2BSDE with uncertain horizon and application to stochastic control in erratic environments

Abstract: We investigate the existence and uniqueness of non-Markovian second-order backward stochastic differential equations with an uncertain terminal horizon and establish comparison principles under the assumption that the driver is Lipschitz continuous. The terminal time is both random and exogenous, and it may not be adapted to the Brownian filtration, leading to a singular jump in the 2BSDE decomposition. We also provide a connection between this new class of 2BSDE and a fully nonlinear PDE in a Markovian setting. Our theoretical results are applied to non-Markovian stochastic control problems in two settings: (1) when an agent seeks to maximize utility from a payoff received at an uncertain terminal time by controlling both the drift and volatility of a diffusion process; and (2) when the agent contends with volatility uncertainty stemming from an external source, referred to as Nature, and optimizes the drift in a worst-case scenario for the ambiguous volatility. We term this class of problems erratic stochastic control, reflecting the dual uncertainty in both model parameters and the timing of the terminal horizon.