\ & ( \int_{e_m } e^p[\ d_{2,\varepsilon , l}^{(m , k)}(t , x)^2 ] p(t , x^ * , dx))^{1/2}\nonumber \\ \leqq    & c l^{-(1-\delta)/2 } ( t_k - t)^{-(1+\delta)(\tilde{n}+1)\ell_0/4 } 1 _ { [ 0 , t_k- \varepsilon ) } ( t),\label{d2}\\ \text{and } \nonumber\\ &   \int_{e_m }   d_{3,\varepsilon , l}^{(m , k)}(t , x ) |g(t , x)| p(t , x^ * , dx )   \leqq c ( t_k - t)1_{[t_k - \varepsilon , t_k ) } ( t ) . \label{d3}\end{aligned}\ ] ] for any @xmath409 , l \geqq 1,$ ] @xmath315 and @xmath410    _ proof . _ equation ( [ d3 ] ) follows from lemma [ disclem1 ] . so we will show ( [ d1 ] ) and ( [ d2 ] ) . note that if @xmath411 , both sides are @xmath412 in ( [ d1 ] ) and ( [ d2 ] ) . so we will consider the case @xmath413 . by proposition [ basic ] , we have @xmath414 |g(t , x)| p(t , x^ * , dx)\\ = & \int_{e } e^p[|(q_{t , t_k , \varepsilon}^{(m)}(1-\varphi_{m , k , l})f_{m , k})(x ) -(p_{t_k - t}^{(m ) } ( 1-\varphi_{m , k , l})f_{m , k})(\pi_m(x ) ) | ] |g(t , x)| p(t , x^ * , dx ) \\ \leqq & \int_{e } ( e^p[(q_{t , t_k , \varepsilon}^{(m ) } |(1-\varphi_{m , k , l})f_{m , k}| ) ( \pi_m(x))]\\ & \qquad \qquad \qquad \qquad \qquad \qquad+ ( p_{t_k - t}^{(m ) } |(1-\varphi_{m , k , l})f_{m , k}| ) ( \pi_m(x ) ) ) |g(t , x)| p(t , x^ * , dx)\\ \leqq & 2 \int_{e } ( p_{t_k - t}^{(m ) } ( 1-\varphi_{m , k , l})|f_{m , k}|)(\pi_m(x ) ) g(t , x ) p(t , x^ * , dx).\end{aligned}\ ] ] using hlder s inequality for @xmath415 @xmath416 @xmath417 @xmath418 @xmath419 @xmath418 @xmath420 @xmath421 we used @xmath422 in the last inequality . so we have equation ( [ d1 ] ) . + next we will show equation ( [ d2 ] ) . noting that from @xmath423 and @xmath137 , @xmath424 since propsition [ exset ] , @xmath425 . and by proposition [ basic ] , we have @xmath426\\ \leqq & 1_{b^{(m , k)}(t , \delta , l ) } \frac{1}{l } \int_{e_m } \frac{|\varphi_{m , k , l}f_{m , k}(\tilde{y}_m)|^2p^{(m)}(t_k - t , \tilde{x}_m,\tilde{y}_m)^2}{q_{t , t_k}^{(m , l,\omega)}(\tilde{y}_m)}d\tilde{y}_m\\ \leqq & \frac{2}{l } \int_{e_m } \frac{|\varphi_{m , k , l}f_{m , k}(\tilde{y}_m)|^2}{p^{(m)}(t_k , \pi_m(x^*),\tilde{y}_m ) } p^{(m)}(t_k - t , \tilde{x}_m,\tilde{y}_m)^2 dy.\end{aligned}\ ] ] then we have @xmath427 p(t , x^ * , dx))^{1/2 } \\ \leqq & ( \int_{e }   e^p[1_{b^{(m , k)}(t , \delta , l)}e^p [ |(q_{t , t_k , \varepsilon}^{(m)}\varphi_{m , k , l}f_{m , k})(x ) -(p_{t_k - t}^{(m ) } \varphi_{m , k , l}f_{m , k})(x ) |^2| \mathcal{f}_t ] ] p(t , x^ * , dx))^{1/2}\\ \leqq & ( \frac{2}{l } \int_{e_m } \frac{|\varphi_{m , k , l}f_{m , k}(y)|^2}{p^{(m)}(t_k , \pi_m(x^*),\tilde{y}_m ) }   ( \int_{e } p^{(m)}(t_k - t , \tilde{x}_m,\tilde{y}_m)^{(1-\delta)+(1+\delta ) } p(t , x^ * , dx))d\tilde{y}_m)^{1/2}\\ \leqq & ( \frac{2}{l } \int_{e_m } \frac{|\varphi_{m , k , l}f_{m , k}(y)|^2}{p^{(m)}(t_k , \pi_m(x^*),\tilde{y}_m)^{\delta } }   ( \int_{e_m } p^{(m)}(t , \pi_m(x^ * ) , \tilde{x}_m ) p^{(m)}(t_k - t , \tilde{x}_m,\tilde{y}_m)^{(1+\delta)/\delta}d\tilde{x}_m)^{\delta}d\tilde{y}_m)^{1/2}.\end{aligned}\ ] ] let @xmath428 from lemma [ heatkernel ] , there exists a constant @xmath251 such that @xmath429 we set @xmath430 as @xmath431}\max_{\substack{m=1,\ldots , m,\\ k=1,\ldots , k}}(\int_{e_m } h_m(x)^{-(\tilde{n}_m+1)\ell_0(1+\delta)/\delta}(1+|x|^2)^{q(1+\delta)/\delta}p^{(m)}(t , \tilde{x}_m^*,x)dx)^{\delta/2}\\ & \quad \quad \times ( \int_e|g(t , x)|dx)^{1/2}.