$\ovalbox{\tt\small REJECT}$ SDE 1 1 SDE ;1) SDE 2) Burgers Model SDE $([4],[5],[7], [8])$ 1.1 SDE SDE (cf.[4],[5]) SDE $\{$ : $dx_

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1 $\ovalbox{\tt\small REJECT}$ SDE 1 1 SDE ;1) SDE 2) Burgers Model SDE $([4],[5],[7], [8])$ 1.1 SDE SDE (cf.[4],[5]) SDE $dx_{t}=a(t, X_{t}, u)dt+b(t, x_{t}, u)dwt$, $X_{0}=\xi(\omega)$ $\sim u_{0}(x)d_{x}$ W. $u(t, x),$ $u_{0}(x)$ $\xi(\omega)$ pdf SDE $\partial_{t}u+\partial_{x}\{a(t, X, u)u\}=\frac{1}{2}\partial_{x}^{2}\{b^{2}(t, X, u)u\}$, $u(0, x)=u_{0}(x)$, $1_{\mathrm{S}}$ -ogawa@tkanazawa-u.ac.jp $(t, x)\in R_{+}\cross R^{1}$ (1) (2)

2 . 47 $0$ ( ) $\mathrm{e}\mathrm{g}$ ( E.Puckett, D.Talay, M.Bossy, S.Ogawa, M.Chauvin, ARouault etc.) SDE 1.2 SDE (2) SDE(I). ([4]) $dx_{t}=a(t, X_{t} u)dt+b(t, X_{t} u)dw_{t}$, $X_{0}=\xi(\omega)$ $\sim u_{0}(x)d_{x}$ (3) $c(t, xg)=c(t, x, \int F(x-y)g(y)dy)(c(\cdot)=a(\cdot), b(\cdot)$ $g(x)$ pdf) $F(x)$ $u(t, x)$ (3) $X_{t}$ pdf

3 48 $\partial_{t}u+\partial_{x}\{a(t, x u)u\}=\frac{1}{2}\partial_{x}^{2}\{b^{2}(t, x u)u\}$, $(t, x)\in R_{+}\cross R^{1}$ $u(0, x)=u0(x)$. (4), 2 PDE (4) SDE (3) ([4], [5], [6], [7]), (2) Burgers $3_{\text{ }}4$ 2 2 (A.1) $a(t, x, y),$ $b(t, x, y)$ $(x, y)$ 1 (of linear growth) (A.2) $a(t, x, y),$ $b(t, x, y)$ $t$ $x,$ $y\in R^{1}$ Lipschitz (A.3) $F(x)$ -class \tau $C^{2}$ $u_{0}(x)$ $p\geq 1$, $E \xi ^{2p}<+\infty$ SDE (3) $(X_{t}, u(t, x))$ 2.1 Euler-Maruyama Euler-Maruyama SDE (3) $t_{k}=k\cdot h,$ $h=t/n$ $\overline{x}_{kt_{k}}=\overline{x}$ $t=t_{k}$

