Adaptive Weak Approximation of Diffusions with Jumps

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 erro...

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Bibliographic Details
Published in:SIAM journal on numerical analysis 2008-01, Vol.46 (4), p.1732-1768
Main Authors: Mordecki, E., Szepessy, A., Tempone, R., Zouraris, G. E.
Format: Article
Language:English
Subjects:
a posteriori error estimates
A posteriori knowledge
Accuracy
Algorithms
Applied mathematics
Approximation
backward dual functions
Computational mathematics
Control theory
Determinism
Differential equations
diffusions with jumps
error control
Error rates
Errors in statistics
Estimates
Euler-Maruyama method
Eulers method
Platens
Sampling errors
Scholarships & fellowships
Science
Securities prices
weak approximation
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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 [Comm. Pure Appl. Math., 54 (2001), pp. 1169-1214]. 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.
ISSN:0036-1429
1095-7170
1095-7170
DOI:10.1137/060669632