Discrete time stochastic impulse control with delay
Said Hamadène, Boualem Djehiche
TL;DR
This work analyzes infinite-horizon impulse control in discrete time with a fixed execution delay $Δ$, allowing interventions to take effect after a lag and producing non-Markovian dynamics between impulses. It develops a probabilistic framework based on Snell envelopes to characterize optimal strategies under both risk-neutral and risk-sensitive utilities, proving the existence of optimal strategies and bounded $\varepsilon$-optimal strategies. For the risk-neutral case, it constructs an iterative scheme with processes $Y^n_k(\nu,\xi)$ that converge to an optimal value $Y_k(\nu,\xi)$, yielding $\delta^*$ with $J(\delta^*)$ maximal; it also shows that a finite-impulse approximation is always possible. For the risk-sensitive case, it extends the machinery to an exponential utility via $V^n_k(\nu,\xi)$, establishing a parallel existence result with $V_0(0,0)=J(\delta^*)$, and revealing a robust approach to delayed decision-making in stochastic systems with potential applications across finance, inventory, and energy management, while suggesting future work on random or state-dependent delays.
Abstract
We study a class of infinite-horizon impulse control problems with execution delay in discrete time. Using probabilistic methods, particularly the notion of the Snell envelope of processes, we construct an optimal strategy among all admissible strategies for both risk-neutral and risk-sensitive utility functions. Furthermore, we establish the existence of bounded $ε$-optimal strategies. This framework provides a robust approach to handling execution delays in discrete-time stochastic systems.