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Markov Jump Processes Approximating a Non-Symmetric Generalized Diffusion

Applied Mathematics & Optimization volume 64, pages 101–133 (2011)Cite this article

Abstract

Consider a non-symmetric generalized diffusion X(⋅) in ℝd 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 ⊂ℝd, which converge in distribution in the Skorokhod space D([0,∞),ℝd) 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 ℝd and piece-wise continuous functions a ij on ℝ2 the construction and principal algorithm are described enabling an easy implementation into a computer code.

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  1. Dept. of Mathematics, University of Zagreb, Bijenička 30, 10002, Zagreb, Croatia

    Nedžad Limić

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  1. Nedžad Limić
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Correspondence to Nedžad Limić.

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Communicating Editor: P. Del Moral.

Supported by grant 0037014 of the Ministry of Science, Higher Education and Sports, Croatia.

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Limić, N. Markov Jump Processes Approximating a Non-Symmetric Generalized Diffusion. Appl Math Optim 64, 101–133 (2011). https://doi.org/10.1007/s00245-011-9133-1

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  • DOI: https://doi.org/10.1007/s00245-011-9133-1

Keywords

  • Symmetric diffusion
  • Approximation of diffusion
  • Simulation of diffusion
  • Divergence form operators