Stochastic control for a class of nonlinear kernels and applications

TL;DR

This paper develops a stochastic control framework for nonlinear kernels via BSDEs, establishing a dynamic programming principle, and applies it to second order BSDEs, super-hedging duality, and path-dependent PDEs.

Contribution

It introduces a dynamic programming principle for control of nonlinear kernels and applies it to solve problems in second order BSDEs, super-hedging, and PPDEs.

Findings

Proved a dynamic programming principle for nonlinear kernel control.

Established well-posedness for second order BSDEs without regularity assumptions.

Derived a super-hedging duality in uncertain financial markets.

Abstract

We consider a stochastic control problem for a class of nonlinear kernels. More precisely, our problem of interest consists in the optimisation, over a set of possibly non-dominated probability measures, of solutions of backward stochastic differential equations (BSDEs). Since BSDEs are nonlinear generalisations of the traditional (linear) expectations, this problem can be understood as stochastic control of a family of nonlinear expectations, or equivalently of nonlinear kernels. Our first main contribution is to prove a dynamic programming principle for this control problem in an abstract setting, which we then use to provide a semi-martingale characterisation of the value function. We next explore several applications of our results. We first obtain a wellposedness result for second order BSDEs (as introduced in [86]) which does not require any regularity assumption on the terminal condition and the generator. Then we prove a nonlinear optional decomposition in a robust setting, extending recent results of [71], which we then use to obtain a super-hedging duality in uncertain, incomplete and nonlinear financial markets. Finally, we relate, under additional regularity assumptions, the value function to a viscosity solution of an appropriate path-dependent partial differential equation (PPDE).