Markov Jump Processes Approximating a Non-Symmetric Generalized Diffusion

Markov Jump Processes Approximating a Non-Symmetric Generalized Diffusion Consider a non-symmetric generalized diffusion X (⋅) in ℝ d determined by the differential operator $A(\mbox{\boldmath{$x$}})=-\sum_{ij}\partial_{i}a_{ij}(\mbox{\boldmath{$x$}})\partial_{j} +\sum_{i} b_{i}(\mbox{\boldmath{$x$}})\partial_{i}$ . In this paper the diffusion process is approximated by Markov jump processes X n (⋅), in homogeneous and isotropic grids G n ⊂ℝ d , which converge in distribution in the Skorokhod space D ((0,∞),ℝ d ) to the diffusion X (⋅). The generators of X n (⋅) are constructed explicitly. Due to the homogeneity and isotropy of grids, the proposed method for d ≥3 can be applied to processes for which the diffusion tensor $\{a_{ij}(\mbox{\boldmath{$x$}})\}_{11}^{dd}$ fulfills an additional condition. The proposed construction offers a simple method for simulation of sample paths of non-symmetric generalized diffusion. Simulations are carried out in terms of jump processes X n (⋅). For piece-wise constant functions a ij on ℝ d and piece-wise continuous functions a ij on ℝ 2 the construction and principal algorithm are described enabling an easy implementation into a computer code. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Applied Mathematics and Optimization Springer Journals

Markov Jump Processes Approximating a Non-Symmetric Generalized Diffusion

, Volume 64 (1) – Aug 1, 2011
33 pages

Publisher
Springer-Verlag
Subject
Mathematics; Calculus of Variations and Optimal Control; Optimization; Mathematical Methods in Physics; Theoretical, Mathematical and Computational Physics; Systems Theory, Control; Numerical and Computational Physics
ISSN
0095-4616
eISSN
1432-0606
D.O.I.
10.1007/s00245-011-9133-1
Publisher site
See Article on Publisher Site

Abstract

Consider a non-symmetric generalized diffusion X (⋅) in ℝ d determined by the differential operator $A(\mbox{\boldmath{$x$}})=-\sum_{ij}\partial_{i}a_{ij}(\mbox{\boldmath{$x$}})\partial_{j} +\sum_{i} b_{i}(\mbox{\boldmath{$x$}})\partial_{i}$ . In this paper the diffusion process is approximated by Markov jump processes X n (⋅), in homogeneous and isotropic grids G n ⊂ℝ d , which converge in distribution in the Skorokhod space D ((0,∞),ℝ d ) to the diffusion X (⋅). The generators of X n (⋅) are constructed explicitly. Due to the homogeneity and isotropy of grids, the proposed method for d ≥3 can be applied to processes for which the diffusion tensor $\{a_{ij}(\mbox{\boldmath{$x$}})\}_{11}^{dd}$ fulfills an additional condition. The proposed construction offers a simple method for simulation of sample paths of non-symmetric generalized diffusion. Simulations are carried out in terms of jump processes X n (⋅). For piece-wise constant functions a ij on ℝ d and piece-wise continuous functions a ij on ℝ 2 the construction and principal algorithm are described enabling an easy implementation into a computer code.

Journal

Applied Mathematics and OptimizationSpringer Journals

Published: Aug 1, 2011

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