On approximation of solutions of multidimensional SDE’s with reflecting boundary conditions. (English) Zbl 0799.60055

The author considers the following \(d\)-dimensional stochastic differential equation on \(D\), with reflecting boundary conditions \[ X^ i_ t=X^ i_ 0+\sum^ d_{j=1} \int^ t_ 0 f_{ij}(X_ s)dW^ j_ s+ \int^ t_ 0 g_ i(X_ s)ds+K^ i_ t;\quad t>0,\;1\leq i\leq d, \] where \((W^ 1_ t,\ldots,W^ d_ t)\) is a \(d\)-dimensional Wiener process, \(X_ t=(X^ 1_ t,\ldots,X^ d_ t)\) is a reflecting process on \(D\cup\partial D\) and \((K^ 1_ t,\ldots,K^ d_ t)\) is a bounded variation process with variation \(| K|_ t\) increasing only when \(X_ t\in\partial D\). For the Euler and Euler-Peano time- discretization schemes of approximations of \(X\) he investigates the rate of convergence. The authors results generalize results of R. J. Chitashvili and N. L. Lazrieva [Stochastics 5, 255-309 (1981; Zbl 0479.60062)], C. N. Kinkladze [Thesis, Tbilisi, 1983] and D. Lepingle [Abstract, 1st Workshop on Stochastic Numerics, September 7-12, Gosen (1992)] who considered more simple domain \(D\).


60H20 Stochastic integral equations
60H99 Stochastic analysis
60F15 Strong limit theorems


Zbl 0479.60062
Full Text: DOI


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