Abstract
This paper considers a mean-field type stochastic control problem where the dynamics is governed by a forward and backward stochastic differential equation (SDE) driven by Levy processes and the information available to the controller is possibly less than the overall information. All the system coefficients and the objective performance functional are allowed to be random, possibly non-Markovian. Malliavin calculus is employed to derive a maximum principle for the optimal control of such a system where the adjoint process is explicitly expressed.
Highlights
IntroductionIn contrast to the stochastic control problem (e.g. [1] [2]) which is studied in the complete information case (and [1] with the Brownian motion case only), the performance functional that we will investigate involves the mean of functionals of the state variables ( the name mean-field)
In contrast to the stochastic control problem (e.g. [1] [2]) which is studied in the complete information case, the performance functional that we will investigate involves the mean of functionals of the state variables
This paper considers a mean-field type stochastic control problem where the dynamics is governed by a forward and backward stochastic differential equation (SDE) driven by Lévy processes and the information available to the controller is possibly less than the overall information
Summary
In contrast to the stochastic control problem (e.g. [1] [2]) which is studied in the complete information case (and [1] with the Brownian motion case only), the performance functional that we will investigate involves the mean of functionals of the state variables ( the name mean-field). [1] [2]) which is studied in the complete information case (and [1] with the Brownian motion case only), the performance functional that we will investigate involves the mean of functionals of the state variables ( the name mean-field) Problems of this type occur in many applications; for example in a continuous-time Markowitz’s mean-variance portfolio selection model where the variance term involves a quadratic function of the expectation. Since we allow the coefficients ( b,σ ,γ , g, f and h2 as follows) to be the stochastic processes and because our control must be partial information adapted, this problem is not of Markovian type and cannot be solved by dynamic programming even if the mean term were not present. [7] presents various versions of the maximum principle for optimal control (not mean-field type) of forward-backward stochastic differential equations with jumps and a Malliavin calculus approach which allow us to handle non-Markovian system.
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