I have an SDE $$X_t := (A_tX_t+a_t)dt+(S_tX_t+\sigma_t))dB_t,$$ $X_0=x_0$ and $A,a,\sigma,S$ are continuous stochastic processes, $B$ is a BM.
Now if I define: $$Y_t:=e^{(\int_0^tA_sds+\int_0^tS_sdB_s-\frac{1}{2}\int_0^tS_s^2ds)}$$ and $$Z_t:=x_0+\int_0^t\frac{a_s-S_s\sigma_s}{Y_s}ds+\int_0^t\frac{\sigma_s}{Y_s}dB_s.$$
How can I show that $X_t:=Y_tZ_t$ is a strong solution to the above SDE?
I was thinking of using Ito's product rule which gives me $$X_t=Y_tZ_t=Y_0Z_0+\int_0^tZ_dY_s + \int_0^tY_sdZ_s+<Z,Y>_t$$ and then showing that this equals my equation. But I don't know how to calculate the RHS.