Analytic solutions to a Stochastic Differential Equation

Question

I want to solve an SDE as follows: $$ dX_t = \alpha(\beta - X_t)dt + dB_t,\quad X_0 = x_0 $$ where $\alpha$, $\beta$ are positive constants and B_t is a Browian motion independent to $X_t$.Is there a way to solve it? Thank in advance for any help or some references to this kind of equation.

Answer 1

Let assume this $$ Y_t = X_t\mathrm{e}^{\alpha t} $$ then using Ito we find $$ dY_t = \alpha X_t\mathrm{e}^{\alpha t} dt + \mathrm{e}^{\alpha t}dX_t $$ with some rearrangement we can find $$ \mathrm{e}^{-\alpha t} dY_t = dX_t + \alpha X_t dt $$ which looks awfully like what you have as the original SDE above except we have $$ dX_t = \alpha\beta dt -\alpha X_t dt + dB_t $$ thus we can easily see $$ \mathrm{e}^{-\alpha t} dY_t = \alpha\beta dt + dB_t $$ thus $$ dY_t = \alpha\beta \mathrm{e}^{\alpha t} dt + \mathrm{e}^{\alpha t}dB_t $$ integrating we get $$ Y_t = Y_0 + \beta\left(\mathrm{e}^{\alpha t} - 1\right) + \int_0^t\mathrm{e}^{\alpha s}dB_s $$ or $$ X_t = x_0 + \beta\left(1-\mathrm{e}^{-\alpha t}\right) + \int_0^t\mathrm{e}^{\alpha (s-t)}dB_s $$