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Conditioning a Brownian bridge

HardStochastic Processes~8m

Let WtW_t be a standard Brownian motion and define the Brownian bridge Xt=Wt−t W1X_t = W_t - t\,W_1 on [0,1][0,1]. Compute E[X1/2∣X3/4=3]\mathbb{E}\big[X_{1/2} \mid X_{3/4} = 3\big].

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  1. Brownian-bridge covariance is Cov⁡(Xs,Xt)=s(1−t)\operatorname{Cov}(X_s,X_t)=s(1-t) for s≤ts\le t.
  2. Gaussian conditional mean =Cov⁡(X1/2,X3/4)Var⁡(X3/4)⋅3=\dfrac{\operatorname{Cov}(X_{1/2},X_{3/4})}{\operatorname{Var}(X_{3/4})}\cdot 3.

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Asked at: Jane Street, Two Sigma

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