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

Said Hamadène, Ibtissam Hdhiri

TL;DR

This paper studies stochastic impulse control with a fixed execution delay $Δ$, where impulse sizes are chosen at execution time. It develops finite- and infinite-horizon frameworks for both risk-neutral and risk-sensitive utilities, proving the existence of optimal strategies and providing explicit iterative constructions. The core methodology combines reflected backward stochastic differential equations and Snell envelopes to encode the delay and post-delay impulse opportunities, yielding implementable optimal policies. The work also connects to swing options by modeling decisions that are prepared at one time but executed after a delay, offering a rigorous blueprint for valuation and hedging in delayed impulse-control problems.

Abstract

In this paper, we consider a class of stochastic impulse control problem when there is a fixed delay $Δ$ 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.

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

TL;DR

This paper studies stochastic impulse control with a fixed execution delay , where impulse sizes are chosen at execution time. It develops finite- and infinite-horizon frameworks for both risk-neutral and risk-sensitive utilities, proving the existence of optimal strategies and providing explicit iterative constructions. The core methodology combines reflected backward stochastic differential equations and Snell envelopes to encode the delay and post-delay impulse opportunities, yielding implementable optimal policies. The work also connects to swing options by modeling decisions that are prepared at one time but executed after a delay, offering a rigorous blueprint for valuation and hedging in delayed impulse-control problems.

Abstract

In this paper, we consider a class of stochastic impulse control problem when there is a fixed delay 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.
Paper Structure (14 sections, 19 theorems, 168 equations)

This paper contains 14 sections, 19 theorems, 168 equations.

Table of Contents

  1. Introduction
  2. The finite horizon setting
  3. Preliminaries and description of the model
  4. The risk neutral impulse control problem
  5. Construction of the optimal strategy. An iterative scheme via reflected BSDEs.
  6. The optimal strategy
  7. The risk sensitive case
  8. The iterative scheme
  9. The optimal strategy
  10. The infinite horizon case
  11. Construction of the iterative scheme
  12. Appendix
  13. Optional and predictable projections
  14. Snell envelope of a process

Key Result

Proposition 2.1

Let $\mathfrak{X}=[\frac{T}{\Delta}]$. We then have:

Theorems & Definitions (34)

  • Remark 2.1
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • Remark 2.2
  • Remark 2.3
  • Proposition 2.4
  • ...and 24 more