Milstein scheme for stochastic differential equation with Markovian switching and Lévy noise

First, we establish the Itô's formula for stochastic differential equations with Markovian switching and Lévy noise which is then used to derive a first-order scheme (Milstein scheme). The moment stability of the Milstein scheme is investigated and its rate of convergence is proved to be 1.0 wh...

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Bibliographic Details
Published in:Journal of mathematical analysis and applications 2024-08, Vol.536 (1), p.128175, Article 128175
Main Authors: Vashistha, Divyanshu, Kumar, Chaman
Format: Article
Language:English
Subjects:
Itô's formula
Milstein scheme
Rate of convergence
SDEs with Markovian switching and Lévy noise
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Summary:First, we establish the Itô's formula for stochastic differential equations with Markovian switching and Lévy noise which is then used to derive a first-order scheme (Milstein scheme). The moment stability of the Milstein scheme is investigated and its rate of convergence is proved to be 1.0 when the coefficients and their derivatives are Lipschitz continuous. The intertwining of the cádlág (solution) process with the discontinuous dynamics of the underlying Markov chain gives rise to several challenges and new techniques are developed to deal with them. We also provide a discussion on the practical implementation of the proposed Milstein scheme under the diffusion and jump commutativity conditions.
ISSN:0022-247X
1096-0813
DOI:10.1016/j.jmaa.2024.128175