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 (λ(a, ·))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 Rq . We do not assume that the family (λ(a, ·))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.

Backward SDE representation for stochastic control problems with nondominated controlled intensity / S. Choukroun, A. Cosso. - In: THE ANNALS OF APPLIED PROBABILITY. - ISSN 1050-5164. - 26:2(2016 Apr), pp. 1208-1259. [10.1214/15-AAP1115]

Backward SDE representation for stochastic control problems with nondominated controlled intensity

A. Cosso
Ultimo
2016

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 (λ(a, ·))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 Rq . We do not assume that the family (λ(a, ·))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.
BSDE with jumps; Conditionally poisson random measure; Constrained BSDE; Controlled intensity; Hamilton-jacobi-bellman equation; Nonlinear integro-PDE; Viscosity solution; Statistics and Probability; Statistics; Probability and Uncertainty
Settore MAT/06 - Probabilita' e Statistica Matematica
apr-2016
http://projecteuclid.org/download/pdfview_1/euclid.aoap/1458651831
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2434/931976
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