Markov chains under nonlinear expectation

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.

Stopper-Controller Games embedded in Single-Player Control Problems

“…Next, we want to have a look into a non-linear framework for continuous-time Markov chains, following [8].…”

“…By [8,Theorem 1.3], the HJB equation for v becomes the backward ODE (6.4) ∂ ∂t v(t) + sup Q∈S Qv(t) = 0, v(T ) = g, for t ∈ [0, T ]. However, Q is in general not local and Section 3 are not applicable here, even not for the linear case where S is a singleton.…”

“…Since S is bounded, Picard-Lindelöf theorem yields that (6.4) has a unique solution which is C 1 . In [8] also the case with ̺ ≤ 0 is covered.…”