Abstract
It is proven that, for any deterministic $L^2\lbrack 0,1\rbrack$ function $\phi(t)$, $E\bigg(\exp\int^1_0\phi(t)dw_t\bigg\arrowvert \|w\| < \varepsilon\bigg) \rightarrow 1\,\text{as}\,\varepsilon \rightarrow 0,$ where $w_t$ is a standard Brownian motion and $\|\cdot\|$ is any "reasonable" norm on $C_0\lbrack 0,1\rbrack$. Applications to the computation of Onsager-Machlup functionals are pointed out.
Citation
Larry A. Shepp. Ofer Zeitouni. "A Note on Conditional Exponential Moments and Onsager-Machlup Functionals." Ann. Probab. 20 (2) 652 - 654, April, 1992. https://doi.org/10.1214/aop/1176989796
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