Lukasz Stettner

Risk-Sensitive Control of Discrete-Time Markov Processes with Infinite Horizon

In this paper we study existence of solutions to the Bellman equation corresponding to risk-sensitive ergodic control of discrete-time Markov processes using three different approaches. Also, for particular classes of systems, asymptotics for vanishing risk factor is investigated, showing that in the limit the optimal value for an average cost per unit time is obtained.

INFINITE HORIZON RISK SENSITIVE CONTROL OF DISCRETE TIME MARKOV PROCESSES UNDER MINORIZATION PROPERTY

Risk sensitive control of Markov processes satisfying the minorization property is studied using splitting techniques. Existence of solutions to the multiplicative Poisson equation is shown. Approximation by uniformly ergodic controlled Markov processes is introduced, which allows us to show the existence of solutions to the infinite horizon risk sensitive Bellman equation.

Risk sensitive portfolio optimization

Abstract. In the paper discrete time portfolio selection with maximization of the risk sensitized growth rate with and without transaction costs is considered.

Option Pricing in Discrete‐Time Incomplete Market Models

Various aspects of pricing of contingent claims in discrete time for incomplete market models are studied. Formulas for prices with proportional transaction costs are obtained. Some results concerning pricing with concave transaction costs are shown. Pricing by the expected utility of terminal wealth is also considered.

On the compactness method in general ergodic impulsive control of markov processes

In this paper the impulsive control of Feller Markov processes on compact state space with long run average cost criterion is studied. Under the assumption of compactness of the resolvent operator the optimal strategies corresponding to general cost for impulses are constructed. Also the case of purely jump Markov processes is considered

Almost self-optimizing strategies for the adaptive control of diffusion processes

The ergodic control of a multidimensional diffusion process described by a stochastic differential equation that has some unknown parameters appearing in the drift is investigated. The invariant measure of the diffusion process is shown to be a continuous function of the unknown parameters. For the optimal ergodic cost for the known system, an almost optimal adaptive control is constructed for the unknown system.

Penalty Method for Finite Horizon Stopping Problems

In this paper we use a penalty method to approximate a number of stopping problems over a finite horizon. In particular we prove existence and continuity of the value function corresponding to Dynkin games over a finite horizon. Since stopping problems can be studied in the context of Dynkin games, as a by-product we obtain continuity of the optimal stopping value. We then study Dynkin games with delayed stopping and finally impulse control. In each case the value function is approximated by a solution to a suitable penalty equation.

Risk sensitive control of discrete time partially observed markov processes with infinite horizon

In this paper existence of solutions to the Bellman equation corresponding to risk sensitive control of partially observed discrete time Markov processes is shown; this in turn leads to the existence of optimal strategies. The method used in the paper is based on discounted risk sensitive approximation

Parameter continuity of the ergodic cost for a growth optimal portfolio with proportional transaction costs

Some results are given for a continuous time long run growth optimal portfolio that has proportional costs consisting of the sum of a fixed proportional cost and a cost that is proportional to the volume of each transaction. An obligatory portfolio diversification is given that requires at least a small portion of the wealth be invested in each asset. It is assumed that the price of each asset is obtained from a Levy noise stochastic equation whose coefficients depend on an unknown parameter from a compact set. It is shown that the optimal cost is a continuous function of the unknown parameter.

On the poisson equation and optimal stopping of ergodic markov processes

Optimal stopping problems for Feller-Markov processes when the discount factor α→0 , and α=0 are studied under the assumption that the Markov semigroup is quasicompact. An ergodic structure of Markov process and solution of a so-called Poisson equation is found.