4 49 $X_{t_{k}}$ $\overline{x}_{k+1}=\overline{x}_{k}+a(t_{k},\overline{x}_{k} \mathrm{m}\overline{u}(t_{k}))\cdot h+b(t_{k}, \overline{x}_{k} \mathrm{m}\overline{u}(t_{k}))\cdot\triangle kw$, $0\leq k\leq N-1$, $\overline{x}_{0}=\xi(\omega)$, $\overline{u}(t_{k}, x)$ $\llcorner \mathrm{b}\text{ }\triangle_{k}w=w(t_{k+1})-w(t_{k})$ $\overline{x}_{k}$ pdf $\mathrm{m}\overline{u}(t_{k})$ Carlo estimator $N_{0}$ $\{\overline{x}_{k}(\omega^{i}), 1\leq i\leq N_{0}\}$ (5) Monte estimator estimator (M) $g(x)d_{x}$ $N_{0}$ $\exists_{\eta(>}0$ ) such that $E[ \int\epsilon(x)\phi(x)dx]^{2}p=c_{g},\psi N_{0^{-}}.2p\eta$ for $\forall_{\phi}\in C_{0}^{1}$, $\epsilon(x)=\mathrm{m}g(x)-g(x)$. $C_{g,\phi}$ $\phi$ $g(x)$, $1$ (A. ) $-(\mathrm{a}.3),$ $(\mathrm{m})$ Theorem 2. 1 $\{\overline{x}_{k}\}$ ([6]) Euler-Maruyama (5) $ \mathit{1}/\mathit{2}^{f}$ - $\max_{k}e \overline{x}_{k}-x_{t_{k}} ^{2p}=C\{N_{0^{-}}2p\eta+h^{p}\}$ (6) ( 1) ([4]) (M.2) (M.2) $E \int \mathrm{m}g(x)-g(x) ^{2}pd_{X}=O(N_{0}^{-2p\eta})$. ( 2) Mil stein scheme (cf. ogawa [5], [6] $)$ 2

5 $\text{ }$ 50 $N,$ $N_{0}$, $N_{0}=N^{\beta}$ for $\exists_{\beta}$, $\max_{k}e \overline{x}_{k}-x_{t_{k}} ^{2p}=C\cdot h^{2p(\beta\wedge 1/2)}\eta$, SDE PDE (4) kernel methodl (cf. ([6])) $g(x)d_{x}$ $\{\zeta(\omega^{i}), 1\leq i\leq N_{0}\}$ $g(x)$ I $N_{0}$ $\mathrm{m}_{k}^{\delta}g(x)=_{\overline{n}}\sum_{0i=1}k_{\delta}(x-\zeta(\omega^{i}))$, $K_{\delta}(x)=\underline{1}_{K(^{X})}$ $\delta$ $K(x)$ \iota J.\supset g \emptyset *X \xi 7 \tau $K(x)$ $\text{ ^{}\delta}\text{ }\mathrm{l}_{\text{ }}\mathrm{a}\text{ }$ (K.1) $\int K$. $(x)d_{x}=1$, $\int xk(x) d_{x}<+\infty$. (K.2) $r_{1}>0$,, $K_{r_{1}}= \sup_{y}\frac{ 1-(FK)(y) }{ y ^{r_{1}}}<+\infty$, eu $FK(y)$ va $K(x)\text{ }$ Fourier image. $K(x)$ $\mathrm{m}_{k}^{\delta}\overline{u}(t_{k}, x)$, $u(t_{k}, x)$ Proposition 2. 1 ([6]) (4) $u(t, x)$ $r>0$ $U_{r}= \sup_{t}\int_{-\infty}^{\infty} y^{r}(fu)(t, y) 2dy<+\infty$.

6 51 $\mathrm{m}_{k}^{\delta}\overline{u}(t_{k}, x)$ 2 IMSE $(t_{k},$ $N_{0,)}\delta$ $=$ $E \int.\mathrm{m}_{k}^{\delta}\overline{u}(tk, x)-u(t_{k}, x) 2dX(1\geq r)$, \mp \acute$\int$ IMSE $(tk, N_{0}, \delta)\leq\frac{k^{2}}{n_{0}\delta}+(k_{r}u_{r})^{22r-3}\delta+\delta K/2E xt_{k}-\overline{x}_{k} ^{2}$ (7) $K^{2}= \int K(x) 2dX$, $K^{\prime 2}= \int K (X) 2dX$. Proposition2. 1 Theorem2. 1 Theorem 2. 2 ([6]) Proposition 2. 1 $\mathrm{m}_{k}^{\delta}u(tk, x)$ $\delta$ window width pammeter $\min_{\delta}\min_{k}imse(t_{k}, N^{\beta}, \delta)\leq 3[E_{r}(\delta_{*})+\delta_{*}^{-1}K^{2\beta}N^{-}]=O(N^{-4r}/(2r+3))$, $E_{r}( \delta)=(k_{r}u_{r})^{22}\delta r+ck2\delta-3n^{-}2-t^{\backslash }\delta_{*}=(\frac{3ck^{2}}{2r(k_{r}u_{r}n)^{2}})^{1}/(2r+3)$. 2.2 $\{\overline{x}_{k}(\omega^{i})\}$ (Q.1) estimator $\frac{1}{n_{0}}\sum_{1i=}^{n_{0}}\delta_{\overline{x^{i}}k}$ estimator (Q.2) (Q.3) (1) (2)? $F(x)$ $\delta_{0}(x)$? Burgers equation (i.e. $a(t, x, y)=y,$ $b(t, x, y)=1,$ $F(x)=\delta(x)))$?

