First Name: Shen
Department Dept of Applied Finance and Actuarial Studies
Supervisor(s): Doctor Tak Kuen (Ken) Siu , Doctor Xian Zhou
A Stochastic Maximum Principle for Mean-Field Models with Jumps and its Application to Finance
The aims of this paper are to establish necessary and sufficient stochastic maximum principles for optimal control of a jump-diffusion mean-field system and to apply the principles to discuss an important problem in mathematical finance, namely, the mean-variance portfolio selection problem.
The stochastic maximum principles established are new to the optimal control literature. Using our newly established principles, we can solve the mean-variance problem directly without embedding it into an auxiliary problem as in the traditional approach.
We first prove the existence and uniqueness of the solution to a mean-field backward stochastic differential equation by a version of the contractive mapping theorem. Then we establish necessary and sufficient stochastic maximum principles for a mean-field model with Poisson random jumps using Benssousan (1982)’s convex perturbation method.
An application of the necessary stochastic maximum principle to a bi-criteria mean -variance portfolio selection problem is discussed.
Findings include the existence and uniqueness result for a jump-diffusion mean-field BSDE under the Lipschitz condition and the necessary and sufficient maximum principles under the convexity assumption. We also apply our results to the mean-variance problem, whose solution coincides with those obtained in the existing literature.
Mean-field model; Backward stochastic differential equations; Poisson jumps; Stochastic maximum principles; Mean-variance portfolio selection.