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Simulation of a space-time bounded diffusion. (English) Zbl 0964.60065

Most of the so far known stochastic-numerical methods rely on a deterministic time-discretization of corresponding stochastic processes. In contrast to that fact, the authors present space-time bounded approximations of initial value problems for $d$-dimensional ordinary stochastic differential equations (SDE)

$dX={\chi }_{{\tau }_{t,x}>s}b\left(s,{X}_{s}\right)ds+{\chi }_{{\tau }_{t,x}>s}\sigma \left(s,X\right)dW\left(s\right),\phantom{\rule{1.em}{0ex}}X\left(t\right)={X}_{t,x}=x\in {R}^{d}$

in a bounded domain $Q=\left[{t}_{0},{t}_{1}\right)×G\subset {R}^{d+1}$, where $X,b$ are $d$-dimensional vectors, $\sigma$ is a $d×d$-matrix, $W={\left(W\left(s\right)\right)}_{s\ge {t}_{0}}$ represents a $d$-dimensional standard Wiener process, and the stopping time ${\tau }_{t,x}$ is the first-passage time of the process $\left(s,{X}_{t,x}\left(s\right)\right)$, $s\ge t$, to ${\Gamma }=\overline{Q}\setminus Q$. The coefficients ${b}^{i}\left(s,x\right)$ and ${\sigma }^{i,j}\left(s,x\right)$, $\left(s,x\right)\in \overline{Q}$, and the boundary $\partial G$ are assumed to be sufficiently smooth and the strict ellipticity condition is imposed on $a\left(s,x\right)=\sigma \left(s,x\right){\sigma }^{T}\left(s,x\right)$. The proposed algorithm is based on a space-time discretization using random walks over boundaries of small space-time parallelepipeds. Corresponding convergence theorems and their proofs are given. A method of approximate search for exit points of space-time diffusions from a bounded domain is presented.

This work continues a series of papers initiated by the first author [see, for example, Stochastics Stochastics Rep. 56, No. 1-2, 103-125 (1996; Zbl 0888.60048) and ibid. 64, No. 3-4, 211-233 (1998)]. For those readers who prefer to read about the original works, the latter two citations are highly recommended, where the idea of space-time discretizations in conjunction with the construction of random walks over boundaries has already been explained, and related mean square approximation theorems are found as well there. The special value of this paper may be seen in the simulation results on which the authors report at the last ten pages. Thus, the choice of the title of this paper is misleading a little bit in view of the anticipative and innovative expectations of the potentially interested reader. The paper is recommended for those readers who are interested in a theoretical justification, pitfalls, advantages of stochastic-numerical methods including simulation studies for the approximation of deterministic initial-boundary value problems for parabolic equations under the condition of strong ellipticity on $\overline{Q}$.

##### MSC:
 60H10 Stochastic ordinary differential equations 65C30 Stochastic differential and integral equations 60J60 Diffusion processes 60H35 Computational methods for stochastic equations 65C05 Monte Carlo methods 60H30 Applications of stochastic analysis