I am new to stochastic differential equations. I would like to solve something like this:
$dx_t = (A - a x_t) dt + (b) dZ_t$
where:
$A = \frac{ - \delta k }{\delta + a} $
The solution is:
$x_t = e^{-at} x_0 + (1 - e^{-at}) \frac{A}{a} + b \int_0^t e^{a(u-t)} dZ_t$
Could someone help me understand what integrating factor I should use?
Consider the integration factor $W_t = e^{a t} x_t$. By Ito's lemma: \begin{align} dW_t &= a e^{a t} x_t dt + e^{a t} dx_t \\ dW_t &= a e^{a t} x_t dt + e^{a t} ((A - a x_t) dt + b dZ_t) \nonumber \\ dW_t &= (a e^{a t} x_t + A e^{a t} - a e^{a t} x_t) dt + b e^{a t} dZ_t \nonumber \\ dW_t &= A e^{a t} dt + b e^{a t} dZ_t \nonumber \end{align} Integrating both sides: \begin{align} \int_0^t d(e^{a u} x_u) &= A \int_0^t e^{a u} dt + b \int_0^t e^{a u} dZ_u \\ e^{a t} x_t &= x_0 + \frac{A}{a} (e^{a t} - 1) + b \int_0^t e^{a u} dZ_u \nonumber \\ x_t &= e^{-a t} x_0 + \frac{A}{a} (1 - e^{- a t}) + b \int_0^t e^{a (u-t)} dZ_u \nonumber \end{align}