solving a two dimensional stochastic differential equation

Question

This is exercise 5.8 from SDE by Oksendal.

Solve the two dimensional SDE:

$dX_1(t)=X_2(t)dt + \alpha dB_1(t)$

$dX_2(t)= -X_1(t)dt + \beta dB_2(t)$

Where $(B_1(t),B_2(t))$ is two dimensional Brownian motion and $\alpha , \beta$ are constants.

Is there any hint how to do it?

Thanks a lot.

Answer 1

According to the cited example 5.1.3 you got to consider the increments of the process $$ Y_t=e^{-Jt}X_t \ \text{ with }\ X=\pmatrix{X_1\\X_2},\ J=\pmatrix{0&1\\-1&0}. $$