Seierstad Sufficient Conditions for Stochastic Optimal Control Problems with Infinite Horizon
Anton O. Belyakov, Yuri M. Kabanov, Ivan A. Terekhov, Maxim M. Savinov
TL;DR
This paper extends Seierstad’s deterministic infinite-horizon sufficiency framework to stochastic optimal control by formulating dual variables through backward stochastic differential equations (BSDEs) associated with the Hamilton-Pontryagin function $\mathcal{H}$. Under a concavity condition on $\mathcal{H}$ for large times and with a limiting condition on $\mathcal{H}_u$ against deviations in the control, the authors derive sufficient (and, in the linear case, necessary) conditions for weakly overtaking and overtaking optimality over an infinite horizon. The core method combines Itô calculus on the dual process with the BSDE representation to bound the cost difference $\Delta J_T$ and take limits as $T\to\infty$. Two examples illustrate how the conditions can be checked in stochastic growth/investment models; in particular, linear $\mathcal{H}$ yields exact necessity, aligning with the deterministic Seierstad theory and expanding its stochastic applicability. Overall, the work provides a practical, BSDE-based sufficiency framework for infinite-horizon stochastic control problems with potential economic applications.
Abstract
In this note we consider a problem of stochastic optimal control with the infinite-time horizon. We present analogues of the Seierstad sufficient conditions of overtaking optimality based on the dual variables stochastic described by BSDEs appeared in the Bismut-Pontryagin maximum principle.