I am new to stochastic differential equations. I would like to solve something like this:
$dx_t = (a) x_t dt + (b) dZ_t$
The solution is:
$x_t = e^{at} x_0 + b e^{at} \int_0^t e^{-au} dZ_u$
I would like to understand the steps to find the solution. All examples I have managed to find are of the type:
$dx_t = (a) x_t dt + (b) x_t dZ_t$
Could someone help me understand the steps or direct me towards a useful resource?
To solve the SDE, consider $W_t = e^{- a t} x_t$. 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 x_t dt + b dZ_t) \nonumber \\ dW_t &= (- a e^{- a t} x_t + a e^{- a t} x_t) dt + b e^{- a t} dZ_t \nonumber \\ dW_t &= b e^{- a t} dZ_t \nonumber \end{align} Integrating both sides: \begin{align} \int_0^t d(e^{- a u} x_u) &= b \int_0^t e^{- a u} dZ_u \\ e^{- a t} x_t &= x_0 + b \int_0^t e^{- a u} dZ_u \nonumber \\ x_t &= e^{a t} x_0 + e^{a t} b \int_0^t e^{- a u} dZ_u \nonumber \end{align}