Euler approximation and stability of the solution to stochastic differential equations with jumps under pathwise uniqueness

Nasroallah, Kaoutar; Ouknine, Youssef

doi:10.1080/17442508.2022.2071107

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“…In the case of presence of jumps, under standard assumptions, classical and jump-adapted Itô-Taylor approximations and Runge-Kutta methods are studied, e.g., in [11,12,44,2]. Approximation results for jump-diffusion SDEs under non-standard assumptions can be found, e.g., in [18,19,20,7,10,9,56,5,29,4,24,36]. Asymptotically optimal approximation rates are proven in [46,23,47,48,52,22].…”

Section: Introductionmentioning

confidence: 99%

A higher order approximation method for jump-diffusion SDEs with discontinuous drift coefficient

Przybyłowicz¹,

Schwarz²,

Szölgyenyi³

2022

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We present the first higher-order approximation scheme for solutions of jump-diffusion stochastic differential equations with discontinuous drift. For this transformation-based jump-adapted quasi-Milstein scheme we prove L p -convergence order 3/4. To obtain this result, we prove that under slightly stronger assumptions (but still weaker than anything known before) a related jump-adapted quasi-Milstein scheme has convergence order 3/4 -in a special case even order 1. Order 3/4 is conjectured to be optimal.

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