Stochastic control for a class of nonlinear kernels and applications
Possamaï, Dylan; Tan, Xiaolu; Zhou, Chao (2018), Stochastic control for a class of nonlinear kernels and applications, Annals of Probability, 46, 1, p. 551-603. 10.1214/17-AOP1191
TypeArticle accepté pour publication ou publié
Journal nameAnnals of Probability
Institute of Mathematical Statistics
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Abstract (EN)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 ) 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 , 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).
Subjects / KeywordsStochastic control; measurable selection; non-linear kernels; second order; BSDEs; path-dependent PDEs; robust superhedging
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