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Pathwise convergence rate for numerical solutions of stochastic differential equations. (English) Zbl 1246.65018

By an approach involving embedding in a new probability space, this paper derives the rate of pathwise weak convergence of the weak Euler-Maruyama scheme \[ x^\varepsilon_{n+1}= x^\varepsilon_n+\varepsilon f(x^\varepsilon_n)+ \sqrt{\varepsilon} \sigma(x^\varepsilon_n) \xi_{n+1},\quad x^\varepsilon_0= x_0 \] for approximating the solution of the stochastic differential equation \[ dX(t)= f(X(t))\,dt+ \sigma(X(t))\,dB(t),\quad X(0)= x+0, \] where \(B(t)\) is a standard Brownian motion.

MSC:

65C30 Numerical solutions to stochastic differential and integral equations
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60H35 Computational methods for stochastic equations (aspects of stochastic analysis)
34F05 Ordinary differential equations and systems with randomness
65L05 Numerical methods for initial value problems involving ordinary differential equations
65L20 Stability and convergence of numerical methods for ordinary differential equations
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