arXiv subjects

Łukasz Stettner

Publications and source records attributed to Łukasz Stettner.

At least 19 recordsLinked to original sources

arXiv2026

Policy stability and ultimate stationarity in discounted risk-sensitive stochastic control

We study discrete-time Markov Decision Processes (MDPs) on finite state-action spaces and analyze the stability of optimal policies and value functions in the long-run discounted risk-sensitive objective setting. Our analysis addresses robustness with respect to perturbations of the risk-aversion parameter and the discount factor, the emergence of ultimate stationarity, and the interaction between discounted and averaged formulations under suitable mixing assumptions. We further investigate limiting regimes associated with vanishing discount and vanishing risk sensitivity, and discuss the role of Blackwell-type stability properties in the discounted setting. Finally, we provide numerical illustrations that highlight the intrinsic non-stationarity of optimal discounted risk-sensitive policies.

math.OC↗

arXiv2026

Long-run risk-sensitive portfolio optimisation with proportional transaction costs and log Lévy asset prices

We study a long-run risk-sensitive portfolio problem with proportional transaction costs in a continuous-time market whose log-prices are given as a Lévy process, and rebalancing is possible only at random moments of investment opportunities. Using a Schauder fixed-point argument, we solve the ergodic Bellman equation without any mixing assumptions. Under additional full-support and growth conditions, the solution to the Bellman equation is unique, with the unique continuous maximiser. We also prove vanishing risk-aversion asymptotics towards the risk-neutral (Kelly) problem and convergence of a dyadic-time-grid approximation of the random intervention moments. Numerical examples illustrate the results.

math.OC↗

arXiv2026

Blackwell optimality in risk-sensitive stochastic control

In this paper, we consider a discrete-time Markov Decision Process (MDP) on a finite state-action space with a long-run risk-sensitive criterion used as the objective function. We discuss the concept of Blackwell optimality and comment on intricacies which arise when the risk-neutral expectation is replaced by the risk-sensitive entropy. Also, we show the relation between the Blackwell optimality and ultimate stationarity and provide an illustrative example that helps to better understand the structural difference between these two concepts.

math.OC↗

arXiv2025

Long run control of nonhomogeneous Markov processes

In the paper average reward per unit time and average risk sensitive reward functionals are considered for controlled nonhomogeneous Markov processes. Existence of solutions to suitable Bellman equations is shown. Continuity of the value functions with respect to risk parameter is also proved. Finally stability of functionals with respect to pointwise convergence of Markov controls is studied.

math.OC↗

arXiv2025

Discrete time shadow price revisited

In the paper discrete time shadow price is constructed for the market with several assets with given bid and ask prices. Shadow price is the price such that the problem of optimal utility from terminal wealth on the market without transaction costs gives the same value function as in the case of bid and ask prices. In the paper we solve first static problem for two assets and then we construct the shadow price for dynamic model. Finally we present construction of the shadow price for several assets.

math.OC↗

arXiv2024

Blackwell optimality and policy stability for long-run risk sensitive stochastic control

This paper analyzes the stability of optimal policies in the long-run stochastic control framework with an averaged risk-sensitive criterion for discrete-time MDPs on finite state-action space. In particular, we study the robustness of optimal controls when perturbations to the risk-aversion parameter are applied, and investigate the Blackwell property, together with its link to the risk-sensitive vanishing discount approximation framework. Finally, we present examples that help to better understand the intricacies of the risk-sensitive control framework.

math.OC↗

arXiv2024

Existence of bounded solutions to multiplicative Poisson equations under mixing property

In this paper we study the problem of Multiplicative Poisson Equation (MPE) bounded solution existence in the generic discrete-time setting. Assuming mixing and boundedness of the risk-reward function, we investigate what conditions should be imposed on the underlying non-controlled probability kernel or the reward function in order for the MPE bounded solution to always exists. In particular, we consolidate span-norm framework based results and derive an explicit sharp bound that needs to be imposed on the cost function to guarantee the bounded solution existence under mixing. Also, we study the properties which the probability kernel must satisfy to ensure existence of bounded MPE for any generic risk-reward function and characterise process behaviour in the complement of the invariant measure support. Finally, we present numerous examples and stochastic-dominance based arguments that help to better understand the intricacies that emerge when the ergodic risk-neutral mean operator is replaced with ergodic risk-sensitive entropy.

math.PR↗

arXiv2023

A novel scaling approach for unbiased adjustment of risk estimators

The assessment of risk based on historical data faces many challenges, in particular due to the limited amount of available data, lack of stationarity, and heavy tails. While estimation on a short-term horizon for less extreme percentiles tends to be reasonably accurate, extending it to longer time horizons or extreme percentiles poses significant difficulties. The application of theoretical risk scaling laws to address this issue has been extensively explored in the literature. This paper presents a novel approach to scaling a given risk estimator, ensuring that the estimated capital reserve is robust and conservatively estimates the risk. We develop a simple statistical framework that allows efficient risk scaling and has a direct link to backtesting performance. Our method allows time scaling beyond the conventional square-root-of-time rule, enables risk transfers, such as those involved in economic capital allocation, and could be used for unbiased risk estimation in small sample settings. To demonstrate the effectiveness of our approach, we provide various examples related to the estimation of value-at-risk and expected shortfall together with a short empirical study analysing the impact of our method.

