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Adaptive weak approximation of diffusions with jumps. (English) Zbl 1169.65302

Summary: This work develops adaptive time stepping algorithms for the approximation of a functional of a diffusion with jumps based on a jump augmented Monte Carlo Euler-Maruyama method, which achieve a prescribed precision. The main result is the derivation of new expansions for the time discretization error, with computable leading order term in a posteriori form, which are based on stochastic flows and discrete dual backward functions. Combined with proper estimation of the statistical error, they lead to efficient and accurate computation of global error estimates, extending the results by A. Szepessy, R. Tempone and G. E. Zouraris [Commun. Pure Appl. Math. 54, No. 10, 1169–1214 (2001; Zbl 1024.60028)]. Adaptive algorithms for either deterministic or trajectory-dependent time stepping are proposed. Numerical examples show the performance of the proposed error approximations and the adaptive schemes.

MSC:

65C30 Numerical solutions to stochastic differential and integral equations
65Y20 Complexity and performance of numerical algorithms
65L50 Mesh generation, refinement, and adaptive methods for ordinary differential equations
60H35 Computational methods for stochastic equations (aspects of stochastic analysis)
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)

Citations:

Zbl 1024.60028
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