Long-Run Risk Sensitive Dyadic Impulse Control

Abstract

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.

Risk sensitive optimal stopping

“…Also, it should be noted that optimal stopping problems might be linked to the optimal impulse control framework. In particular, we refer to [17,8,15] and [4,10,13] for risk-sensitive and risk-neutral impulse control framework discussion, respectively.…”

“…In fact, in the companion paper [9], we show how to exploit this link and use results from this paper in order to prove the existence of a solution to the continuous time impulse control Bellman equation using its discrete time dyadic approximations. This shows how to apply probabilistic approach and dyadic framework from [15] in the continuous time setting. Note that in [16] a similar link is established for the classical risk-neutral case.…”

Long-run risk sensitive impulse control

“…Expectation E (x,V ) is defined on a probability space corresponding to the controlled process; see Section 2 for details. We refer to [13,16,19] where similar framework has been studied.…”

“…The main aim of this paper is to find optimal control that minimise (1.1). The optimal strategy is constructed as a limit of dyadic impulse strategies by exploiting results from [16] and [6], i.e. by combining discrete time existence results that are based on the span-contraction approach with regularity properties of risk sensitive optimal stopping value functions.…”

“…This paper is organised as follows: In Section 2 we establish the framework and the linked notation that will be used throughout the paper. Section 3 discusses the dyadic case; this section incorporates the results from [16]. In Section 4 we consider finite time horizon version of the problem.…”