A solution technique for Lévy driven long term average impulse control problems

Christensen, Sören; Sohr, Tobias

doi:10.1016/j.spa.2020.07.016

Cited by 10 publications

(6 citation statements)

References 26 publications

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“…The results of the present article have applications in optimal control problems based on MAPs, see e.g. the recent article [19] for the more particular case of a Lévy driven impulse control problem. There, the generator of the ascending ladder height process is decisive for determining optimal threshold times of a desired reflection strategy.…”

Section: Introduction 1background and Aims Of The Articlementioning

confidence: 83%

Stability of overshoots of Markov additive processes

Döring¹,

Lukas²

2021

Preprint

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We prove precise stability results for overshoots of Markov additive processes (MAPs) with finite modulating space. Our approach is based on the Markovian nature of overshoots of MAPs whose mixing and ergodic properties are investigated in terms of the characteristics of the MAP. On our way we extend fluctuation theory of MAPs, contributing among others to the understanding of the Wiener-Hopf factorization for MAPs by generalizing Vigon's équations amicales inversés known for Lévy processes. Using the Lamperti transformation the results can be applied to self-similar Markov processes. Among many possible applications, we study the mixing behavior of stable processes sampled at first hitting times as a concrete example.

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Section: Introduction 1background and Aims Of The Articlementioning

confidence: 83%

Stability of overshoots of Markov additive processes

Döring¹,

Lukas²

2021

Preprint

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“…for some given set B Ď S. We remark that this class of problems naturally arises in problems of optimal control that are based on stopping, see -in a continuous time framework -e.g. Christensen and Sohr [2020].…”

Section: ˜8 źmentioning

confidence: 99%

Flexible forward improvement iteration for infinite time horizon Markovian optimal stopping problems

Christensen¹,

Irle²,

Lemburg³

2021

Preprint

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In this paper, we propose an extension of the forward improvement iteration algorithm, originally introduced in Irle [2006] and recently reconsidered in Miclo and Villeneuve [2021]. The main new ingredient is a flexible window parameter describing the look-ahead distance in the improvement step. We consider the framework of a Markovian optimal stopping problem in discrete time with random discounting and infinite time horizon. We prove convergence and show that the additional flexibility may significantly reduce the runtime.

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“…In most cases, solving an ergodic control problem is based on a dynamic programming approach that reduces a problem to finding a proper solution to an optimality equation called Hamilton-Jacobi-Bellman (HJB) equation. HJB equations are nonlinear and sometimes non-local differential equations that are solvable analytically only in the limited cases [11][12][13][14][15] and must be handled numerically in general. Therefore, the core of this approach is to find or approximate solutions to the HJB equations.…”

Section: Introductionmentioning

confidence: 99%

Analytical and numerical solutions to ergodic control problems arising in environmental management

Yoshioka

Tsujimura

Yaegashi

2022

Math Methods in App Sciences

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Environmental management optimizing a long-run objective is an ergodic control problem whose resolution can be achieved by solving an associated nonlocal Hamilton-Jacobi-Bellman (HJB) equation having an effective Hamiltonian. Focusing on sediment storage management as a modern engineering problem, we formulate, analyze, and compute a new ergodic control problem under discrete observations: a simple but non-trivial mathematical problem. We give optimality and comparison results of the corresponding HJB equation having unique non-smoothness and discontinuity. To numerically compute HJB equations, we propose a new fast-sweep method resorting to neither pseudo-time integration nor vanishing discount. The optimal policy and the effective Hamiltonian are then computed simultaneously. Convergence rate of numerical solutions is computationally analyzed. An advanced robust control counterpart where the dynamics involve uncertainties is also numerically considered.

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A solution technique for Lévy driven long term average impulse control problems

Cited by 10 publications

References 26 publications

Stability of overshoots of Markov additive processes

Stability of overshoots of Markov additive processes

Flexible forward improvement iteration for infinite time horizon Markovian optimal stopping problems

Analytical and numerical solutions to ergodic control problems arising in environmental management

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