Backward SDE Representation for Stochastic Control Problems with Non Dominated Controlled Intensity

TL;DR

This paper develops a backward stochastic differential equation (BSDE) framework with jumps to represent complex stochastic control problems involving non-dominated jump intensities and degenerate diffusions, extending the nonlinear Feynman-Kac formula.

Contribution

It introduces a novel class of BSDEs with jumps and partial constraints to represent control problems with non-dominated intensities, providing existence, uniqueness, and a viscosity solution link.

Findings

Established existence and uniqueness of minimal BSDE solutions.

Proved the minimal BSDE solution corresponds to the viscosity solution of the PDE.

Extended the nonlinear Feynman-Kac formula to non-dominated jump intensities.

Abstract

We are interested in stochastic control problems coming from mathematical finance and, in particular, related to model uncertainty, where the uncertainty affects both volatility and intensity. This kind of stochastic control problems is associated to a fully nonlinear integro-partial differential equation, which has the peculiarity that the measure $(\lambda(a,\cdot))_a$ characterizing the jump part is not fixed but depends on a parameter $a$ which lives in a compact set $A$ of some Euclidean space $\R^q$. We do not assume that the family $(\lambda(a,\cdot))_a$ is dominated. Moreover, the diffusive part can be degenerate. Our aim is to give a BSDE representation, known as nonlinear Feynman-Kac formula, for the value function associated to these control problems. For this reason, we introduce a class of backward stochastic differential equations with jumps and partially constrained diffusive part. We look for the minimal solution to this family of BSDEs, for which we prove uniqueness and existence by means of a penalization argument. We then show that the minimal solution to our BSDE provides the unique viscosity solution to our fully nonlinear integro-partial differential equation.