Strong convergence of split-step backward Euler method for stochastic differential equations with non-smooth drift.

*(English)*Zbl 1246.65010Under less restrictive assumptions on the drift coefficient \(f\) than is customary, the split-step backward Euler method is shown to converge strongly with order \(1/2\) to the solution of the Ito stochastic differential equation
\[
dX(t)= f(t,X(t))\,dt+ g(t,X(r))\,dW(t),\quad 0\leq t\leq T,\quad X(0)= X_0.
\]
Numerical results are presented that verify that this accuracy is achieved for three examples. Also under even less restrictive assumptions on \(f\) , order \(1/4\) strong convergence to the solution is proved.

Reviewer: Melvin D. Lax (Long Beach)

##### 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 |

65L20 | Stability and convergence of numerical methods for ordinary differential equations |

##### Keywords:

stochastic differential equations; non-smooth drift; split-step backward Euler method; Euler; Maruyama method; one-sided Lipschitz condition; convergence; Ito stochastic differential equation; numerical results
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\textit{A. F. Bastani} and \textit{M. Tahmasebi}, J. Comput. Appl. Math. 236, No. 7, 1903--1918 (2012; Zbl 1246.65010)

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