Optimal stochastic impulse control problem with delay with actions decided at the execution time

TL;DR

This paper studies a stochastic impulse control problem with a fixed delay between decision and execution, introducing a model where impulse sizes are decided at execution time, applicable to real-world problems like swing option pricing.

Contribution

It develops a new framework for impulse control with delay, allowing impulse sizes to be decided at execution, and proves the existence of optimal strategies using reflected BSDEs and Snell envelopes.

Findings

Existence of optimal strategies for finite and infinite horizons.

Extension of impulse control models to include decision timing at execution.

Application to real-world problems like swing options pricing.

Abstract

In this paper, we consider a class of stochastic impulse control problem when there is a fixed delay $\Delta$ between the decision and execution times. The dynamics of the controlled system between two impulses is an arbitrary adapted stochastic process. Unlike the most existing literature, we consider the problem when the impulse sizes are decided at the execution time in both risk-neutral and risk-sensitive cases. This model fits more, in the real life, for some problems such as the pricing of swing options. The horizon T of the problem can be finite or infinite. In each case we show the existence of an optimal strategy. The main tools we use are the notions of reflected Backward Stochastic Differential Equations (BSDEs for short) and the Snell envelope of processes.