Solution of SDE that is almost geometric

Question

Let the SDE be defined by $$dS_t=\frac{l-S_t}{\tau} dt+mS_t dB_t$$

I tried solving it using the given general solution on wikipedia page with $c_t= \frac{l}{\tau}$, $a(t)=\frac{-1}{\tau}$, $m=b(t)$, $d(t)=0$. Then $\Phi$ equals $$\exp\left( \frac{t}{\tau} - \frac{m^2t}{2} +m W_t\right)$$ and so $$S_t=-\tau S_0\exp\left( \frac{t}{\tau} - \frac{m^2t}{2} +m W_t\right)-\tau S_0e^{-t/\tau}$$

Is this correct? where can I read more about this type of SDE? https://en.wikipedia.org/wiki/Stochastic_differential_equation

Answer 1

You can use the general formula from here Solution to General Linear SDE for $dX_t = \big( a(t) X_t + b(t) \big) dt + \big( g(t) X_t \big) dB_t,$

\begin{align*} X_t =& \exp\left( t\left( \frac{\ell}{\tau}- \frac{1}{2\tau^{2}}\right) + \frac{-1}{\tau}B_{t}\right) \\ &\cdot \left(X_0+ \int_0^{t} \frac{\ell}{\tau}\exp\left( -s \left( \frac{\ell}{\tau}- \frac{1}{2\tau^{2}}\right) + \frac{1}{\tau}B_{s}\right)\mathrm{d}s\right). \end{align*}