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Lower error bounds and optimality of approximation for jump-diffusion SDEs with discontinuous drift.

Paweł Przybyłowicz¹, Verena Schwarz², Michaela Szölgyenyi²

¹Faculty of Applied Mathematics, AGH University of Krakow, Al. Mickiewicza 30, 30-059 Krakow, Poland.

BIT. Numerical Mathematics

|September 12, 2024

Summary

Sharp lower error bounds were proven for numerical methods approximating jump-diffusion stochastic differential equations (SDEs) with discontinuous drift. The findings demonstrate the optimality of specific jump-adapted schemes for these complex SDEs.

Keywords:

Discontinuous drift Jump-adapted scheme Jump-diffusion stochastic differential equations Lower bounds Optimality of approximation schemes

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Area of Science:

Numerical Analysis
Stochastic Processes
Computational Mathematics

Background:

Stochastic differential equations (SDEs) are crucial for modeling systems with inherent randomness.
Jump-diffusion SDEs incorporate sudden, discontinuous changes, posing significant challenges for numerical approximation.
Discontinuous drift terms further complicate the accurate numerical solution of these equations.

Purpose of the Study:

To establish sharp lower error bounds for numerical methods applied to jump-diffusion SDEs with discontinuous drift.
To analyze the performance of both non-adaptive and jump-adapted approximation schemes.
To determine the theoretical limits of accuracy for these numerical methods.

Main Methods:

Derivation of analytical lower error bounds using mathematical analysis.
Investigation of approximation schemes, including non-adaptive and jump-adapted methods.
Comparative analysis of error bounds across different approximation strategies.

Main Results:

Sharp lower error bounds of order 3/4 are proven for numerical approximations of jump-diffusion SDEs with discontinuous drift.
These bounds apply to both non-adaptive and jump-adapted approximation schemes.
The results confirm the optimality of the transformation-based jump-adapted quasi-Milstein scheme.

Conclusions:

The established lower error bounds provide a fundamental understanding of the best possible accuracy for numerical methods in this domain.
The optimality of the transformation-based jump-adapted quasi-Milstein scheme is theoretically validated.
These findings guide the development and selection of efficient numerical techniques for jump-diffusion SDEs.