What is the correlation between 2 Brownian motions at time $t_1$ and $t_2$? Assume that they are jointly normal with correlation $\rho$.
Why does it matter about $t_1$ and $t_2$? The question doesn't suggest that they start off at different times.
2026-03-27 17:51:36.1774633896
Correlation Between Two Brownian Motions
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I suspect you (or whoever gave you the problem) are talking about a bivariate Markov process where, conditional on $(B_1(t_1), B_2(t_1))$, $(B_1(t_2), B_2(t_2))$ for $t_2 > t_1$ are jointly normal with mean $(B_1(t_1),B_2(t_1))$ and covariance matrix $$ (t_2-t_1) \pmatrix{\sigma_1^2 & \sigma_1 \sigma_2 \rho\cr \sigma_1 \sigma_2 \rho & \sigma_2^2}$$
Note that $(B_1(t_2)-B_1(t_1)$ and $B_2(t_2)-B_2(t_1)$ are independent of $B_1(t_1)$ and $B_2(t_1)$. So, assuming the process starts with $B_1(0)= B_2(0)=0$, $$\text{Cov}(B_1(t_1), B_2(t_2)) = \text{Cov}(B_1(t_1),B_2(t_1)) = t_1 \sigma_1 \sigma_2 \rho$$ Divide by the product of the standard deviations and you have the correlation.