Stochastic Control of Addiction with State-Dependent Jump Relapse

2026/07/27 by Dounia Aissi, Ioana Ciotir, Dan Goreac +1
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Abstract

We study a continuous-time rational addiction model where addiction capital follows a piecewise deterministic Markov process with state-dependent jumps capturing relapse and recovery. The instantaneous utility combines consumption and addiction capital via a power specification, leading to Hamilton-Jacobi-Bellman (HJB) equations with nonlocal jump terms. In the capped, bounded-control case we obtain a unique bounded viscosity solution and prove that optimal policies are bang-bang, switching between minimal and maximal consumption, while in the uncapped case we derive explicit linear feedback controls and closed-form value functions in several parameter regimes.

Citations

• Asymptotic Control for a Class of Piecewise Deterministic Markov Processes Associated to Temperate Viruses
• McKean-Vlasov Ito-Skorohod equations, and nonlinear diffusions with discrete jump sets

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