Well-posedness of a system of SDEs driven by jump random measures

TL;DR

This paper proves the well-posedness of a class of stochastic differential equations with jumps, non-Lipschitz coefficients, and interdependent components, using comparison theorems and approximation techniques.

Contribution

It introduces a novel approach to establish existence, uniqueness, and comparison properties for complex SDE systems with jumps and non-Lipschitz coefficients.

Findings

Established pathwise uniqueness for the SDE systems.

Constructed non-negative, integrable solutions via monotone limits.

Extended the framework to include time-inhomogeneous drifts and applications in finance.

Abstract

We establish well-posedness for a class of systems of SDEs with non-Lipschitz coefficients in the diffusion and jump terms and with two sources of interdependence: a monotone function of all the components in the drift of each SDE and the correlation between the driving Brownian motions and jump random measures. Pathwise uniqueness is derived by employing some standard techniques. Then, we use a comparison theorem along with our uniqueness result to construct non-negative, $L^1$-integrable c\`adl\`ag solutions as monotone limits of solutions to approximating SDEs, allowing for time-inhomogeneous drift terms to be included. Our approach allows also for a comparison property to be established for the solutions to the systems we investigate. The applicability of certain systems in financial modeling is also discussed.