Volume 3, Issue 2
Convergence and Stability of Implicit Methods for Jump-diffusion Systems

D. J. Higham & P. E. Kloeden

DOI:

Int. J. Numer. Anal. Mod., 3 (2006), pp. 125-140

Published online: 2006-03

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  • Abstract

A class of implicit methods is introduced for Ito stochastic difference equations with Poisson-driven jumps. A convergence proof shows that these implicit methods share the same finite time strong convergence rate as the explicit Euler-Maruyama scheme. A mean-square linear stability analysis shows that implicitness offers benefits, and a natural analogue of mean-square A-stability is studied. Weak variants are also considered and their stability analyzed.

  • Keywords

A-stability backward Euler Euler-Maruyama linear stability Poisson process stochastic differential equation strong convergence theta method trapezoidal rule

  • AMS Subject Headings

65C30 65L20 60H10

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{IJNAM-3-125, author = {D. J. Higham and P. E. Kloeden}, title = {Convergence and Stability of Implicit Methods for Jump-diffusion Systems}, journal = {International Journal of Numerical Analysis and Modeling}, year = {2006}, volume = {3}, number = {2}, pages = {125--140}, abstract = {A class of implicit methods is introduced for Ito stochastic difference equations with Poisson-driven jumps. A convergence proof shows that these implicit methods share the same finite time strong convergence rate as the explicit Euler-Maruyama scheme. A mean-square linear stability analysis shows that implicitness offers benefits, and a natural analogue of mean-square A-stability is studied. Weak variants are also considered and their stability analyzed. }, issn = {2617-8710}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/ijnam/893.html} }
TY - JOUR T1 - Convergence and Stability of Implicit Methods for Jump-diffusion Systems AU - D. J. Higham & P. E. Kloeden JO - International Journal of Numerical Analysis and Modeling VL - 2 SP - 125 EP - 140 PY - 2006 DA - 2006/03 SN - 3 DO - http://dor.org/ UR - https://global-sci.org/intro/article_detail/ijnam/893.html KW - A-stability KW - backward Euler KW - Euler-Maruyama KW - linear stability KW - Poisson process KW - stochastic differential equation KW - strong convergence KW - theta method KW - trapezoidal rule AB - A class of implicit methods is introduced for Ito stochastic difference equations with Poisson-driven jumps. A convergence proof shows that these implicit methods share the same finite time strong convergence rate as the explicit Euler-Maruyama scheme. A mean-square linear stability analysis shows that implicitness offers benefits, and a natural analogue of mean-square A-stability is studied. Weak variants are also considered and their stability analyzed.
D. J. Higham & P. E. Kloeden. (1970). Convergence and Stability of Implicit Methods for Jump-diffusion Systems. International Journal of Numerical Analysis and Modeling. 3 (2). 125-140. doi:
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