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Discretization and simulation of stochastic differential equations. (English) Zbl 0554.60062
Let \(X_ t\) be a solution of the following ItĂ´-type stochastic differential equation \(dX_ t=b(X_ t)dt+\sigma (X_ t)dW_ t\). The authors consider approximation schemes of the type \(X_{t_{i+1}}=f(x_{t_ i},W.)\) for the processes \(X_ t\). They discuss both pathwise and mean-square convergence of several approximation schemes and estimate the speed of convergence and errors.
As far as the pathwise approximation is concerned the results are connected to the second authors paper, Stochastics 9, 275-306 (1983; Zbl 0512.60041). In the second part the authors discuss schemes for which \(\limsup_{n\to \infty}n^ 2E(| X^ n_ t-X_ t|^ 2)=C_ t\) holds and compare this with several approximations suggested in the literature. Especially the results from many doctorial theses are mentioned which are not published in regular journals. There are finally some numerical results by Monte-Carlo-approximations and stability discussions.
Reviewer: M.Breger

MSC:
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
65C05 Monte Carlo methods
65L05 Numerical methods for initial value problems involving ordinary differential equations
93E25 Computational methods in stochastic control (MSC2010)
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