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 pro- gramming 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 addi- tional regularity assumptions, the value function to a viscosity solution of an appropriate path-dependent partial differential equation (PPDE).
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).
R
optimization
optimisation
R
non-linear generalizations
nonlinear generalisations
R
pro- gramming
programming
R
semimartingale characterization
semi-martingale characterisation
R
76
86
R
non-linear
nonlinear
R
63
71
R
superhedging
super-hedging
R
non-linear
nonlinear
R
addi- tional
additional