MP and DPP for Mean-Variance Portfolio Selection Problem with Poisson Jumps, Recursive Utility and Their Relationship
Qiyue Zhang, Jingtao Shi
TL;DR
The paper tackles mean-variance portfolio optimization under Poisson jumps with recursive utility, casting the problem as an FBSDEP and solving it via both the maximum principle (MP) and dynamic programming principle (DPP). It derives a state-feedback optimal investment strategy and a dynamic efficient frontier, supported by explicit ODEs for auxiliary functions and a demonstration that MP and DPP yield the same control. A comparative analysis with classical frontiers shows how recursive utility and jumps reshape the risk–return landscape. The work also removes the mean-wealth constraint through embedding and provides a clear link between the two fundamental stochastic control approaches.
Abstract
In this paper, the mean-variance portfolio selection problem with Poisson jumps are studied, where the recursive utility is given by the solution to a backward stochastic differential equation with Poisson jumps. Both the maximum principle and dynamic programming principle are applied to solve this problem, and their relationship is also investigated. The optimal portfolio and efficient frontier of Markowitz's type are derived using both methods. A comparison of efficient frontiers obtained in this paper and in the framework without jumps is conducted.