A Simple Integral Equation Approach for Optimal Investment Stopping Problems with Partial Information

In this paper, we study a finite horizon optimal investment stopping problem with an unobservable random variable for the return of a risky asset. Using the Bayesian filter and the dual control approach, we transform the original primal problem into a dual finite horizon optimal stopping problem, which results in the dual value function satisfying a variational inequality with two state variables. For a class of utility functions that includes power utility and non–hyperbolic absolute risk aversion utility, we show that the free boundary satisfies a Volterra-type nonlinear integral equation with expectation over the joint distribution of the dual state process and the filtered probability process, and we simplify and solve the integral equation with the dimension reduction and backward recursive methods. We also construct two simple closed-form approximations for the free boundary using its asymptotic properties and show their accuracy and efficiency with numerical examples. Furthermore, we demonstrate that different model parameters may lead to one, two, or no free boundaries with a simple example.

Funding: J. Xing was supported by the National Natural Science Foundation of China [Grant 12101151] and the Youth Foundation of Guizhou University of Finance and Economics [Grant 2022KYQN10]. J. Ma was supported by the National Natural Science Foundation of China [Grant 12071373]. H. Zheng was supported by the Engineering and Physical Sciences Research Council of the United Kingdom [Grant EP/V008331/1].