Dual formulation of second order target problems

TL;DR

This paper introduces a new dual formulation for second order stochastic target problems by modifying the reference probability, linking them to backward stochastic differential equations and fully nonlinear PDEs.

Contribution

It presents a novel dual formulation for second order stochastic target problems, extending their connection to backward SDEs and nonlinear PDEs.

Findings

Dual formulation as supremum of backward SDE solutions

Connection to fully nonlinear parabolic PDEs in the Markov case

Stochastic representation for scalar second order parabolic equations

Abstract

This paper provides a new formulation of second order stochastic target problems introduced in [SIAM J. Control Optim. 48 (2009) 2344-2365] by modifying the reference probability so as to allow for different scales. This new ingredient enables us to prove a dual formulation of the target problem as the supremum of the solutions of standard backward stochastic differential equations. In particular, in the Markov case, the dual problem is known to be connected to a fully nonlinear, parabolic partial differential equation and this connection can be viewed as a stochastic representation for all nonlinear, scalar, second order, parabolic equations with a convex Hessian dependence.