\end{aligned}\ ] ] @xmath430 is bounded by proposition 3 of @xcite . then since @xmath432 , we have @xmath433^{1/2}|g(t , x)| p(t , x^ * , dx)\\ \leqq & \frac{c_1}{l } \int_{e_m } p^{(m)}(t_k , \tilde{x}_m^*,\tilde{y}_m)^{-\delta } |\varphi_{m , k , l}f_{m , k}(\tilde{y}_m)|^2(1+|\tilde{y}_m|^2)^{-q(1+\delta)}d\tilde{y}_m   ( t_k - t)^{-(1+\delta)(n+1)\ell_0/2 } , \\ \leqq & c_1l^{-(1-\delta)}(t_k - t)^{-(1+\delta)(\tilde{n}_m+1)l_0/2 }   \int_{e_m } |f_{m , k}(\tilde{y}_m)|^2(1+|\tilde{y}_m|^2)^{-q(1+\delta)}d\tilde{y}_m .\end{aligned}\ ] ] since @xmath434 and @xmath24 is lipschitz continuous , @xmath435 so we have the assertion . + let @xmath436 and @xmath437 let @xmath438 and @xmath439 be @xmath440    [ dd ] there exists a constant @xmath382 such that @xmath441 and , @xmath442 for any @xmath384 . _ let us take @xmath443 as @xmath444 if @xmath445 $ ] and @xmath446 then @xmath447 so notice that @xmath448 so we have @xmath449 because @xmath450 for @xmath451 . + on the other hand , @xmath452 and @xmath453 so we have equation ( [ sum1 ] ) . + next we show equation ( [ sum2 ] ) . @xmath454 where @xmath455 @xmath456 @xmath457 @xmath458 note that ( [ disceps ] ) and @xmath459 we have for @xmath460 , @xmath461 @xmath462 we have the followings similarly . @xmath463 so we obtain ( [ sum2 ] ) . [ convergence ] there exists a constant @xmath306 such that @xmath464 \leqq c\left ( l^{-(1-\delta)/2 }   \hat{e}\left(\varepsilon , ( 1+\delta)(\tilde{n}+1)\ell_0/4 \right ) + \varepsilon^2\right ) , l\geqq 1.\ ] ]    _ proof . _ @xmath464\ ] ] @xmath465 @xmath466|.\ ] ] then by schwartz s inequality , @xmath467|\ ] ] @xmath468\ ] ] @xmath469 @xmath470 \right)^{1/2 } 1_{[0 , t_k-\varepsilon)}\ ] ] @xmath471^{1/2 } + c(t_k - t)1_{[t_k-\varepsilon , t_k)}\ ] ] by proposition [ key ] @xmath464\ ] ] @xmath472 by proposition[dd ] , we have the assertion . [ improve ] let @xmath473 and @xmath474 . then we have @xmath475    _ proof . _ if @xmath476 , then @xmath477 so we see that @xmath478 @xmath479 @xmath480    let @xmath481 suppose that there is @xmath142 $ ] and @xmath482 such that @xmath483 @xmath484 then there exists a constant @xmath251 , @xmath485 , such that @xmath486 and @xmath487 @xmath488 for @xmath489    _ proof . _ in this proof , we denote @xmath490 by @xmath491 for simplicity . let @xmath492 let @xmath493 be given by @xmath494 and let @xmath495 be given by @xmath496 then @xmath497 @xmath498|,\ ] ] @xmath499 since @xmath500 is @xmath501 measurable . + applying lemma [ improve ] to @xmath502 and @xmath503 we have @xmath504 where @xmath505,\\ & i_2=1_{\tilde{b}_{\varepsilon } } \sum_{i=0}^{n-1}(t_{i+1}-t_i ) \theta e^{\mu } [ |g(t_i , x(t_i))||1_{\{f_{p , i}(x(t_i)| \leqq \theta \}}].\end{aligned}\ ] ] by hlder s inequality , @xmath506^{\delta } \mu ( |f_{p , i}(x(t_i)| \leqq \theta)^{1-\delta } \leqq   c \theta \mu ( |f_{p , i}(x(t_i)| \leqq \theta).\ ] ] from the assumption , we have @xmath507 next we will estimate @xmath508 . @xmath509 @xmath510   .\end{aligned}\ ] ] @xmath508 is dominated by @xmath511 where @xmath512,\ ] ] @xmath513 @xmath514,\ ] ] @xmath515 @xmath516.