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Volume 42, Issue 1
Strong Convergence of Jump-Adapted Implicit Milstein Method for a Class of Nonlinear Jump-Diffusion Problems

Xu Yang & Weidong Zhao

DOI: 10.4208/jcm.2206-m2021-0354

J. Comp. Math., 42 (2024), pp. 248-270.

Published online: 2023-12

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  • Abstract

In this paper, we study the strong convergence of a jump-adapted implicit Milstein method for a class of jump-diffusion stochastic differential equations with non-globally Lipschitz drift coefficients. Compared with the regular methods, the jump-adapted methods can significantly reduce the complexity of higher order methods, which makes them easily implementable for scenario simulation. However, due to the fact that jump-adapted time discretization is path dependent and the stepsize is not uniform, this makes the numerical analysis of jump-adapted methods much more involved, especially in the non-globally Lipschitz setting. We provide a rigorous strong convergence analysis of the considered jump-adapted implicit Milstein method by developing some novel analysis techniques and optimal rate with order one is also successfully recovered. Numerical experiments are carried out to verify the theoretical findings.

  • Keywords

Jump-diffusion, Jump-adapted implicit Milstein method, Poisson jumps, Strong convergence rate, Non-Lipschitz coefficients.

  • AMS Subject Headings

60H35, 65C30, 65L20

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{JCM-42-248, author = {Yang , Xu and Zhao , Weidong}, title = {Strong Convergence of Jump-Adapted Implicit Milstein Method for a Class of Nonlinear Jump-Diffusion Problems}, journal = {Journal of Computational Mathematics}, year = {2023}, volume = {42}, number = {1}, pages = {248--270}, abstract = {

In this paper, we study the strong convergence of a jump-adapted implicit Milstein method for a class of jump-diffusion stochastic differential equations with non-globally Lipschitz drift coefficients. Compared with the regular methods, the jump-adapted methods can significantly reduce the complexity of higher order methods, which makes them easily implementable for scenario simulation. However, due to the fact that jump-adapted time discretization is path dependent and the stepsize is not uniform, this makes the numerical analysis of jump-adapted methods much more involved, especially in the non-globally Lipschitz setting. We provide a rigorous strong convergence analysis of the considered jump-adapted implicit Milstein method by developing some novel analysis techniques and optimal rate with order one is also successfully recovered. Numerical experiments are carried out to verify the theoretical findings.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.2206-m2021-0354}, url = {http://global-sci.org/intro/article_detail/jcm/22159.html} }
TY - JOUR T1 - Strong Convergence of Jump-Adapted Implicit Milstein Method for a Class of Nonlinear Jump-Diffusion Problems AU - Yang , Xu AU - Zhao , Weidong JO - Journal of Computational Mathematics VL - 1 SP - 248 EP - 270 PY - 2023 DA - 2023/12 SN - 42 DO - http://doi.org/10.4208/jcm.2206-m2021-0354 UR - https://global-sci.org/intro/article_detail/jcm/22159.html KW - Jump-diffusion, Jump-adapted implicit Milstein method, Poisson jumps, Strong convergence rate, Non-Lipschitz coefficients. AB -

In this paper, we study the strong convergence of a jump-adapted implicit Milstein method for a class of jump-diffusion stochastic differential equations with non-globally Lipschitz drift coefficients. Compared with the regular methods, the jump-adapted methods can significantly reduce the complexity of higher order methods, which makes them easily implementable for scenario simulation. However, due to the fact that jump-adapted time discretization is path dependent and the stepsize is not uniform, this makes the numerical analysis of jump-adapted methods much more involved, especially in the non-globally Lipschitz setting. We provide a rigorous strong convergence analysis of the considered jump-adapted implicit Milstein method by developing some novel analysis techniques and optimal rate with order one is also successfully recovered. Numerical experiments are carried out to verify the theoretical findings.

Xu Yang & Weidong Zhao. (2023). Strong Convergence of Jump-Adapted Implicit Milstein Method for a Class of Nonlinear Jump-Diffusion Problems. Journal of Computational Mathematics. 42 (1). 248-270. doi:10.4208/jcm.2206-m2021-0354
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