{"paper":{"title":"Karhunen-Loeve expansions of alpha-Wiener bridges","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Endre Igloi, Matyas Barczy","submitted_at":"2010-07-17T07:58:38Z","abstract_excerpt":"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 t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1007.2904","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}