Infinite horizon impulse control problem with jumps and continuous switching costs

Monique Pontier,Hani Abidi,Rim Amami

doi:10.1108/ajms-10-2020-0088

Abstract

PurposeThe purpose of this paper is to show the existence results for adapted solutions of infinite horizon doubly reflected backward stochastic differential equations with jumps. These results are applied to get the existence of an optimal impulse control strategy for an infinite horizon impulse control problem.Design/methodology/approachThe main methods used to achieve the objectives of this paper are the properties of the Snell envelope which reduce the problem of impulse control to the existence of a pair of right continuous left limited processes. Some numerical results are provided to show the main results.FindingsIn this paper, the authors found the existence of a couple of processes via the notion of doubly reflected backward stochastic differential equation to prove the existence of an optimal strategy which maximizes the expected profit of a firm in an infinite horizon problem with jumps.Originality/valueIn this paper, the authors found new tools in stochastic analysis. They extend to the infinite horizon case the results of doubly reflected backward stochastic differential equations with jumps. Then the authors prove the existence of processes using Envelope Snell to find an optimal strategy of our control problem.

Highlights

The main motivation of this paper is to prove the existence of an optimal strategy which maximizes the expected profit of a firm in an infinite horizon problem with jumps
Let a Brownian motion ðWtÞt≥0 and an independent Poisson measure μðdt; deÞ defined on a probability space ðΩ; A; PÞ and let F be the right continuous complete filtration generated by the pair ðW ; μÞ
The existence of ðY 1; Y 2Þ is established via the notion of doubly reflected backward stochastic differential equation. Another interest of our work is to extend to the infinite horizon case the results of doubly reflected backward stochastic differential equations with jumps

Summary

Introduction

The main motivation of this paper is to prove the existence of an optimal strategy which maximizes the expected profit of a firm in an infinite horizon problem with jumps. 4. Reflected BSDE with jumps and infinite horizon the results from [10] are extended to infinite horizon reflected backward stochastic differential equations with general jumps, showing existence and uniqueness of an infinite horizon solution, imposing additional assumptions on the drift function and using appropriate estimates of the process Y. Since Y S; Y T ∈ C2; Z S; Z T ∈ H2 and V S; V T ∈ L2; the third line in (16) is a martingale; taking the expectation of both sides with α 1⁄4 2C yields for any t ≤ T

ZsT ds þ

Double barrier reflected BSDE with jumps and infinite horizon

Application to the impulse control problem with infinite horizon