De Finetti's control problem with a concave bound on the control rate

TL;DR

This paper addresses De Finetti's control problem with a concave bound on control rates, deriving an explicit value function and establishing the optimality of a generalized mean-reverting strategy for nonlinear Ornstein-Uhlenbeck processes.

Contribution

It introduces a novel control problem with a concave bound on control rates and provides explicit solutions, extending previous cases with constant or linear bounds.

Findings

Explicit expression for the value function obtained

Generalized mean-reverting strategy proven optimal

Includes special cases with constant and linear bounds

Abstract

We consider De Finetti's control problem for absolutely continuous strategies with control rates bounded by a concave function and prove that a generalized mean-reverting strategy is optimal. In order to solve this problem, we need to deal with a nonlinear Ornstein-Uhlenbeck process. Despite the level of generality of the bound imposed on the rate, an explicit expression for the value function is obtained up to the evaluation of two functions.This optimal control problem has those with control rates bounded by a constant and a linear function, respectively, as special cases.