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$q$-Wiener and $(\alpha,q)$-Ornstein--Uhlenbeck Processes. A Generalization of Known Processes

P. J. Szabł owski

2012
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Theory of Probability and its Applications
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We collect, scattered through literature, as well as we prove some new properties of two Markov processes that in many ways resemble Wiener and Ornstein--Uhlenbeck processes. Although processes considered in this paper were defined either in non-commutative probability context or through quadratic harnesses we define them once more as so to say 'continuous time ' generalization of a simple, symmetric, discrete time process satisfying simple conditions imposed on the form of its first two
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... onal moments. The finite dimensional distributions of the first one (say X=(X_t)_t≥0 called q-Wiener) depends on one parameter q∈(-1,1] and of the second one (say Y=(Y_t)_t called (α,q)- Ornstein--Uhlenbeck) on two parameters (α,q)∈(0,∞)×(-1,1]. The first one resembles Wiener process in the sense that for q=1 it is Wiener process but also that for |q|<1 and ≥1: t^n/2H_n(X_t/|q), where (H_n)_n≥0 are the so called q-Hermite polynomials, are martingales. It does not have however neither independent increments not allows continuous sample path modification. The second one resembles Ornstein--Uhlenbeck process. For q=1 it is a classical OU process. For |q|<1 it is also stationary with correlation function equal to exp(-α|t-s|) and has many properties resembling those of its classical version. We think that these process are fascinating objects to study posing many interesting, open questions.

doi:10.1137/s0040585x97985674
fatcat:yfrjzpp6xjcrhndvlv6d6xfy3q