zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
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=χ τ t,x >s b(s,X s )ds+χ τ t,x >s σ(s,X)dW(s),X(t)=X t,x =xR d

in a bounded domain Q=[t 0 ,t 1 )×GR d+1 , where X,b are d-dimensional vectors, σ is a d×d-matrix, W=(W(s)) st 0 represents a d-dimensional standard Wiener process, and the stopping time τ t,x is the first-passage time of the process (s,X t,x (s)), st, to Γ=Q ¯Q. The coefficients b i (s,x) and σ i,j (s,x), (s,x)Q ¯, and the boundary G are assumed to be sufficiently smooth and the strict ellipticity condition is imposed on a(s,x)=σ(s,x)σ 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.

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 Q ¯.

MSC:
60H10Stochastic ordinary differential equations
65C30Stochastic differential and integral equations
60J60Diffusion processes
60H35Computational methods for stochastic equations
65C05Monte Carlo methods
60H30Applications of stochastic analysis