I have a second-order differential equation that has been put in the following form:
$$dx = y dt$$ $$dy = -ax dt+ c \sin{(\omega t)} dt - xdB_t$$
where $dB_t = W_t dt$, $B_t$ being the Brownian motion.
I have found methods to solve the system of the general form (in the book by Henderson and Plaschko):
$$dX_k = a_k(\textbf{X},t)dt + h_k(t)X_k B_t$$
[for $k=1,2,...,n$]
But this is Not the Coupled case as my problem has.
Please guide me on how to solve this or provide any reference where I can get it. Thank you!
Using the method of Henderson Plaschko, we can derive the following results,
$$x(t) = x(0) cos(t\sqrt{a}) + \frac{y(0)}{\sqrt{a}} sin(t\sqrt{a}) + \frac{c}{\sqrt{a}} \int ^t _0 sin((t-s)\sqrt{a}) sin(\omega s)ds - \frac{1}{\sqrt{a}} \int ^t _0 x(s) sin((t-s)\sqrt{a}) dB_s$$
$$y(t) = -\sqrt{a}x(0) sin(t\sqrt{a}) + y(0) cos(t\sqrt{a}) + c \int ^t _0 cos((t-s)\sqrt{a}) sin(\omega s)ds - \int ^t _0 x(s) cos((t-s)\sqrt{a}) dB_s$$
With the mean values we can write:
$$x(t) = \langle x(t) \rangle - g_1(t)dB_s$$
$$y(t) = \langle y(t) \rangle - g_2(t)dB_s$$
Giving rise to the variance matrix [note: $dB^2_s = ds$ ]:
$$V_{11} (t) = \frac{1}{a} \int ^t _0 \langle x^2(s) \rangle sin^2((t-s)\sqrt{a}) ds$$
$$V_{12} (t) = V_{21} (t) = \frac{1}{2 \sqrt{a}} \int ^t _0 \langle x^2(s) \rangle sin(2(t-s)\sqrt{a}) ds$$
$$V_{22} (t) = \int ^t _0 \langle x^2(s) \rangle cos^2((t-s)\sqrt{a}) ds$$
Form here we can proceed as given by the book, which is quite a lengthy process. Thank you.