7 52 3 (Q.1) $-$ estimator $\mathrm{m}\overline{u}(t_{k}, x)$ SDE 1 ( ) SDE solver line (cf. [3]) $N_{0}$ SDE $O(N_{0^{-}})1/2$ ( 1992 ) SDE 1 $N_{0}$ $dx_{t}^{i}=a(t, X^{t}, \overline{n_{0}}j\neq\sum Fi(X^{i}-tx_{t}^{j}))dt$ $+b(t,$ $x^{i},$ I $N_{0}$ $- \sum F(x_{t}^{i}-X_{t}j))dW^{i}t$ ${}^{t}n\mathit{0}j\neq i$ (8) $\xi^{i}$ $X_{0}^{i}=$, $1\leq i\leq N_{0}$, $W_{t}=(W^{1}, W^{2}, \cdots, W^{N}0)$ $u_{0}(x)dx$ $\{\xi^{i}\}$ No- i.i.d. Brownian PDE (4) SDE (3)

8 53 SDE SDE Boltzmann (cf. D.Talay, L.Tubaro, P.Bernard, M.Bossy [11], [1] $)$ 3.1 SDE SDE (8) Euler-Mauyama scheme $\neg X_{k+},$ $=a(t_{kk}, \overline{x}^{i}, \frac{1}{n_{0}}\sum_{j\neq i}^{n_{0_{f(x_{k}}}}\neg-\overline{x})j)kh$ $+b(t_{k},x_{k} \neg, \frac{1}{n_{0}}\sum^{n0_{f(^{\neg}}}j\neq ix_{k^{-}}\overline{x}^{j}k))\triangle_{k}w^{i}$ (9) $\neg X_{0}=$ $\xi^{i}$, $1\leq i\leq N_{0}$. $\neg \mathrm{y}_{t_{k+1}}=$ $a(t_{k},\mathrm{y}_{t}\cdot\overline{u}^{i}\neg k(t_{k}))h+b(t_{k},\mathrm{y}_{t_{k}}; \overline{u}^{i}(\neg tk))\triangle_{k}w^{i}$ $\overline{u}^{i}(t_{k}, X)=\neg \mathrm{y}_{t_{k+1}}\emptyset pdf_{0}$ (10) $\neg \mathrm{y}_{0}=$ $\xi^{i}$, $1\leq i\leq N_{0}$ $\{\overline{\mathrm{y}}_{t_{k}}^{1}\}$ $X_{t_{k}}$ 1/2 ( $\mathrm{c}\mathrm{f}.[4]^{)}$ $E \{\max_{k}1\leq\leq N X_{t_{k}}-\overline{Y}t_{k}1 ^{2}\}\leq C\cdot h$. AKohatsu Kohatsu-Ogawa [8] $E \{\max_{k} \overline{\mathrm{y}}_{t_{k^{-}}}^{1}\overline{x}_{k}12 \}\leq C\cdot N_{0}^{-1}$ $\{\overline{x}_{k}^{1},1\leq k\leq\ovalbox{\tt\small REJECT}^{1}\}$ 1\sim /2- (Q.1)

