Kac-Ornstein-Uhlenbeck Processes: Stationary Distributions and Exponential Functionals

Nikita Ratanov ()
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Nikita Ratanov: Chelyabinsk State University

Methodology and Computing in Applied Probability, 2022, vol. 24, issue 4, 2703-2721

Abstract: Abstract We study Ornstein-Uhlenbeck processes whose parameters are modulated by an external two-state Markov process. The conditional means of such a process for a given modulation follow an analogue of the Langevin equation, which is controlled by a pair of telegraph processes. Stationary distributions of such processes are described. The relationship between stationary distributions and the distribution of the corresponding exponential functional is also discussed. Of interest is the limiting behaviour of these processes under conditions similar to the Kac scaling. Limiting processes turn out to be different classes of ordinary Ornstein-Uhlenbeck processes.

Keywords: Ornstein-Uhlenbeck process; Asymmetric telegraph process; Invariant distribution; Exponential functional; Kac’s scaling; 60G44; 60J75; 60K99 (search for similar items in EconPapers)
Date: 2022
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DOI: 10.1007/s11009-022-09956-z

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