Optimal stochastic control problem under model uncertainty with non-entropic penalty

TL;DR

This paper investigates a stochastic control problem under model uncertainty using two types of penalties, providing a characterization of the value process through quadratic backward stochastic differential equations.

Contribution

It introduces a unified framework for model uncertainty with general penalties and characterizes the value process in the case of consistent time penalties.

Findings

Characterization of the value process via quadratic BSDEs.

Analysis of two penalty types: f-divergence and consistent time.

Solution uniqueness for the control problem.

Abstract

In this paper, a stochastic control problem under model uncertainty with general penalty term is studied. Two types of penalties are considered. The first one is of type f-divergence penalty treated in the general framework of a continuous filtration. The second one called consistent time penalty studied in the context of a Brownian filtration. In the case of consistent time penalty, we characterize the value process of our stochastic control problem as the unique solution of a class of quadratic backward stochastic differential equation with unbounded terminal condition.