Karhunen-Loeve expansions of alpha-Wiener bridges

Abstract

We study Karhunen-Loeve expansions of the process $(X_t^{(\alpha )}{)}_{t\in [0,T)}$ given by the stochastic differential equation $d{X}_t^{(\alpha )}=-\frac{\alpha }{T-t}{X}_t^{(\alpha )}dt+d{B}_t,$ $t\in [0,T),$ with an initial condition $X_0^{(\alpha )}=0,$ where $\alpha >0,$ $T\in (0,\infty )$ and $(B_t{)}_{t\ge 0}$ is a standard Wiener process. This process is called an $\alpha$-Wiener bridge or a scaled Brownian bridge, and in the special case of $\alpha =1$ the usual Wiener bridge. We present weighted and unweighted Karhunen-Loeve expansions of $X^{(\alpha )}$. As applications, we calculate the Laplace transform and the distribution function of the $L^2[0,T]$-norm square of $X^{(\alpha )}$ studying also its asymptotic behavior (large and small deviation).

32 pages, a new remark is added for applications on small deviation probabilities

References in corpus (2)

• Expansions for Gaussian processes and Parseval frames
• Explicit formulas for Laplace transforms of certain functionals of some time inhomogeneous diffusions