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Numerical solutions of stochastic differential delay equations under local Lipschitz condition. (English) Zbl 1015.65002

Under certain hypotheses, which include the less restrictive assumption that \(f,g\) satisfy a local (rather than global) Lipschitz condition, a theorem is proved establishing convergence of Euler-Maruyama approximate solutions to the solution of the stochastic differential delay equation with variable delay \[ dx(t)= f\biggl(x,\bigl( \delta(t)\bigr)\biggr) dt+g\biggl(x(t), x \bigl(\delta(t) \bigr)\biggr) dB(t) \] where \(B\) is an \(m\)-dimensional Brownian motion.

MSC:

65C30 Numerical solutions to stochastic differential and integral equations
65L20 Stability and convergence of numerical methods for ordinary differential equations
65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations
34F05 Ordinary differential equations and systems with randomness
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60H35 Computational methods for stochastic equations (aspects of stochastic analysis)
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