Affiliation:
1. Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China
2. School of Management, Shandong University, Jinan 250100, China
Abstract
In the theory of portfolio selection, there are few methods that effectively address the combined challenge of insider information and model uncertainty, despite numerous methods proposed for each individually. This paper studies the problem of the robust optimal investment for an insider under model uncertainty. To address this, we extend the Itô formula for forward integrals by Malliavin calculus, and use it to establish an implicit anticipating stochastic differential game model for the robust optimal investment. Since traditional stochastic control theory proves inadequate for solving anticipating control problems, we introduce a new approach. First, we employ the variational method to convert the original problem into a nonanticipative stochastic differential game problem. Then we use the stochastic maximum principle to derive the Hamiltonian system governing the robust optimal investment. In cases where the insider information filtration is of the initial enlargement type, we derive the closed-form expression for the investment by using the white noise theory when the insider is ’small’. When the insider is ’large’, we articulate a quadratic backward stochastic differential equation characterization of the investment. We present the numerical result and conduct an economic analysis of the optimal strategy across various scenarios.
Subject
General Mathematics,Engineering (miscellaneous),Computer Science (miscellaneous)
Reference51 articles.
1. Lifetime portfolio selection under uncertainty: The continuous-time case;Merton;Rev. Econ. Stat.,1969
2. Optimum consumption and portfolio rules in a continuous-time model;Merton;J. Econ. Theory,1971
3. Karatzas, I., and Shreve, S.E. (1991). Brownian Motion and Stochastic Calculus, Springer.
4. Yong, J.M., and Zhou, X.Y. (1999). Stochastic Controls: Hamiltonian Systems and HJB Equations, Springer.
5. Karatzas, I., and Shreve, S.E. (1998). Methods of Mathematical Finance, Springer.