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

Discounted Risk-Sensitive Optimal Control of Switching Diffusions: Viscosity Solution and Numerical Approximation

“…If we can verify that v * and v * are suband supersolutions of the HJB Equation ( 14), respectively, then the result that follows is associated with the comparison result. In fact, as in [27], we can show that…”

“…However, there is still an open problem. As to the existence of optimal control, in the risk-neutral case [20,27], the occupation measure method is usually used. By introducing the occupation measure method, one can pose the risk-neutral optimal control problem as a convex optimization problem.…”

“…For example, [26] applies the switching diffusions to the ecosystems and let the discrete part of the state represent the random environment. Within the framework of financial applications, the discrete part of the state is usually used to capture the market environment, say bull or bear, see [27].…”

“…To solve such problem, similar to the risk neutral case, see [20,27], we should find suitable characterizations to the value function W(θ, x, α) and the optimal control u * (•). Due to the dynamic program principle, such characterizations are usually given via the associated optimality equation, or the HJB equation.…”

“…We will see that such equation is a second-order partial differential equation. The viscosity solution is one of the commonly used weak solutions for this kind of equation; we recommend [27][28][29] and the reference therein for readers who are not familiar with the concept of viscosity solutions. In particular, the development of viscosity solutions is briefly introduced in reference [28].…”