Martingale Nature and Laws of the Iterated Logarithm for Markov Processes of Pure-Jump Type
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Martingale Nature and Laws of the Iterated Logarithm for Markov Processes of Pure-Jump Type Yuichi Shiozawa1 · Jian Wang2 Received: 5 November 2019 / Revised: 29 July 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract We present sufficient conditions, in terms of the jumping kernels, for two large classes of conservative Markov processes of pure-jump type to be purely discontinuous martingales with finite second moment. As an application, we establish the law of the iterated logarithm for sample paths of the associated processes. Keywords Feller process · Hunt process · Lower bounded semi-Dirichlet form · Martingale · Jumping kernel · Law of the iterated logarithm Mathematics Subject Classification 60J75 · 47G20 · 60G52
1 Introduction It is well known that any symmetric Lévy process with finite first moment possesses the martingale property because of the independent increments property. Apart from Lévy processes, the martingale property was studied for a one-dimensional diffusion process with natural scale (see [9,17] and references therein). Note that this process is a time-changed Brownian motion and thus possesses the local martingale property (see, e.g., [13, Proposition V.1.5]). In [9,17], a necessary and sufficient condition is given for this process to be a martingale by adopting the Feller theory.
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Jian Wang [email protected] Yuichi Shiozawa [email protected]
1
Department of Mathematics, Graduate School of Science, Osaka University, Toyonaka 560-0043, Japan
2
College of Mathematics and Informatics, Fujian Key Laboratory of Mathematical Analysis and Applications (FJKLMAA), Center for Applied Mathematics of Fujian Province (FJNU), Fujian Normal University, Fuzhou 350007, People’s Republic of China
123
Journal of Theoretical Probability
To the best of our knowledge, except for these Markov processes mentioned above, answers are not available in the literature to the following “fundamental”question — when does a Markov process become a martingale? The aim of this paper is to present explicit sufficient conditions for two large classes of jump processes to be purely discontinuous martingales with finite second moment in terms of jumping kernels. As an application, we show Khintchine’s law of the iterated logarithm (LIL) for two classes of non-symmetric jump processes. We also provide examples of non-symmetric jump processes which are purely discontinuous martingales with finite second moment and satisfy the LIL. To derive the martingale property for a jump process, we apply two different approaches: One is based on the infinitesimal generator along with the moments calculus of the process, and the other relies on the componentwise decomposition of the process with the aid of the semimartingale theory ([7, Chapter II, Section 2]). The assumptions of our paper are mild. For example, condition (2.3) (or (3.4)) means the existence of the second moment for the jumping kernel (which seems to be necessary for the LIL), while condition (2.6) (or (3.3)) rough
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