System of Stochastic Diff Eq

Question

How can I solve the system of stochastic differential equation $$dX_{1}=X_{2}dt+adW_{1}$$ $$dX_{2}=-X_{1}dt+bdW_{1}$$

Answer 1

Let $$ X_1(t) = A(t) \sin (t) + B(t) \cos (t)\\ X_2(t) = A(t) \cos (t) - B(t) \sin (t) $$ then thanks to the Ito fornula: \begin{align} dX_1(t) &= dA(t) \sin (t) + A(t) \cos (t) dt + dB(t) \cos (t) - B(t)\sin(t) dt \\ &= X_2(t)dt + dA(t) \sin (t) + dB(t) \cos (t) \\ dX_2(t) &= dA(t) \cos (t) - A(t)\sin (t) dt - dB(t) \sin (t) - B(t) \cos (t) dt \\ &= -X_1(t) + dA(t) \cos (t) - dB(t) \sin (t) \\ \end{align} then you are left with \begin{align} adW_1(t) &= dA(t) \sin (t) + dB(t) \cos (t) \\ bdW_1(t) &= dA(t) \cos (t) - dB(t) \sin (t) \\ \implies dA(t) &= \left[ a \sin(t) + b \cos(t) \right] dW_1(t) \\ dB(t) &= \left[ a \cos(t) - b \sin(t) \right] dW_1(t) \\ A(t) &= A_0 + \int_0^t \left[ a \sin(s) + b \cos(s) \right] dW_1(s) \\ B(t) &= B_0 + \int_0^t \left[ a \cos(s) - b \sin(s) \right] dW_1(s) \end{align}

Answer 2

I did that like that

Let $f(t)$ and $g(t)$ be integrating factors By Ito's Formula $$d(f(t)X1(t))=f(t)dX1(t)+f'(t)X1(t)dt$$ $$d(f(t)X1(t))=(f(t)X2(t)+f'(t)X1(t))dt+af(t)dW_{1}(t)$$ Similarly $$d(g(t)X2(t))=(-g(t)X1(t)+g'(t)X2(t))dt+bg(t)dW_{2}(t)$$

Need to find $f(t)$and $g(t)$ such that $f(t)X2(t)+f'(t)X1(t)=0$ and $-g(t)X1(t)+g'(t)X2(t)$

So, $f(t)=sint$ and $g(t)=cost$

therefore $$d(sintX1(t))=asintdW_{1}(t)$$ $$X1(t)=acosct\int_{0}^{t} sinsdW_{1}(s)$$ and $$d(CosX2(t))=bcostdW_{2}(t)$$ $$X2(t)=X2(0)sect+bsect\int_{0}^{t} cossdW_{2}(s)$$

Is there anything which I did wrong?