Optimal decision under ambiguity for diffusion processes

TL;DR

This paper studies stochastic optimization for ambiguity-averse decision makers, focusing on optimal stopping problems under drift ambiguity and potential process crashes, using geometric and game-theoretic methods.

Contribution

It introduces a novel approach to optimal stopping under ambiguity for diffusion processes, including cases with potential crashes, by reducing problems to geometric and game-theoretic formulations.

Findings

Reduced optimal stopping under drift ambiguity to geometric problems.

Solved optimal stopping with process crashes using combined stopping and Dynkin game.

Provided examples illustrating the application of the methods.

Abstract

In this paper we consider stochastic optimization problems for an ambiguity averse decision maker who is uncertain about the parameters of the underlying process. In a first part we consider problems of optimal stopping under drift ambiguity for one-dimensional diffusion processes. Analogously to the case of ordinary optimal stopping problems for one-dimensional Brownian motions we reduce the problem to the geometric problem of finding the smallest majorant of the reward function in a two-parameter function space. In a second part we solve optimal stopping problems when the underlying process may crash down. These problems are reduced to one optimal stopping problem and one Dynkin game. Examples are discussed.