Second-order BSDEs with jumps: Formulation and uniqueness

TL;DR

This paper introduces a new formulation of second-order backward stochastic differential equations with jumps (2BSDEJs), extending previous continuous models by incorporating measure-dependent jump compensators, and proves the uniqueness of solutions.

Contribution

It generalizes existing 2BSDE frameworks to include jumps with measure-dependent compensators, providing a universal solution concept and a representation that guarantees uniqueness.

Findings

Defined a new notion of 2BSDEJs with measure-dependent compensators

Established a universal solution framework for 2BSDEJs

Proved the representation and uniqueness of solutions

Abstract

In this paper, we define a notion of second-order backward stochastic differential equations with jumps (2BSDEJs for short), which generalizes the continuous case considered by Soner, Touzi and Zhang [Probab. Theory Related Fields 153 (2012) 149-190]. However, on the contrary to their formulation, where they can define pathwise the density of quadratic variation of the canonical process, in our setting, the compensator of the jump measure associated to the jumps of the canonical process, which is the counterpart of the density in the continuous case, depends on the underlying probability measures. Then in our formulation of 2BSDEJs, the generator of the 2BSDEJs depends also on the underlying probability measures through the compensator. But the solution to the 2BSDEJs can still be defined universally. Moreover, we obtain a representation of the $Y$ component of a solution of a 2BSDEJ as a supremum of solutions of standard backward SDEs with jumps, which ensures the uniqueness of the solution.