\ ] ] @xmath517\ ] ] from proposition [ d12 ] , @xmath518=\sum_{i=0}^{n-1 } ( t_{i+1}-t_i ) \sum_{\substack{m=1,\ldots , m,\\ k ; t_k \geqq t_{i+1 } } } \int_{e } e^p[d_{1,\varepsilon , l}^{(m , k)}(t,\tilde{x}_m ) ]    @xmath519 @xmath520 + next , we will estimate @xmath521 . by hlder s inequality @xmath522^{1/2}\ ] ] @xmath523^{\delta/2}e^{\mu}[1_{\{\sum_{m=1}^m \sum_{k : t_k\geqq t_{i+1 } } ( d_{1,\varepsilon , l}^{(m , k)}(t_i , \pi_m(x(t_i ) + d_{3,\varepsilon , l}^{(m , k)}(t_i , \pi_m ( x(t_i ) ) > \theta/2\}}]^{(1-\delta)/2},\ ] ] @xmath524^{1/2}\ ] ] @xmath525)^{(1-\delta)/2}.\ ] ] so we have @xmath526 \leqq   c\sqrt{\frac{2}{\theta } } \sum_{i=0}^{n-1}(t_{i+1}-t_i )    e^p[\sum_{\substack{m=1,\ldots , m,\\k : t_k\geqq t_{i+1 } } }   \int_{e_m } d_{2,\varepsilon , l}^{(m , k)}(t_i , x)^2 p^{(m)}(t_i , \tilde{x}_m^ * , x)dx ] ^{1/2}\\ & \times e^p [ \sum_{\substack{m=1,\ldots , m,\\k : t_k\geqq t_{i+1 } } } \int_{e_m } ( d_{1,\varepsilon , l}^{(m , k)}(t_i , x ) + d_{3,\varepsilon , l}^{(m , k)}(t_i , x ) ) p^{(m)}(t_i , \tilde{x}_m^ * , x)dx ] ^{(1-\delta)/2}\\ \leqq & c \theta^{-(1-\delta)/2 } \sum_{i=0}^{n-1}(t_{i+1}-t_i )   \left ( \sum_{k : t_k\geqq t_{i+1 } } \phi^{(k)}(t_i , \varepsilon ; l^{-(1-\delta)^2/2 } , ( 1-\delta^2)(\tilde{n}+1)\ell_0/4 , 0 , 0)\right ) \\ & \times \left(\sum_{k : t_k\geqq t_{i+1 } } \phi^{(k)}(t_i , \varepsilon ; l^{-(1-\delta)^4/2 } , 0 , 1 , ( 1-\delta)/2)\right).\end{aligned}\ ] ] by proposition [ dd ] , @xmath527\leqq c\theta^{-1/2 }   \left ( l^{-(1-\delta)^4 } \hat{e}\left(\varepsilon , ( 1-\delta^2)(\tilde{n}+1)\ell_0/4\right ) + l^{-(1-\delta)^2/2}\varepsilon^{3(1-\delta)/2 } \right).\ ] ] similarly , we have @xmath528 \leqq   c \theta^{-(1-\delta)/2 } \sum_{i=0}^{n-1}(t_{i+1}-t_i )    e[\sum_{\substack{m=1,\ldots , m,\\k : t_k\geqq t_{i+1 } } }   \int_{e_m } d_{2,\varepsilon , l}^{(k)}(t_i , x)^2 p^{(k)}(t_i , \tilde{x}_k^ * , x)dx ] ^{(1-\delta)}\\ \leqq & c \theta^{-(1-\delta)/2 } \sum_{i=0}^{n-1}(t_{i+1}-t_i )   ( \sum_{k : t_k\geqq t_{i+1 } } \phi^{(k)}(t_i , \varepsilon ; l^{-(1-\delta)^2/2 } , ( 1-\delta^2)(\tilde{n}+1)\ell_0/4 , 0,0 ) ) ^2\\    \leqq & c\theta^{-1 } l^{-(1-\delta)^2 } \hat{e}\left(\varepsilon,(1-\delta^2)(\tilde{n}+1)\ell_0/2 \right).\end{aligned}\ ] ] it follows easily that @xmath529\leqq c\varepsilon^2.\ ] ] notice that @xmath530 we have @xmath531 \leqq c\left(\theta^{\gamma+1}+\theta^{-1 }    l^{-(1-\delta)^2/2 } \left ( l^{-(1-\delta)^2/2 } \hat{e}\left(\varepsilon,(1-\delta^2)(\tilde{n}+1)\ell_0/2 \right)+\varepsilon^{3(1-\delta)/2 } \right ) \right).\ ] ]    in particular if we take @xmath532 as @xmath533 then we have @xmath531 \leqq    c\left ( l^{-(1-\delta)^2/2 } \left ( l^{-(1-\delta)^2/2 } \hat{e}\left(\varepsilon,(1-\delta^2)(\tilde{n}+1)\ell_0/2 \right)+\varepsilon^{3(1-\delta)/2 } \right )   \right)^{(1+\gamma)/(2+\gamma ) } .\ ] ]    let @xmath534 be @xmath535 from proposition [ exset ] , we have @xmath536 @xmath537 and @xmath487 @xmath538    theorem [ main2 ] and theorem [ main1 ] follow from the theorem [ convergence ] and theorem [ improve ] . let @xmath539 be 1 dimensional brownian motion . let @xmath540 let @xmath541 be @xmath542\vee0 ) \ dt ] = \frac{2}{3 \sqrt{2\pi}}.