Karhunen-Loeve expansions of alpha-Wiener bridges

Details Karhunen-Loeve expansions of alpha-Wiener bridges Karhunen-Loeve expansions of alpha-Wiener bridges

2010-07-17

arxiv.org/abs/1007.2904v2

Central European Journal of Mathematics, Vol. 9, No. 1, 2011, 65-84

Mathematics

Probability

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

Scientific paper

We study Karhunen-Loeve expansions of the process $(X_t^{(\alpha)})_{t\in[0,T)}$ given by the stochastic differential equation $dX_t^{(\alpha)} = -\frac\alpha{T-t} X_t^{(\alpha)} dt+ dB_t,$ $t\in[0,T),$ with an initial condition $X_0^{(\alpha)}=0,$ where $\alpha>0,$ $T\in(0,\infty)$ and $(B_t)_{t\geq 0}$ 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).

Affiliated with

Also associated with

No associations

Rating

Karhunen-Loeve expansions of alpha-Wiener bridges does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Karhunen-Loeve expansions of alpha-Wiener bridges, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Karhunen-Loeve expansions of alpha-Wiener bridges will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-658783