The Annals of Applied Probability

Backward SDE representation for stochastic control problems with nondominated controlled intensity

Sébastien Choukroun and Andrea Cosso

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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 problem 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 $\mathbb{R}^{q}$. 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 a nonlinear Feynman–Kac formula, for the value function associated with these control problems. For this reason, we introduce a class of backward stochastic differential equations with jumps and a 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.

Article information

Source
Ann. Appl. Probab., Volume 26, Number 2 (2016), 1208-1259.

Dates
Received: May 2014
Revised: January 2015
First available in Project Euclid: 22 March 2016

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1458651831

Digital Object Identifier
doi:10.1214/15-AAP1115

Mathematical Reviews number (MathSciNet)
MR3476636

Zentralblatt MATH identifier
1339.60085

Subjects
Primary: 60H10: Stochastic ordinary differential equations [See also 34F05] 93E20: Optimal stochastic control
Secondary: 60G57: Random measures

Keywords
BSDE with jumps constrained BSDE controlled intensity conditionally Poisson random measure Hamilton–Jacobi–Bellman equation nonlinear integro-PDE viscosity solution

Citation

Choukroun, Sébastien; Cosso, Andrea. Backward SDE representation for stochastic control problems with nondominated controlled intensity. Ann. Appl. Probab. 26 (2016), no. 2, 1208--1259. doi:10.1214/15-AAP1115. https://projecteuclid.org/euclid.aoap/1458651831


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