Please help me in solving this Stochastic Differential Equation for $Y_t$
$$dY_t = a Y_t dt+ b dX_t \qquad Y(0) = c $$
where $a$ and $b$ are constants. Also find the $\mathbb{E}[Y_t]$ and $\text{var}[Y_t]$.
I have tried to write the Ito Lemma for general $Y_t$ function in the two variables $t$ and $X_t$ and then tried to compare the coeficients term of $dt$ and $dX_t$ but not getting any proper result.
Thanks,
Use the integrating-factor approach, where $\mu_t = e^{-a t}$.
Rewrite the original SDE as: $$ d\left( \mu_t Y_t \right) = \mu_t \, b \, dX_t \, .$$
$Y_t$ can be obtained as $$ Y_t = \frac{b}{\mu_t} \int_0^t \mu_s \, dX_s + \frac{\mu_0 Y_0}{\mu_t} = b\, e^{a t} \int_0^t e^{-a s} dX_s + c \, e^{a t} \, .$$