Stochastic Calculus with Jumps Processes : Theory and Numerical Techniques

TL;DR

This paper studies stochastic differential equations with jumps, proving solution existence, and introduces new numerical schemes with strong convergence and stability analysis, supported by numerical experiments.

Contribution

It introduces the compensated tamed Euler and semi-tamed Euler schemes for SDEs with jumps, proving their strong convergence under non-global Lipschitz conditions.

Findings

CSTM converges strongly with order 0.5 under Lipschitz conditions.

Proposed schemes are stable and effective for SDEs with jumps.

Numerical experiments confirm theoretical convergence and stability results.

Abstract

In this work we consider a stochastic differential equation (SDEs) with jump. We prove the existence and the uniqueness of solution of this equation in the strong sense under global Lipschitz condition. Generally, exact solutions of SDEs are unknowns. The challenge is to approach them numerically. There exist several numerical techniques. In this thesis, we present the compensated stochastic theta method (CSTM) which is already developed in the literature. We prove that under global Lipschitz condition, the CSTM converges strongly with standard order 0.5. We also investigated the stability behaviour of both CSTM and stochastic theta method (STM). Inspired by the tamed Euler scheme developed in [8], we propose a new scheme for SDEs with jumps called compensated tamed Euler scheme. We prove that under non-global Lipschitz condition the compensated tamed Euler scheme converges strongly with standard order 0.5. Inspired by [11], we propose the semi-tamed Euler for SDEs with jumps under non-global Lipschitz condition and prove its strong convergence of order 0.5. This latter result is helpful to prove the strong convergence of the tamed Euler scheme. We analyse the stability behaviours of both tamed and semi-tamed Euler scheme We present also some numerical experiments to illustrate our theoretical results.