A New Class of Backward Stochastic Partial Differential Equations with Jumps and Applications

TL;DR

This paper introduces a new class of high-order vector backward stochastic partial differential equations with jumps, providing methods for proving existence and uniqueness of solutions, and demonstrating applications in finance.

Contribution

It formulates a novel class of B-SPDEs with jumps involving high-order operators and develops a proof technique for their solutions, extending the theory beyond conventional backward stochastic equations.

Findings

Established existence and uniqueness of solutions under Lipschitz and linear growth conditions.

Addressed differentiability challenges due to differential order inconsistencies.

Illustrated applications in financial modeling with nonlinear operators.

Abstract

We formulate a new class of stochastic partial differential equations (SPDEs), named high-order vector backward SPDEs (B-SPDEs) with jumps, which allow the high-order integral-partial differential operators into both drift and diffusion coefficients. Under certain type of Lipschitz and linear growth conditions, we develop a method to prove the existence and uniqueness of adapted solution to these B-SPDEs with jumps. Comparing with the existing discussions on conventional backward stochastic (ordinary) differential equations (BSDEs), we need to handle the differentiability of adapted triplet solution to the B-SPDEs with jumps, which is a subtle part in justifying our main results due to the inconsistency of differential orders on two sides of the B-SPDEs and the partial differential operator appeared in the diffusion coefficient. In addition, we also address the issue about the B-SPDEs under certain Markovian random environment and employ a B-SPDE with strongly nonlinear partial differential operator in the drift coefficient to illustrate the usage of our main results in finance.