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Volume 16, Issue 6
Strong Convergence and Stability of the Semi-Tamed and Tamed Euler Schemes for Stochastic Differential Equations with Jumps under Non-Global Lipschitz Condition

Antoine Tambue & Jean Daniel Mukam

Int. J. Numer. Anal. Mod., 16 (2019), pp. 847-872.

Published online: 2019-08

[An open-access article; the PDF is free to any online user.]

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  • Abstract

We consider the explicit numerical approximations of stochastic differential equations (SDEs) driven by Brownian process and Poisson jump. It is well known that under non-global Lipschitz condition, Euler Explicit method fails to converge strongly to the exact solution of such SDEs without jumps, while implicit Euler method converges but requires much computational efforts. We investigate the strong convergence, the linear and nonlinear exponential stabilities of tamed Euler and semi-tamed methods for stochastic differential equation driven by Brownian process and Poisson jumps, both in compensated and non compensated forms. We prove that under non-global Lipschitz condition and superlinearly growing drift term, these schemes converge strongly with the standard one-half order. Numerical simulations to sustain the theoretical results are provided.

  • Keywords

Stochastic differential equation, strong convergence, linear stability, exponential stability, jump processes, one-sided Lipschitz.

  • AMS Subject Headings

65X

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

antoine.tambue@hvl.no (Antoine Tambue)

jean.d.mukam@aims-senegal.org (Jean Daniel Mukam)

  • BibTex
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  • TXT
@Article{IJNAM-16-847, author = {Tambue , Antoine and Mukam , Jean Daniel}, title = {Strong Convergence and Stability of the Semi-Tamed and Tamed Euler Schemes for Stochastic Differential Equations with Jumps under Non-Global Lipschitz Condition}, journal = {International Journal of Numerical Analysis and Modeling}, year = {2019}, volume = {16}, number = {6}, pages = {847--872}, abstract = {

We consider the explicit numerical approximations of stochastic differential equations (SDEs) driven by Brownian process and Poisson jump. It is well known that under non-global Lipschitz condition, Euler Explicit method fails to converge strongly to the exact solution of such SDEs without jumps, while implicit Euler method converges but requires much computational efforts. We investigate the strong convergence, the linear and nonlinear exponential stabilities of tamed Euler and semi-tamed methods for stochastic differential equation driven by Brownian process and Poisson jumps, both in compensated and non compensated forms. We prove that under non-global Lipschitz condition and superlinearly growing drift term, these schemes converge strongly with the standard one-half order. Numerical simulations to sustain the theoretical results are provided.

}, issn = {2617-8710}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/ijnam/13257.html} }
TY - JOUR T1 - Strong Convergence and Stability of the Semi-Tamed and Tamed Euler Schemes for Stochastic Differential Equations with Jumps under Non-Global Lipschitz Condition AU - Tambue , Antoine AU - Mukam , Jean Daniel JO - International Journal of Numerical Analysis and Modeling VL - 6 SP - 847 EP - 872 PY - 2019 DA - 2019/08 SN - 16 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/ijnam/13257.html KW - Stochastic differential equation, strong convergence, linear stability, exponential stability, jump processes, one-sided Lipschitz. AB -

We consider the explicit numerical approximations of stochastic differential equations (SDEs) driven by Brownian process and Poisson jump. It is well known that under non-global Lipschitz condition, Euler Explicit method fails to converge strongly to the exact solution of such SDEs without jumps, while implicit Euler method converges but requires much computational efforts. We investigate the strong convergence, the linear and nonlinear exponential stabilities of tamed Euler and semi-tamed methods for stochastic differential equation driven by Brownian process and Poisson jumps, both in compensated and non compensated forms. We prove that under non-global Lipschitz condition and superlinearly growing drift term, these schemes converge strongly with the standard one-half order. Numerical simulations to sustain the theoretical results are provided.

Tambue , Antoine and Mukam , Jean Daniel. (2019). Strong Convergence and Stability of the Semi-Tamed and Tamed Euler Schemes for Stochastic Differential Equations with Jumps under Non-Global Lipschitz Condition. International Journal of Numerical Analysis and Modeling. 16 (6). 847-872. doi:
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