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Suppose $\{B_t,t\ge0\}$ be a standard brownian motion and suppose $0\le u\le s, calculate $E(e^{B(t)}|e^{B(u)},0\le u\le s )$.

Attempts: $E(e^{B(t)}|e^{B(u)},0\le u\le s )=E(e^{B(t-u)+B(u)}|e^{B(u)})=E(e^{B(t-u)}|e^{B(u)})E(e^{B(u)}|e^{B(u)})$ but then not sure how to proceed.

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    OP: What happens with this question? Did you manage to write down a full proof, using @Stefan's hints?2013-01-02

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Hint: Write $B_t=B_t-B_s+B_s$ and use that $ E[XY\mid\mathcal{G}]=XE[Y\mid \mathcal{G}] $ if $X$ is $\mathcal{G}$-measurable along with the fact that $ E[X\mid\mathcal{G}]=E[X] $ if $X$ is independent of $\mathcal{G}$.

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    @RobertIsrael: Indeed!2012-12-21