Abstract
In this paper, we pursue the study of second order BSDEs with jumps (2BSDEJs for short) started in an accompanying paper. We prove existence of these equations by a direct method, thus providing complete wellposedness for 2BSDEJs. These equations are a natural candidate for the probabilistic interpretation of some fully non-linear partial integro-differential equations, which is the point of the second part of this work. We prove a non-linear Feynman-Kac formula and show that solutions to 2BSDEJs provide viscosity solutions of the associated PIDEs.
Highlights
Motivated by duality methods and maximum principles for optimal stochastic control, Bismut studied in [6] a linear backward stochastic differential equation (BSDE)
Given a filtered probability space (Ω, F, {Ft}0≤t≤T, P) generated by an Rd-valued Brownian motion B, solving a BSDE with generator g, and terminal condition ξ consists in finding a pair of progressively measurable processes (Y, Z) such that
In the case of a filtered probability space generated by both a Brownian motion B and a Poisson random measure μ with compensator ν, the martingale representation for (EP [ξ|Ft])t≥0 becomes t t
Summary
Motivated by duality methods and maximum principles for optimal stochastic control, Bismut studied in [6] a linear backward stochastic differential equation (BSDE) In their seminal paper [21], Pardoux and Peng generalized such equations to the non-linear Lipschitz case and proved existence and uniqueness results in a Brownian framework. Soner, Touzi and Zhang proved in [24] that Markovian 2BSDEs, are connected in the continuous case to a class of parabolic fully non-linear PDEs. On the other hand, we know that solutions to standard Markovian BSDEJs provide viscosity solutions to some parabolic partial integro-differential equations whose non-local operator is given by a quantity similar to v, ν defined in (3.2) (see [2] for more details). The Appendix is dedicated to the proof of some important technical results needed throughout the paper
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