[Paper Review] Second Order BSDEs with Jumps: Existence and probabilistic representation for fully-nonlinear PIDEs

M. Nabil Kazi-Tani, Dylan Possamaï+1 more|arXiv (Cornell University)|Aug 3, 2012

Stochastic processes and financial applications33 references3 citations

TL;DR

This paper establishes the existence and probabilistic representation of second-order backward stochastic differential equations with jumps (2BSDEJs), providing a complete well-posedness theory through a direct method using regular conditional probability distributions. It proves that solutions to 2BSDEJs correspond to viscosity solutions of fully nonlinear parabolic partial integro-differential equations (PIDEs), extending the nonlinear Feynman-Kac formula to jump-diffusion settings with model uncertainty.

ABSTRACT

In this paper, we pursue the study of second order BSDEs with jumps (2BSDEJs for short) started in our accompanying paper [15]. 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.

Motivation & Objective

• To establish a complete well-posedness theory for second-order backward stochastic differential equations with jumps (2BSDEJs) by proving existence via a direct method.
• To extend the nonlinear Feynman-Kac formula to the jump-diffusion setting, connecting 2BSDEJs to fully nonlinear parabolic partial integro-differential equations (PIDEs).
• To provide a probabilistic interpretation of viscosity solutions to fully nonlinear PIDEs under model uncertainty, particularly in the presence of jumps.
• To generalize the 2BSDE framework from continuous diffusions to jump processes, incorporating both Brownian motion and Poisson random measures.
• To bridge the gap between stochastic control, nonlinear PDEs, and financial mathematics by constructing a unified probabilistic approach for non-Markovian, non-linear PIDEs with jumps.

Proposed method

• Uses a direct existence proof for 2BSDEJs based on regular conditional probability distributions, avoiding reliance on approximation or duality methods.
• Defines 2BSDEJs on a filtered probability space generated by a Brownian motion and a Poisson random measure, with the generator $ g $ depending on the solution and its controls.
• Applies the comparison theorem for standard BSDEs with jumps to derive a priori estimates and stability results.
• Employs a partitioning technique on the state space using countable covers of balls to construct approximating probability measures and verify the dynamic programming principle.
• Utilizes the essential supremum representation of 2BSDEJs over a class of non-dominated probability measures to link to viscosity solutions of PIDEs.
• Applies stability and comparison results for BSDEs with jumps to handle lower semicontinuity and limiting arguments for stopping times.

Experimental results

Research questions

• RQ1 Can a complete well-posedness theory for second-order BSDEs with jumps be established via a direct method, without relying on duality or approximation?
• RQ2 Do solutions to 2BSDEJs provide a probabilistic representation for viscosity solutions of fully nonlinear PIDEs involving jump components?
• RQ3 How does the inclusion of jumps in the underlying process affect the representation of solutions to fully nonlinear PDEs in the context of model uncertainty?
• RQ4 Can the nonlinear Feynman-Kac formula be extended to the case of jump-diffusion processes with non-Lipschitz generators?
• RQ5 What is the role of regular conditional probability distributions in constructing solutions to 2BSDEJs under non-dominated probability measures?

Key findings

• The paper establishes the existence of solutions to 2BSDEJs through a direct method, providing a complete well-posedness theory for this class of equations.
• Solutions to 2BSDEJs are shown to be viscosity solutions of the associated fully nonlinear PIDEs, extending the classical nonlinear Feynman-Kac formula to the jump-diffusion setting.
• The probabilistic representation holds under model uncertainty, with the solution defined as the essential supremum over a class of non-dominated probability measures.
• The method relies on constructing approximating probability measures via partitioning of the state space and using comparison theorems for BSDEs with jumps to derive stability and convergence.
• Lower semicontinuity of the value function $ u $ in $ (t,x) $ from the right in $ t $ is sufficient to extend the representation to general stopping times via limiting arguments.
• The results generalize prior work on 2BSDEs in the continuous case and provide a rigorous foundation for probabilistic methods in fully nonlinear PIDEs with jumps.