Simple Search

Query:

Enter a query and click »Search«...

Format:

Display: entries per page entries

Zbl 0964.60065
Milstein, G.N.; Tretyakov, M.V.
Simulation of a space-time bounded diffusion. (English)
[J] Ann. Appl. Probab. 9, No.3, 732-779 (1999). ISSN 1050-5164

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) $$dX= \chi_{\tau_{t,x}> s}b(s,X_s) ds + \chi_{\tau_{t,x}> s} \sigma (s,X) dW(s),\quad X(t)=X_{t,x} = x \in R^d$$ in a bounded domain $Q = [t_0,t_1) \times G \subset R^{d+1}$, where $X,b$ are $d$-dimensional vectors, $\sigma$ is a $d \times d$-matrix, $W=(W(s))_{s \ge t_0}$ represents a $d$-dimensional standard Wiener process, and the stopping time $\tau_{t,x}$ is the first-passage time of the process $(s,X_{t,x}(s))$, $s\ge t$, to $\Gamma = \overline{Q}\setminus Q$. The coefficients $b^i(s,x)$ and $\sigma^{i,j} (s,x)$, $(s,x) \in \overline{Q}$, and the boundary $\partial G$ are assumed to be sufficiently smooth and the strict ellipticity condition is imposed on $a(s,x) = \sigma (s,x) \sigma^T (s,x)$. 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. \par 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 $\overline{Q}$.
[Henri Schurz (Minneapolis)]

MSC 2000:: *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 Appl. of stochastic analysis

Keywords: Monte Carlo methods; stochastic differential equations; numerical methods for SDE; space-time discretization; mean square approximation; simulation studies; approximation of exit times of space-time Brownian motion; random walk over boundaries; convergence results; random algorithms; numerical methods for Dirichlet problems; numerical approximation of initial-boundary value problems for parabolic equations

Citations: Zbl 0888.60048

Comment on this Item PDF MathML XML ASCII DVI PS BibTeX Online Ordering Article Journal

	Scientific prize winners of the ICM 2010
	Overhang
	Lie groups, physics and geometry. An introduction for physicists, engineers and chemists.

Simple Search

Zentralblatt MATH Berlin [Germany]