2020
DOI: 10.1111/mafi.12289
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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, time‐homogeneous Markov chains under convex expectations, and a class of nonlinear ordinary d… Show more

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Cited by 12 publications
(13 citation statements)
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“…However, using a similar approach to the one by Nisio on the space of bounded and uniformly continuous functions, Denk et al [3] proved the existence of a least upper bound for semigroups related to Lévy processes. In [9] the approach by Nisio has been used to characterize the generators of Markov chains with finite state space under Date: June 12, 2019. Financial support through the German Research Foundation via CRC 1283 "Taming Uncertainty" is gratefully acknowledged.…”
Section: Introduction and Main Resultsmentioning
confidence: 99%
“…However, using a similar approach to the one by Nisio on the space of bounded and uniformly continuous functions, Denk et al [3] proved the existence of a least upper bound for semigroups related to Lévy processes. In [9] the approach by Nisio has been used to characterize the generators of Markov chains with finite state space under Date: June 12, 2019. Financial support through the German Research Foundation via CRC 1283 "Taming Uncertainty" is gratefully acknowledged.…”
Section: Introduction and Main Resultsmentioning
confidence: 99%
“…(9); the following result, taken from [13,Proposition 3.42], formalises this. 3 For any matrix 𝑀 and real number 𝜇, the matrix exponential 𝑒 𝜇𝑀 of 𝜇𝑀 is…”
Section: Markovianity and Homogeneitymentioning
confidence: 99%
“…Recently, several authors have independently proposed generalisations of Markovian jump processesalso called continuous-time Markov chains or Markov processes -that provide an elegant way of dealing with parameter uncertainty [1][2][3]. Whereas a (homogeneous) Markovian jump process is uniquely defined by its rate matrix and initial probability mass function, these 'imprecise' generalisations allow for partially specified parameters: they are defined through sets of rate matrices and/or sets of initial probability mass functions.…”
Section: Introductionmentioning
confidence: 99%
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