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Markov chains under nonlinear expectation
Author(s) -
Nendel Max
Publication year - 2021
Publication title -
mathematical finance
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.98
H-Index - 81
eISSN - 1467-9965
pISSN - 0960-1627
DOI - 10.1111/mafi.12289
Subject(s) - markov chain , mathematics , nonlinear system , legendre transformation , operator (biology) , generator (circuit theory) , convex combination , representation (politics) , markov process , convex optimization , mathematical optimization , regular polygon , mathematical analysis , power (physics) , biochemistry , statistics , physics , chemistry , repressor , quantum mechanics , politics , political science , transcription factor , law , gene , geometry
In this paper, we consider continuous‐time Markov chains with a finite state space under nonlinear expectations. We define so‐called Q‐operator s 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‐operator s in terms of a positive maximum principle, a dual representation by means of Q ‐matrices, time‐homogeneous Markov chains under convex expectations, and a class of 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 an explicit primal and dual representation of convex semigroups arising from Markov chains under convex expectations via the Fenchel–Legendre transformation of the 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.