1 Fe b 20 11 General α - Wiener bridges
نویسنده
چکیده
An α-Wiener bridge is a one-parameter generalization of the usual Wiener bridge, where the parameter α > 0 represents a mean reversion force to zero. We generalize the notion of α-Wiener bridges to continuous functions α : [0, T ) → R. We show that if the limit limt↑T α(t) exists and is positive, then a general α-Wiener bridge is in fact a bridge in the sense that it converges to 0 at time T with probability one. Further, under the condition limt↑T α(t) 6= 1 we show that the law of the general α-Wiener bridge can not coincide with the law of any non time-homogeneous Ornstein-Uhlenbeck type bridge. In case limt↑T α(t) = 1 we determine all the Ornstein-Uhlenbeck type processes from which one can derive the general α-Wiener bridge by conditioning the original Ornstein-Uhlenbeck type process to be in zero at time T .
منابع مشابه
α-Wiener bridges: singularity of induced measures and sample path properties
Let us consider the process (X (α) t )t∈[0,T ) given by the SDE dX (α) t = − α T−t X (α) t dt+ dBt, t ∈ [0, T ), where α ∈ R, T ∈ (0,∞), and (Bt)t>0 is a standard Wiener process. In case of α > 0 the process X(α) is known as an α-Wiener bridge, in case of α = 1 as the usual Wiener bridge. We prove that for all α, β ∈ R, α 6= β, the probability measures induced by the processes X(α) and X(β) are...
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