Infinite-Horizon Optimal Control of Jump-Diffusion Models for Pollution-Dependent Disasters

Abstract

The paper develops a unified framework for stochastic growth models with environmental risk, in which rare but catastrophic shocks interact with capital accumulation and pollution. The analysis begins with a Poisson process formulation, leading to a Hamilton-Jacobi-Bellman (HJB) equation with jump terms that admits closed-form candidate solutions and yields a composite state variable capturing exposure to rare shocks. The framework is then extended by endogenizing disaster intensity via a nonhomogeneous Poisson process, showing how environmental degradation amplifies macroeconomic risk and strengthens incentives for abatement. A further extension introduces pollution diffusion alongside state-dependent jump intensity, yielding a tractable jump-diffusion HJB that decomposes naturally into capital and pollution components under power-type value functions. Finally, a formulation in terms of Poisson random measures unifies the dynamics, makes arrivals and compensators explicit, and accommodates state-dependent magnitudes. Together, these results establish rigorous verification theorems and viscosity-solution characterizations for the associated integro-differential HJB equations, highlight how vulnerability emerges endogenously from the joint evolution of capital and pollution, and show that the prospect of rare, state-dependent disasters fundamentally reshapes optimal intertemporal trade-offs.

Theorems & Definitions (26)

• Remark 2.1
• Remark 2.2
• Remark 3.2
• Remark 4.1
• Proposition 4.2: Envelope identities
• proof
• Proposition 4.3: Envelope identities with state--dependent hazard
• proof
• Theorem 5.1: Verification -- Jump--control growth--environment model
• proof