Remarks on Risk-Sensitive Control Problems

The main purpose of this paper is to investigate the asymptotic behavior of the discounted risk-sensitive control problem for periodic diffusion processes when the discount factor α goes to zero. If uα(θ, x) denotes the optimal cost function, being the risk factor, then it is shown that limα→0αuα(θ, x) = ξ(θ) where ξ(θ) is the average on ]0, θ[ of the optimal cost of the (usual) in nite horizon risk-sensitive control problem

On the Optimal Reward Function of the Continuous Time Multiarmed Bandit Problem

The optimal reward function associated with the so-called multiarmed bandit problem for general Markov-Feller processes is considered. It is shown that this optimal reward function has a simple expression (product form) in terms of individual stopping problems, without any smoothness properties of the optimal reward function neither for the global problem nor for the individual stopping problems. Some results relative to a related problem with switching cost are obtained

Green and Poisson Functions with Wentzell Boundary Conditions

We discuss the construction and estimates of the Green and Poisson functions associated with a parabolic second order integro-di erential operator with Wentzell boundary conditions

Reflected Diffusion Processes with Jumps

A stochastic differential equation of Wiener-Poisson type is considered in a d-dimensional bounded region. By using a penalization argument on the domain, we are able to prove the existence and uniqueness of solutions in the strong sense. The main assumptions are Lipschitzian coefficients, either convex or smooth domains and a regular outward reflecting direction. As a direct consequence, it is verified that the reflected diffusion process with jumps depends on the initial date in a Lipschitz fashion

Generalized Lame-Clapeyron Solution for a One-Phase Source Stefan Problem

In this paper we obtain a generalized Lamé-Clapeyron solution for a one-phase Stefan problem with a particular type of sources. Necessary and sufficient conditions are given in order to characterize the source term which provides a unique solution. Some estimates on the free boundary and the temperature are presented. In particular, asymptotic expansions are given for small Stefan number and source

Optimal Control of Stochastic Integrals and Hamilton-Jacobi-Bellman Equations, II

We consider the solution of a stochastic integral control problem, and we study its regularity. In particular, we characterize the optimal cost as the maximum solution of ∀v ∈ V, A(v)u ≤ ƒ(v) in D\u27(Ο), u = 0 on ∂Ο, u ∈ W1,∞(Ο), where A(v) is a uniformly elliptic second order operator and V is the set of the values of the control

On an Investment-Consumption Model with Transaction Costs

This paper considers the optimal consumption and investment policy for an investor who has available one bank account paying a fixed interest rate and n risky assets whose prices are log-normal diffusions. We suppose that transactions between the assets incur a cost proportional to the size of the transaction. The problem is to maximize the total utility of consumption. Dynamic programming leads to a variational inequality for the value function. Existence and uniqueness of a viscosity solution are proved. The variational inequality is solved by using a numerical algorithm based on policies, iterations, and multigrid methods. Numerical results are displayed for n = 1 and n = 2

On the Impulse Control of Jump Diffusions

Regularity of the impulse control problem for a nondegenerate n-dimensional jump diffusion with infinite activity and finite variation jumps was recently examined in [M. H. A. Davis, X. Guo, and G. Wu, SIAM J. Control Optim., 48 (2010), pp. 5276–5293]. Here we extend the analysis to include infinite activity and infinite variation jumps. More specifically, we show that the value function u of the impulse control problem satisfies u ∈ Wloc2,p(Rn)