Markov Chain Approximation of Pure Jump Processes
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Markov Chain Approximation of Pure Jump Processes Ante Mimica · Nikola Sandri´c1 · René L. Schilling2
Received: 15 December 2016 / Accepted: 27 March 2018 © Springer Science+Business Media B.V., part of Springer Nature 2018
Abstract In this paper we discuss weak convergence of continuous-time Markov chains to a non-symmetric pure jump process. We approach this problem using Dirichlet forms as well as semimartingales. As an application, we discuss how to approximate a given Markov process by Markov chains. Keywords Non-symmetric Dirichlet form · Non-symmetric Hunt process · Markov chain · Mosco convergence · Semimartingale · Semimartingale characteristics · Weak convergence Mathematics Subject Classification (2010) 60J25 · 60J27 · 60J75
1 Introduction Let Xn , n ∈ N be a sequence of continuous-time Markov chains where Xn takes values on the lattice n−1 Zd , and let X be a Markov process on Rd . We are interested in the following two questions: (i) Under which conditions does {Xn }n∈N converge weakly to some (non-symmetric) Markov process? (ii) Can a given Markov process X be approximated (in the sense of weak convergence) by a sequence of Markov chains? Ante Mimica is deaceased, 1981-01-20–2016-06-09.
B N. Sandri´c
[email protected] A. Mimica url: https://web.math.pmf.unizg.hr/~amimica/ R.L. Schilling [email protected]
1
Faculty of Civil Engineering, University of Zagreb, 10000 Zagreb, Croatia
2
Institut für Mathematische Stochastik, Fachrichtung Mathematik, TU Dresden, 01062 Dresden, Germany
A. Mimica et al.
These questions have a long history. If X is a diffusion process determined by a generator in non-divergence form these problems have been studied in [29] using martingale problems. The key ingredient in this approach is that the domain of the corresponding generator is rich enough, i.e. containing the test functions Cc∞ (Rd ). On the other hand, if the generator of X is given in divergence form, it is a delicate matter to find non-trivial functions in its domain. In order to overcome this problem, one resorts to an L2 -setting and the theory of Dirichlet forms; for example, [32] solve these problems for symmetric diffusion processes X using Dirichlet forms. The main assumptions are certain uniform regularity conditions and the boundedness of the range of the conductances of the approximating Markov chains. These results are further extended in [3], where the uniform regularity condition is relaxed and the conductances may have unbounded range. Very recently, [8] discusses these questions for a non-symmetric diffusion process X. Let us also mention that the problem of approximation of a reflected Brownian motion on a bounded domain in Rd is studied in [2]. As far as we know, the paper [14] is among the first papers studying the approximation of a jump process X. In this work the authors investigate convergence to and approximation of a symmetric jump process X whose jump kernel is comparable to the jump kernel of a symmetric stable Lévy process. These results have been extended in [4], whe
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