q-fin.RM↗

arXiv2023

Topology-Driven Goodness-of-Fit Tests in Arbitrary Dimensions

This paper adopts a tool from computational topology, the Euler characteristic curve (ECC) of a sample, to perform one- and two-sample goodness of fit tests. We call our procedure TopoTests. The presented tests work for samples of arbitrary dimension, having comparable power to the state-of-the-art tests in the one-dimensional case. It is demonstrated that the type I error of TopoTests can be controlled and their type II error vanishes exponentially with increasing sample size. Extensive numerical simulations of TopoTests are conducted to demonstrate their power for samples of various sizes.

stat.ME↗

arXiv2023

Long-run impulse control with generalised discounting

In this paper, we investigate the effects of applying generalised (non-exponential) discounting on a long-run impulse control problem for a Feller-Markov process. We show that the optimal value of the discounted problem is the same as the optimal value of its undiscounted version. Next, we prove that an optimal strategy for the undiscounted discrete time functional is also optimal for the discrete-time discounted criterion and nearly optimal for the continuous-time discounted one. This shows that the discounted problem, being time-inconsistent in nature, admits a time-consistent solution. Also, instead of a complex time-dependent Bellman equation one may consider its simpler time-independent version.

math.OC↗

arXiv2023

Long run stochastic control problems with general discounting

Controlled discrete time Markov processes are studied first with long run general discounting functional. It is shown that optimal strategies for average reward per unit time problem are also optimal for average generally discounting functional. Then long run risk sensitive reward functional with general discounting is considered. When risk factor is positive then optimal value of such reward functional is dominated by the reward functional corresponding to the long run risk sensitive control. In the case of negative risk factor we get an asymptotical result, which says that optimal average reward per unit time control is nearly optimal for long run risk sensitive reward functional with general discounting, assuming that risk factor is close to $0$. For this purpose we show in Appendix upper estimates for large deviations of weighted empirical measures, which are of independent interest.

math.OC↗

arXiv2023

Discrete time risk sensitive control problem

In the paper adapting Krein Rutman theory we show the existence of solutions to the long run risk sensitive control problem for controlled discrete time Markov processes over locally compact separable metric spaces.

math.OC↗

arXiv2023

Asymptotics of impulse control problem with multiplicative reward

We consider a long-run impulse control problem for a generic Markov process with a multiplicative reward functional. We construct a solution to the associated Bellman equation and provide a verification result. The argument is based on the probabilistic properties of the underlying process combined with the Krein-Rutman theorem applied to the specific non-linear operator. Also, it utilises the approximation of the problem in the bounded domain and with the help of the dyadic time-grid.

math.OC↗

arXiv2022

Discrete-time risk sensitive portfolio optimization with proportional transaction costs

In this paper we consider a discrete-time risk sensitive portfolio optimization over a long time horizon with proportional transaction costs. We show that within the log-return i.i.d. framework the solution to a suitable Bellman equation exists under minimal assumptions and can be used to characterize the optimal strategies for both risk-averse and risk-seeking cases. Moreover, using numerical examples, we show how a Bellman equation analysis can be used to construct or refine optimal trading strategies in the presence of transaction costs.

q-fin.PM↗

arXiv2021

Risk-sensitive optimal stopping with unbounded terminal cost function

In this paper we consider an infinite time horizon risk-sensitive optimal stopping problem for a Feller--Markov process with an unbounded terminal cost function. We show that in the unbounded case an associated Bellman equation may have multiple solutions and we give a probabilistic interpretation for the minimal and the maximal one. Also, we show how to approximate them using finite time horizon problems. The analysis, covering both discrete and continuous time case, is supported with illustrative examples.

math.OC↗

arXiv2020

Long-run risk sensitive impulse control

In this paper we consider long-run risk sensitive average cost impulse control applied to a continuous-time Feller-Markov process. Using the probabilistic approach, we show how to get a solution to a suitable continuous-time Bellman equation and link it with the impulse control problem. The optimal strategy for the underlying problem is constructed as a limit of dyadic impulse strategies by exploiting regularity properties of the linked risk sensitive optimal stopping value functions. In particular, this shows that the discretized setting could be used to approximate near optimal strategies for the underlying continuous time control problem, which facilitates the usage of the standard approximation tools. For completeness, we present examples of processes that could be embedded into our framework.

math.OC↗

arXiv2019

Risk sensitive optimal stopping

In this paper we consider discrete and continuous time risk sensitive optimal stopping problem. Using suitable properties of the underlying Feller-Markov process we prove continuity of the optimal stopping value function and provide formula for the optimal stopping policy. Next, we show how to link continuous time framework with its discrete time analogue. By considering suitable approximations, we obtain uniform convergence of the corresponding value functions.

math.OC↗

arXiv2019

Long-run risk sensitive dyadic impulse control

In this paper long-run risk sensitive optimisation problem is studied with dyadic impulse control applied to continuous-time Feller-Markov process. In contrast to the existing literature, focus is put on unbounded and non-uniformly ergodic case by adapting the weight norm approach. In particular, it is shown how to combine geometric drift with local minorisation property in order to extend local span-contraction approach when the process as well as the linked reward/cost functions are unbounded. For any predefined risk-aversion parameter, the existence of solution to suitable Bellman equation is shown and linked to the underlying stochastic control problem. For completeness, examples of uncontrolled processes that satisfy the geometric drift assumption are provided.

math.OC↗