We consider the infinite horizon risk-sensitive problem for nondegenerate diffusions with a compact action space, and controlled through the drift. We only impose a structural assumption on the running cost function, namely near-monotonicity, and show that there always exists a solution to the risk-sensitive Hamilton–Jacobi–Bellman (HJB) equation, and that any minimizer in the Hamiltonian is op...

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...

In the present paper, we consider nonlinear optimal control problems with constraints on the state of the system. We are interested in the characterization of the value function without any controllability assumption. In the unconstrained case, it is possible to derive a characterization of the value function by means of a Hamilton-Jacobi-Bellman (HJB) equation. This equation expresses the beha...

In many applications (engineering, management, economy) one is led to control problems for stochastic systems : more precisely the state of the system is assumed to be described by the solution of stochastic differential equations and the control enters the coefficients of the equation. Using the dynamic programming principle E. Bellman [6] explained why, at least heuristically, the optimal cos...

In this work we deal with the numerical solution of a Hamilton-Jacobi-Bellman (HJB) equation with infinitely many solutions. To compute the maximal solution – the optimal cost of the original optimal control problem – we present a complete discrete method based on the use of some finite elements and penalization techniques. Mathematics Subject Classification. 49L20, 49L99, 93C15, 65L70. Receive...

We determine the optimal robust investment strategy of an individual who targets at a given rate of consumption and seeks to minimize the probability of lifetime ruin when she does not have perfect confidence in the drift of the risky asset. Using stochastic control, we characterize the value function as the unique classical solution of an associated Hamilton-Jacobi-Bellman (HJB) equation, obta...

In this paper, the piecewise homotopy perturbation method (PHPM) is employed to solve the Hamilton– Jacobi–Bellman (HJB) equation arising in the optimal control problems. The method is a simple modification of the standard homotopy perturbation method (HPM), in which it is treated as an algorithm in a sequence of small intervals (i.e. time step) for finding accurate approximate solutions to the...

1 We present efficient partial differential equation (PDE) methods for continuous time mean2 variance portfolio allocation problems when the underlying risky asset follows a stochastic 3 volatility process. The standard formulation for mean variance optimal portfolio allocation 4 problems gives rise to a two-dimensional non-linear Hamilton-Jacobi-Bellman (HJB) PDE. We 5 use a wide stencil metho...

In this paper, an iterative algorithm to solve a special class of Hamilton-JacobiBellman-Isaacs (HJBI) equations is proposed. By constructing two series of nonnegative functions, we replace the problem of solving an HJBI equation by the problem of solving a sequence of Hamilton-Jacobi-Bellman (HJB) equations whose solutions can be approximated recursively by existing methods. The local converge...

An optimal control law for a general nonlinear system can be obtained by solving Hamilton-Jacobi-Bellman equation. However, it is difficult to obtain an analytical solution of this equation even for a moderately complex system. In this paper, we propose a continuoustime single network adaptive critic scheme for nonlinear control affine systems where the optimal cost-to-go function is approximat...