9 54 Theorem 3. 1 (A. ) $l$ $-(A.\mathit{3})$ $E \{\max_{k} \overline{x}_{k}^{1}-x_{t} ^{2}k\}\leq C(h+N_{0}^{-1})$. ( $p=1$) SDE Euler-Maruyama scheme 1/2 1 SDE (Q.2) (Ogawa,S. Kohatsu-Higa,A[8]) Theorem 3. 2 (A.Kohatsu-Higa&S.Ogawa [8]) Euler- Malyama scheme (9 $\{\overline{x}_{k}^{\perp}, 1\leq k\leq N\}(\mathit{3})$ $X$ 1 $h(x)\in C^{2}$ $\max_{k} E\{h(X_{t_{k}})-h(\overline{X}^{1})k\} \leq C\{\triangle t+\frac{1}{\sqrt{n_{0}}}\}$. 4 Burgers (Q.3) (9) $\overline{u}_{k}^{1}(x)(1\leq$ $\{\overline{x}_{k}^{1}\}-$ $k\leq N)$ (2) $u(t, x)$ $F(x)$ $\delta(x)$ $\overline{u}_{k}^{1}(x)(1\leq k\leq N)$ $u(t, x)$ Burgers equation 4.1 (Q.3) (2) ( $u(t, x)^{)}$ $U(t, x)= \int_{-\infty}^{x}u(t, y)dy$

10 55 Burgers equation $(a(t, x, y)=y,$ $b(t, x, y)=1f(x)=$ $H(x)$ $H(x)=$ Heaviside s function) $dx_{t}=u(t, X_{t})dt+dW_{t}$, $X_{0}=\xi\sim u_{0}(x)dx$, (11) $U(t, x)= \int_{-\infty}^{x}u(t, y)dy$ $u(t, x)$ $X_{t}$ pdf (11) $u(t, x)$ Cauchy $\partial_{t}u+\partial_{x}\{uu\}=\frac{1}{2}\partial_{x}^{2}u$, $(t, x)\in R_{+}\cross R^{1}$ $u(0, x)=u0(x)$. $U(t, x)$ $\partial_{x}\{\partial_{t}u+\frac{1}{2}\partial xu^{2}-\frac{1}{2}\partial^{2}u\}x=0$, $(t, x)\in R_{+}\cross R^{1}$ $U(0, x)=u_{0}(x)= \int_{-\infty}^{x}u_{0}(y)dy$. $u_{0}(x)$ $U(t, x)$ Burgers equation $\partial_{t}u\dotplus U\partial_{x}U=\frac{1}{2}\partial_{x}^{2}U$, $U(0, x)=u_{0}(x)$. $(t, x)\in R_{+}$. $\cross R^{1}$ (12) (12) SDE (11) Burgers equation $H(x)$ SDE (11) $H(x)$ ( $H^{\epsilon}(x)$ $H^{\epsilon}(x)arrow H(x)$ as $6arrow 0$ ) $\overline{x}_{t}^{\epsilon}$ $X$ ( 3) $\mathrm{d}.\mathrm{t}\mathrm{a}\mathrm{l}\mathrm{a}\mathrm{y}[1]$ 1994 M.Sniztman [10], Burgers equatiopn SDE

