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- W1485336502 abstract "We collect and prove some new properties of two Markov processes that in many ways resemble Wiener and Ornstein--Uhlenbeck (OU) processes. Although processes considered in this paper were defined either in a noncommutative probability context or through quadratic harnesses we define them once more as a “continuous time” generalization of a simple, symmetric, discrete time process satisfying simple conditions imposed on the form of its first two conditional moments. The finite dimensional distributions of the first one (say ${bf X=}( X_{t})_{tge 0}$ called $q$-Wiener) depend on one parameter $qin(-1,1]$, and those of the second one (say ${bf Y=}(Y_{t})_{tin{bf R}}$ called $(alpha,q)$-Ornstein--Uhlenbeck) on two parameters $(alpha,q)in(0,infty)times(-1,1]$. The first one resembles the Wiener process in the sense that for $q=1$ it is a Wiener process but also that for $vert qvert <1$ and $forall nge1$: $t^{n/2}H_{n}( X_{t}/sqrt{t},|,q),$ where $(H_{n})_{nge0}$ are the so-called $q$-Hermite polynomials, are martingales. However, it neither has independent increments not allows continuous sample path modification. The second one resembles the OU process. For $q=1$ it is a classical OU process. For $vert qvert <1$ it is also stationary with correlation function equal to $exp(-alpha|t-s|)$ and has many properties resembling those of its classical version. We think that these processes are fascinating objects to study posing many interesting, open questions." @default.
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- W1485336502 date "2012-01-01" @default.
- W1485336502 modified "2023-09-27" @default.
- W1485336502 title "$q$-Wiener and $(alpha,q)$-Ornstein--Uhlenbeck Processes. A Generalization of Known Processes" @default.
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- W1485336502 doi "https://doi.org/10.1137/s0040585x97985674" @default.
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