I have a question regarding finding the joint distribution of two process$$dX_{t}=a_{t}dB_{t}$$$$dY_{t}=b_{t}dW_{t}$$where $B_{t}$ and $W_{t}$ are two Brownian motions with correlated increments, in which $Corr_{t}(dB_{t}, dW_{t}$)=$\rho$.
My question is to find the joint distribution of ($X_{t}$, $Y_{t}$).
I know that both X and Y has normal distribution, but for correlated, how to proceed for finding the joint distribution, I am very confused now..
Any help will be very appreciated
Best Answer
Hint: One needs two means, two variances and one correlation. The OP already knows the value of the means and the variances. For the correlation, note that $$ E[X_tY_t]=E\left[\int_0^ta_s\,\mathrm dB_s\cdot\int_0^tb_s\,\mathrm dW_s\right]=E\left[\int_0^ta_sb_s\,\mathrm d\langle B,W\rangle_s\right] =\rho\cdot \int_0^tE\left[a_sb_s\right]\cdot\mathrm ds. $$