\ ] ] let @xmath357 be the discretization of @xmath541 , such that @xmath543.\ ] ] we approximate @xmath541 as remark [ actual ] , where @xmath544 let @xmath545 be i.i.d sample paths of @xmath546 . we compute @xmath547 and @xmath157 by using of paths @xmath160 . @xmath548 let @xmath549 be another i.i.d sample paths of @xmath546 . we compute @xmath156 by @xmath550 we have @xmath551 when we take @xmath552 we have @xmath553 we also take @xmath554 and @xmath555 we replicate 100 estimators of @xmath556 for each @xmath557 . let `` average @xmath558 '' denote the average and `` standard dviation @xmath558 '' denote the unbiased standard deviation of these 100 estimators of @xmath559 we show the numerical result in table [ nume ] , we show graph of `` average @xmath560 '' and @xmath357 in figure [ fig1 ] and graph of `` standard deviation @xmath561 '' in figure [ fig2 ] . we see in figure [ fig1 ] that both @xmath157 and @xmath156 are close to @xmath357 , but we see in figure [ fig2 ] that @xmath156 is more stable than @xmath562 +               99 avramidis , a.n . , and p. hyden , efficiency improvements for pricing american options with a stochastic mesh , in proceedings of the 1999 winter simulation conference , pp . 344 - 350 broadie , m. , and p. glasserman , a stochastic mesh method for pricing high - dimensional american options j. computational finance , 7 ( 4 ) ( 2004 ) , 35 - 72 . duffie , d. , m. huang , 1996 , swap rates and credit quality , journal of finance , vol . 51 , no . 3 , 921 fujii , f. , a. takahashi derivative pricing under asymmetric and imperfect collateralization , and cva carf working paper f-240 , december 2010 glasserman , p. , `` monte carlo methods in financial engineering '' springer , 2004 , berlin . gregory , j. , counterparty credit risk and credit value adjustment : a continuing challenge for global financial markets , second editio , john wiley & sons , 2012 . kusuoka , s. , malliavin calculus revisited , j. math . tokyo 10(2003 ) , 261 - 277 . kusuoka , s. , and d.w.stroock , applications of malliavin calculus ii , j. fac . tokyo sect . 32(1985),1 - 76 . kusuoka , s. , and y. morimoto , stochastic mesh methods for h " ormander type diffusion processes , preprint . labordere , h . cutting cva s complexity , risk magazine , july , 2012 , pp . liu , g. , and l.j . hong , revisit of stochastic mesh method for pricing american options , operations research letters 37(2009 ) , 411 - 414 . shigekawa , i. , `` stochastic analysis '' , translation of mathematical monographs vol.224 , ams 2000 .
in this paper , the author considers the numerical computation of cva for large systems by mote carlo methods . 
 he introduces two types of stochastic mesh methods for the computations of cva . in the first method , 
 stochastic mesh method is used to obtain the future value of the derivative contracts . in the second method , 
 stochastic mesh method is used only to judge whether future value of the derivative contracts is positive or not . 
 he discusses the rate of convergence to the real cva value of these methods . 
 jel classification : c63 , g12    mathematical subject classification(2010 ) 65c05 , 60g40    keywords : computational finance , option pricing , malliavin calculus ,    stochastic mesh method , cva