11 56 (cf. $\mathrm{o}\mathrm{g}\mathrm{a}\mathrm{w}\mathrm{a}[7])\circ$ D.Talay, M.Bossy Burgers equation [7] Burgers equation D.Talay-MMM Bossy [1] 4.2 SDE of Burgers like processes (11) Burgers $dx_{t}=a(t, X, U)dt+b(t, X, U)dW_{t}$, $X_{0}=\xi\sim u_{0}(_{x})d_{x}$ (13) $U(t, x)= \int^{x}u(t\cdot, y)dy\text{ }u(t, X)=X_{t}\text{ }$ pdfo $a(),$ $b(\cdot)$ $u_{0}(x)$ (A.2), (A.3) (with $p=1$ ) (A.1) (A.1) (A.1) $a(t, x, y),$ $b(t, x, y)$ $[0, T]\cross R^{1}\cross[-1,1]$ (13) $u(t, x)$ $u(t, x)$ $\mathrm{m}\mathrm{c}\mathrm{k}\mathrm{e}\mathrm{a}\mathrm{n}$-vlasov (GMV) $u(0, x)=u0(x)$. $\partial_{t}u+\partial_{x}\{a(t, X, U)u\}=\frac{1}{2}\partial_{x}^{2}\{b^{2}(t, x, U)u\}$ $U(t, x)= \int_{-\infty}^{x}u(t, y)dy$

12 57 $\partial_{t}u+a(t, x, U)\partial_{x}U=\frac{1}{2}\partial_{x}\{b^{2}(t, X, U)\partial_{x}U\}$, $U(0, x) U_{0()}X(= \int_{-\infty}^{x}u_{0}(y)dy)$. (14) (13). Burgers 4.3 $\rho_{\epsilon}(x)$ $\{H^{\epsilon}(x)\}$ $H^{\epsilon}(x)=(H*\rho_{\epsilon})(X)$ $\epsilon(>0)$ molli er $(X^{\epsilon}., u^{\epsilon})$ SDE (3) $F(x)=H^{\epsilon}(X)$ $dx_{t}^{\epsilon}=a_{\epsilon}(t, x_{t}^{\epsilon}, u^{\epsilon})dt+b_{\epsilon}(t, X_{t^{\epsilon}}, u)\epsilon dw$ $t$,. $X_{0}=\xi(\omega)$ $\sim u_{0}(x)d_{x}$ $u^{\epsilon}(t, x)=$ $X_{t}^{\epsilon}$ pdf of the. (15) $c_{\epsilon}(t, x, g)=c(t, x, \int H^{\epsilon}(x-y)g(y)dy),$ $(c(\cdot)--a(\cdot), b(\cdot))$ $g(x)$ pdf $X^{\epsilon}$ SDE (13) $a(\cdot),$ $b(\cdot)$ $u_{0}(x)$ $-(\mathrm{a}.3)$ (A. $1$) (H.1) (H.2) Pdf $u^{\epsilon}(t, SDE (15) x)$ $x$ $L^{2}$ $ u^{\epsilon}(t, \cdot) _{2}\leq\psi(t)$ estimate, \psi ( $(0, T]$ (GMV) 1 Ll- $\sup_{t} u(t, \cdot) _{1}<\infty$

13 58 ( ) $b(t, x, y)=1$ $a(t, x, y)$ (A.1), (A.2) $1$ $(\mathrm{h}.2)$ (H. ), (H.1) S.Meleard-S Roelly [2] Lemma (H.2) (GMV) a(t, $x,$ $y$) $y$ Lipschitz Talay-Bossy [1] Theorem 4. 1([7]) (H. ) Cauchy (13) $l$ $-(H.\mathit{2})$ (X, $u$) $\{X^{\epsilon}\}$ $U(t, x)$ (14) 4.4 (13) $u(t, x)$ SDE (13) X. pdf ( ) $u(t, x)$ (GMV) (H.2) SDE (13) $X_{t}$ $U(t, x)=$ $\int^{x}u(t, y)dy$ SDE (13) $(t, x)arrow a(t, x, u(t, x)),$ $b(t, X, u(t, x))$ ( $(\mathrm{m}\mathrm{p})$ ) Martingale Problem SDE (13) X. SDE (13) Martingale Problem $(\mathrm{m}\mathrm{p})$ $C([0, T]arrow R^{1})$,?, $\mu$

