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Interview prep / Stochastic Calculus

Variance of Brownian motion at times one and two

Question

For a standard Brownian motion WtW_t, find Var⁡(W1+W2)\operatorname{Var}(W_1+W_2).

Short answer

55. Write W1+W2=2W1+(W2−W1)W_1+W_2=2W_1+(W_2-W_1), a sum of independent pieces, so the variance is 4⋅1+1=54\cdot1+1=5. Equivalently, 1+2+2Cov⁡(W1,W2)=3+2⋅1=51+2+2\operatorname{Cov}(W_1,W_2)=3+2\cdot1=5.

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