We give a short introduction to the stochastic calculus for ItôLévy processes and review briefly the two main methods of optimal control of systems described by such processes: (i) Dynamic programming and the Hamilton-Jacobi-Bellman (HJB) equation (ii) The stochastic maximum principle and its associated backward stochastic differential equation (BSDE). The two methods are illustrated by applica...

In this paper, we consider a joint dynamic pricing and production policy for stochastic inventory system with perishable products. The demand is dependent on the price level of on-hand inventory. Combined control, optimization model that maximizes total discounted profit built. Applying optimal control theory, formulate problem finding schedule as solving Hamilton-Jacobi-Bellman (HJB) equation....

In this paper, a new analytical method to find a near-optimal high gain controller for the non-minimum phase affine nonlinear systems is introduced. This controller is derived based on the closed form solution of the Hamilton-Jacobi-Bellman (HJB) equation associated with the cheap control problem. This methodology employs an algebraic equation with parametric coefficients for the systems with s...

This paper is concerned with Sobolev weak solution of Hamilton-Jacobi-Bellman (HJB) equation. This equation is derived from the dynamic programming principle in the study of the stochastic optimal control problem. Adopting Doob-Meyer decomposition theorem as one of main tool, we prove that the optimal value function is the unique Sobolev weak solution of the corresponding HJB equation. For the ...

We propose multigrid methods for solving the discrete algebraic equations arising from the discretization of the second order Hamilton–Jacobi–Bellman (HJB) and Hamilton– Jacobi–Bellman–Isaacs (HJBI) equations. We propose a damped-relaxation method as a smoother for multigrid. In contrast with the standard policy iteration, the proposed damped-relaxation scheme is convergent for both HJB and HJB...

Portfolio optimization problems on a finite time horizon under proportional transaction costs are considered. The objective is to maximize the expected utility of the terminal wealth. The ensuing non-smooth timedependent Hamilton-Jacobi-Bellman (HJB) equation is solved by regularization and the application of a semi-smooth Newton method. Discretization in space is carried out by finite differen...

In this paper, we consider the optimal investment problem for an insurer who has n dependent lines of business. The surplus process of the insurer is described by a n-dimensional compound Poisson risk process. Moreover, the insurer is allowed to invest in a risk-free asset and a risky asset whose price process follows the constant elasticity of variance (CEV) model. The investment objective is ...

This paper investigates a mean-variance portfolio selection problem in continuous time with fixed and proportional transaction costs. Utilizing the dynamic programming, the Hamilton-Jacobi-Bellman (HJB) equation is derived, and the explicit closed form solution is obtained. Furthermore, the optimal strategies and efficient frontiers are also proposed for the original mean-variance problem. Nume...

This paper is concerned with Sobolev weak solution of Hamilton-Jacobi-Bellman (HJB) equation. This equation is derived from the dynamic programming principle in the study of the stochastic optimal control problem. Adopting Doob-Meyer decomposition theorem as one of main tool, we prove that the optimal value function is the unique Sobolev weak solution of the corresponding HJB equation. For the ...

This paper is concerned with Sobolev weak solution of Hamilton-Jacobi-Bellman (HJB) equation. This equation is derived from the dynamic programming principle in the study of the stochastic optimal control problem. Adopting Doob-Meyer decomposition theorem as one of main tool, we prove that the optimal value function is the unique Sobolev weak solution of the corresponding HJB equation. For the ...