14 $\mathcal{l}$ 59 (a) $\mu_{0}=u_{0}$ $\mu_{s}(0\leq s\leq T)$ $x(s)$ (b) $\phi\in C^{2}$ $\emptyset(x(t))-\emptyset(x(\mathrm{o}))-\int_{0}^{t}\mathcal{l}_{\mu_{s}}\phi(x(s))ds$ $\mu$-martingale, $x(\cdot)$ $C([0, T]arrow R^{1})$ canonical process $\mathcal{l}_{\mu}\phi=\frac{1}{2}b^{2}(t, X, U(t, X))\partial 2\emptyset x(x)+a(t, X, U(t, x))\partial x\emptyset(x)$. $\{X^{\epsilon}.\}$ SDE(13) SDE (15) $\epsilon$ $(X^{\epsilon}, u^{\epsilon})$ $L^{\epsilon}$ infinitesimal generator Martingale $\mu^{\epsilon}$ Problem $L_{\mu}^{\epsilon} \phi=\frac{1}{2}b_{\epsilon}2(t, X, \mu)\partial_{x}^{2}\psi+a_{\epsilon}(t, x, \mu)\partial_{x}\phi$. $\{\mu^{\epsilon}\}$ tight (Proof).. (A.1) $a(t, x, y),$ $b(t, x, y)$ $\{X^{\epsilon}.\}$ equicontinuous $\epsilon$, $E X_{t}^{\epsilon}-X^{\epsilon}s ^{4}\leq C(t-S)^{2}\text{ }$ $\epsilon_{n}arrow 0$ $\{\mu^{\epsilon_{n}}\}$ $\mu^{*}$ (H.1) problem $(\mathrm{m}\mathrm{p})$ $\mu^{*}$ $\mathcal{l}_{\mu^{-}}$ martingale $\mu_{t}^{*},$ $(^{\forall}t)$

15 60 $\mu^{*}$ [1] M.Bossy and D.Talay Convergence rate for the approximation of the limit law of weakly interacting particles, INRIA Research Report 2410, Novembre 1994 $N^{o}$ [2] S.Meleard, S.Roelly-Coppoletta A propagation of chaos result for a system of particles with moderate interaction, Stochastic Processes and Their Applications, 26, , 1987 [3] S.Ogawa, K.Naono Stochastic simulation of nonlinear diffusions, Lectures at Organized session on the numerical simulation of $\prime SDEs, held at the g\mathit{5}$-annual Meeting Indust.Applied Math. of The Japanese Soc. of [4] S.Ogawa $\mathrm{m}\mathrm{o}\dot{\mathrm{n}}$te Carlo simulation of nonlinear diffusion processes $\mathrm{v}\mathrm{o}\mathrm{l}2$ II, Japan J.Indust.Appl.Math.,, No.1, (1996) [5] S.Ogawa Some problems in the numerical simulation of nonlinear SDEs, $\mathrm{x}$ Math. and Computers in Simulation vol., No,xx, (1995) [6] S.Ogawa Density estimation problem in the simulation of nonlinear diffusions, (in Japanese) $Suur\dot{\mathrm{v}}$kagaku Koukyuuroku 850, RIMS Kyoto Univ., 1995 [7] S.Ogawa Problems in the simulation of Burgers like processes, Proceedings of the Workshop on Turbulent Diffusion and Related Problems in Stochastic Numerics, held at $ISM$ in Oct.1996 April 1997.

16 61 [8] A.Kohatsu-Higa, S.Ogawa Rate of convergence of the weak approximation of an Euler type to nonlinear SDEs, Monte Carlo Methods and Applications 1997 [9] E.G.Puckett Convergence of a random particle method to solutions of the Kolmogorov equation, Math. of Comput., 52, 31-45, [10] M.Snitzman Private communication, at the seminar in Z\"urich Univ [11] D.Talay, P,Bernard, L.Rbaro Rate of convergence of a stochastic particle method for the Kolmogorov equation with variable coefficients, Math. of Comput., 63, 1994

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