Markov chains under nonlinear expectation

TL;DR

This paper extends classical Markov chain theory to nonlinear expectations, characterizing convex Q-operators and their generators, and applies the results to compute price bounds under model uncertainty.

Contribution

It introduces convex Q-operators for Markov chains under nonlinear expectations and provides a full characterization and dual representation, extending classical generator theory.

Findings

Characterization of convex Q-operators via a positive maximum principle

Dual representation of convex Q-operators through Q-matrices

Numerical examples computing price bounds under model uncertainty

Abstract

In this paper, we consider continuous-time Markov chains with a finite state space under nonlinear expectations. We define so-called Q-operators as an extension of Q-matrices or rate matrices to a nonlinear setup, where the nonlinearity is due to model uncertainty. The main result gives a full characterization of convex Q-operators in terms of a positive maximum principle, a dual representation by means of Q-matrices, continuous-time Markov chains under convex expectations and nonlinear ordinary differential equations. This extends a classical characterization of generators of Markov chains to the case of model uncertainty in the generator. We further derive a primal and dual representation of the convex semigroup arising from a Markov chain under a convex expectation via the Fenchel-Legendre transformation of its generator. We illustrate the results with several numerical examples, where we compute price bounds for European contingent claims under model uncertainty in terms of the rate matrix.