Stochastic control for a class of nonlinear kernels and applications

Dylan Possamaï (1) , Xiaolu Tan (1) , Chao Zhou (2)

1 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
2 NUS - National University of Singapore

We consider a stochastic control problem for a class of nonlinear kernels. More precisely, our problem of interest consists in the optimization, over a set of possibly non-dominated probability measures, of solutions of backward stochastic differential equations (BSDEs). Since BSDEs are non-linear generalizations 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 semimartingale characterization 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 [76]) which does not require any regularity assumption on the terminal condition and the generator. Then we prove a non-linear optional decomposition in a robust setting, extending recent results of [63], which we then use to obtain a superhedging duality in uncertain, incomplete and non-linear 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).

Mots clés: Stochastic control; measurable selection; non-linear kernels; second order BSDEs; path-dependent PDEs; robust superhedging AMS 2000 subject classifications:

Domaines: Probabilités [math.PR]

Dylan Possamaï, Xiaolu Tan, Chao Zhou. Stochastic control for a class of nonlinear kernels and applications. Annals of Probability, 2018, 46 (1), pp.551-603. ⟨10.1214/17-AOP1191⟩. ⟨hal-01247006⟩