arXiv subjects
Publications and source records attributed to Lukasz Stettner.
In the paper we study continuous time controlled Markov processes using discrete time controlled Markov processes. We consider long run functionals: average reward per unit time or long run risk sensitive functional. We also investigate stability of continuous time functionals with respect to pointwise convergence of Markov controls.
In the paper we study dependence of long run functionals and limit characteristics assuming that Borel measurable Markov controls converge pointwise. We consider two kinds of functionals: average cost per unit time and long run risk sensitive. We impose uniform ergodicity assumption, which is later is relaxed and suitable convergence of controlled transition probabilities.
In this paper we consider impulse control of continuous time Markov processes with average cost per unit time functional. This problem is approximated using impulse control problems stopped at the first exit time from increasing sequence of open sets. We find solution to Bellman equation corresponding to the original problem and show that stopped impulse control problems approximate optimal value of the cost functional.
This paper studies maximisation of an average-cost-per-unit-time ergodic functional over impulse strategies controlling a Feller-Markov process. The uncontrolled process is assumed to be ergodic but, unlike the extant literature, the convergence to invariant measure does not have to be uniformly geometric in total variation norm; in particular, we allow for non-uniform geometric or polynomial convergence. Cost of an impulse may be unbounded, e.g., proportional to the distance the process is shifted. We show that the optimal value does not depend on the initial state and provide optimal or $\ve$-optimal strategies.
We study optimal stopping of Feller-Markov processes to maximise an undiscounted functional consisting of running and terminal rewards. In a finite-time horizon setting, we extend classical results to unbounded rewards. In infinite horizon, we resort to ergodic structure of the underlying process. When the running reward is mildly penalising for delaying stopping (i.e., its expectation under the invariant measure is negative), we show that an optimal stopping time exists and is given in a standard form as the time of first entrance to a closed set. This paper generalises Palczewski, Stettner (2014), Stoch Proc Appl 124(12) 3887-3920, by relaxing boundedness of rewards.
We study an infinite horizon optimal stopping problem which arises naturally in the optimal timing of a firm/project sale or in the valuation of natural resources: the functional to be maximised is a sum of a discounted running reward and a discounted final reward. The running and final rewards as well as the instantaneous interest rate (used for calculating discount factors) depend on a Feller-Markov process modelling the underlying randomness in the world, such as prices of stocks, prices of natural resources (gas, oil, etc.), or factors influencing the interest rate. For years, it had seemed sensible to assume that interest rates were uniformly separated from 0, which is needed for the existing theory to work. However, recent developments in Japan and in Europe showed that interest rates can get arbitrarily close to 0. In this paper we establish the feasibility of the stopping problem, prove the existence of optimal stopping times and a variational characterisation (in the viscosity sense) of the value function when interest rates are NOT uniformly separated from 0. Our results rely on certain ergodic properties of the underlying (non-uniformly) ergodic Markov process. We provide several criteria for diffusions and jump-diffusions.
We explore properties of the value function and existence of optimal stopping times for functionals with discontinuities related to the boundary of an open (possibly unbounded) set $\mathcal{O}$. The stopping horizon is either random, equal to the first exit from the set $\mathcal{O}$, or fixed: finite or infinite. The payoff function is continuous with a possible jump at the boundary of $\mathcal{O}$. Using a generalization of the penalty method we derive a numerical algorithm for approximation of the value function for general Feller-Markov processes and show existence of optimal or $ε$-optimal stopping times.
This paper studies a portfolio optimization problem in a discrete-time Markovian model of a financial market, in which asset price dynamics depend on an external process of economic factors. There are transaction costs with a structure that covers, in particular, the case of fixed plus proportional costs. We prove that there exists a self-financing trading strategy maximizing the average growth rate of the portfolio wealth. We show that this strategy has a Markovian form. Our result is obtained by large deviations estimates on empirical measures of the price process and by a generalization of the vanishing discount method to discontinuous transition operators.
We consider a discrete-time financial market model with finite time horizon and give conditions which guarantee the existence of an optimal strategy for the problem of maximizing expected terminal utility. Equivalent martingale measures are constructed using optimal strategies.