Constrained optimality for controlled switching diffusions with an application to stock purchasing

This work studies the optimal control of switching diffusions with single constraint. The underlying criterion consists of an expected discounted reward function to be maximized and a discounted cost function as the constraint. Firstly, to solve this constrained problem, the original constrained problem should be converted to the unconstrained one, by introducing the Lagrange multiplier. Then it has been shown that the value function to the unconstrained problem is the unique viscosity solution to the optimality equation, also known as the Hamilton–Jacobi–Bellman equation. A verification theorem is also obtained under suitable conditions. Then, the relationship between the optimality results of the original problem and that of the unconstrained problem can be established, by finding the appropriate Lagrange multiplier. Finally, the optimality results obtained have been applied to characterize the stock purchasing problem. Which is formulated as a constrained optimal purchasing (or control) problem, on behalf of the vendee. The purpose is to investigate optimal purchase strategies and give quantitative reference information for stock purchase. Based on the specific model, a two loop approximation scheme is provided to approximate the optimal value function and the optimal control.