Conditioning a Brownian bridge

HardStochastic Processes~8m

Let $W_t$ be a standard Brownian motion and define the Brownian bridge $X_t = W_t - t\,W_1$ on $[0,1]$ . Compute $\mathbb{E}\big[X_{1/2} \mid X_{3/4} = 3\big]$ .

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Brownian-bridge covariance is $\operatorname{Cov}(X_s,X_t)=s(1-t)$ for $s\le t$ .
Gaussian conditional mean $=\dfrac{\operatorname{Cov}(X_{1/2},X_{3/4})}{\operatorname{Var}(X_{3/4})}\cdot 3$ .

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Conditioning a Brownian bridge

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