alpha-Wiener bridges: singularity of induced measures and sample path properties

Abstract

Let us consider the process $(X_t^{(\alpha )}{)}_{t\in [0,T)}$ given by the SDE $d{X}_t^{(\alpha )}=-\frac{\alpha }{T-t}{X}_t^{(\alpha )}dt+d{B}_t$, $t\in [0,T)$, where $\alpha \in R$, $T\in (0,\infty )$, and $(B_t{)}_{t\ge 0}$ is a standard Wiener process. In case of $\alpha >0$ the process $X^{(\alpha )}$ is known as an $\alpha$-Wiener bridge, in case of $\alpha =1$ as the usual Wiener bridge. We prove that for all $\alpha ,\beta \in R$, $\alpha \ne \beta$, the probability measures induced by the processes $X^{(\alpha )}$ and $X^{(\beta )}$ are singular on C[0,T). Further, we investigate regularity properties of $X_t^{(\alpha )}$ as $t\uparrow T$.

20 pages

References in corpus (1)

• Explicit formulas for Laplace transforms of certain functionals of some time inhomogeneous diffusions