Daniel Hernández-Hernández

An optimal consumption model with stochastic volatility

Abstract. We consider an optimal consumption and investment model in continuous time, which is an extension of the original Mertons problem. In the proposed model, the asset prices are affected by correlated economic factors, modelled as diffusion processes. Writing the value function in a special form, it can be seen that another optimal control problem is involved and studying its associated HJB equation smoothness properties of the original value function can be derived as well as optimal policies.

Risk-Sensitive Control of Finite State Machines on an Infinite Horizon II

In this paper we consider robust and risk-sensitive control of discrete time finite state systems on an infinite horizon. The solution of the state feedback robust control problem is characterized in terms of the value of an average cost dynamic game. The risk-sensitive stochastic optimal control problem is solved using the policy iteration algorithm, and the optimal rate is expressed in terms of the value of a stochastic dynamic game with average cost per unit time criterion. By taking a small noise limit, a deterministic dynamic game which is closely related to the robust control problem is obtained.

Optimal Consumption-Investment Problems in Incomplete Markets with Stochastic Coefficients

The goal of this paper is to solve an optimal consumption-investment problem in the context of an incomplete financial market. The model is a generalization of the Black and Scholes diffusion model, where the coefficients of the diffusion modelling the stocks price depend on some stochastic economic factors. Based on the martingale approach, a basic methodology to get the optimal solution is presented. Combining this procedure with stochastic control techniques, explicit solutions for HARA and logarithmic utility functions are obtained.

A characterization of the optimal risk-sensitive average cost in finite controlled Markov chains

This work concerns controlled Markov chains with finite state and action spaces. The transition law satisfies the simultaneous Doeblin condition, and the performance of a control policy is measured by the (long-run) risk-sensitive average cost criterion associated to a positive, but otherwise arbitrary, risk sensitivity coefficient. Within this context, the optimal risk-sensitive average cost is characterized via a minimization problem in a finite-dimensional Euclidean space.

Discounted Approximations for Risk-Sensitive Average Criteria in Markov Decision Chains with Finite State Space

This work concerns Markov decision processes with finite state space and compact action set. The performance of a control policy is measured by a risk-sensitive average cost criterion and, under standard continuity-compactness conditions, it is shown that the discounted approximations converge to the optimal value function, and that the superior and inferior limit average criteria have the same optimal value function. These conclusions are obtained for every nonnull risk-sensitivity coefficient, and regardless of the communication structure induced by the transition law.

An optimal investment strategy with maximal risk aversion and its ruin probability

In this paper we study an optimal investment problem of an insurer when the company has the opportunity to invest in a risky asset using stochastic control techniques. A closed form solution is given when the risk preferences are exponential as well as an estimate of the ruin probability when the optimal strategy is used.

Risk sensitive control of finite state machines on an infinite horizon. I

Robust and risk sensitive control of discrete time, finite state systems on an infinite horizon is considered, with either state or output feedback. The solution of the state feedback robust control problem is characterized in terms of the value of an average cost dynamic game. The risk sensitive stochastic optimal control problem is solved using the policy iteration algorithm, and the optimal rate is expressed in terms of the value of a stochastic dynamic game with average cost per unit time criterion. By taking a small noise limit a deterministic dynamic game is obtained, which is closely related to the robust control problem. For the problem with output feedback (partial state information) the analysis depends on introducing appropriate information states for the risk sensitive and robust control problems.

Optimality of Refraction Strategies for Spectrally Negative Lévy Processes

We revisit a stochastic control problem of optimally modifying the underlying spectrally negative Levy process. A strategy must be absolutely continuous with respect to the Lebesgue measure, and the objective is to minimize the total costs of the running and controlling costs. Under the assumption that the running cost function is convex, we show the optimality of a refraction strategy. We also obtain the convergence of the optimal refraction strategies and the value functions, as the control set is enlarged, to those in the relaxed case without the absolutely continuous assumption. Numerical results are further given to confirm these analytical results.

Solution to the risk-sensitive average optimality equation in communicating Markov decision chains with finite state space: An alternative approach

Abstract. This note concerns Markov decision chains with finite state and action sets. The decision maker is assumed to be risk-averse with constant risk sensitive coefficient λ, and the performance of a control policy is measured by the risk-sensitive average cost criterion. In their seminal paper Howard and Matheson established that, when the whole state space is a communicating class under the action of each stationary policy, then there exists a solution to the optimality equation for every λ>0. This paper presents an alternative proof of this fundamental result, which explicitly highlights the essential role of the communication properties in the analysis of the risk-sensitive average cost criterion.

Robust maximization of consumption with logarithmic utility

We analyze the stochastic control approach to the dynamic maximization of the robust utility of consumption and investment. The robust utility functionals are defined in terms of logarithmic utility and a dynamically consistent convex risk measure. The underlying market is modeled by a diffusion process whose coefficients are driven by an external stochastic factor process. Our main results give conditions on the minimal penalty function of the robust utility functional under which the value function of our problem can be identified with the unique classical solution of a quasilinear PDE within a class of functions satisfying certain growth conditions.