Stochastic Optimal Control Theory Applied in Finance

摘要

In the field of investment, depending on their structures and in order to make the best decisions that are optimal, some companies are subject to some restrictions on their assets. And generally speaking, these constraints concern assets evolving in uncertainty. This paper focuses on studying a financial continuous-time Merton optimal investment problem in the case where there is a reallocation constraint with regard to the risky asset. Under this constraint, a certain rate is fixed such that the stock asset cannot be liquidated sooner than the rate. It is a stochastic control pure investment case for a large investor who faces a discounted infinite time horizon with utility function of only wealth, subject to a risk aversion coefficient. Our main goal is to characterise an optimal trading strategy for investors expecting high returns for low risks. We propose the dynamic programming method whose value function satisfies a nonlinear partial differential equation. Under homotheticity of the value function, a reduction of dimension is used in order to reduce the original two spatial dimensions problem to one dimension in a bounded domain. Numerical approximations are used to study the dynamic programming by finite difference discretisation and the convergence between the finite and the infinite time horizon problem is presented.