Hello everyone,

I am faced with a problem of having two coupled stochastic differential equations which needs to be solved through applying methods of solution w.r.t to corresponding Fokker-Planks equation, the two equations which is in question are given as:

$\displaystyle \frac{\partial x_{1}(X,t)}{\partial t} = -\frac{1}{2}\left(x_{1}(X,t) -x_{2}(X,t)\right) - \frac{1}{2}\xi(t)(x_{1}(X,t)-x_{2}(X,t))$

and

$\displaystyle \frac{\partial x_{2}(X,t)}{\partial t} = -\frac{1}{2}\left(x_{2}(X,t) -x_{1}(X,t)\right) - \frac{1}{2}\xi(t)(x_{2}(X,t)-x_{1}(X,t))$

Where $\displaystyle \xi(t)$ is a gaussian noise process. For which I obtain the following Fokker-Planck equation:

$\displaystyle \frac{\partial P(X,t)}{\partial t} = - \sum \limits_{i=1}^{2} \frac{\partial }{\partial x_{i}}A_{i}(x)P(X,t) + \frac{1}{2} \sum \limits_{i,j=1}^{2} \frac{\partial^2 }{\partial x_{i}x_{j}}B_{i,j}(x)P(X,t)$

With $\displaystyle A_{i} = \begin{pmatrix} -\frac{1}{2}x_{1}+ \frac{1}{2}x_{2} \\ -\frac{1}{2}x_{2} + \frac{1}{2}x_{1}\end{pmatrix}$

and $\displaystyle B_{i,j} = \begin{pmatrix} - \frac{1}{2}(x_{1}(X,t)-x_{2}(X,t)) & 0 \\ 0 & - \frac{1}{2}(x_{2}(X,t)-x_{1}(X,t)) \end{pmatrix}$

Although this is the point in which I get stuck, I have no idea from where I need to proceed now, can anyone give a hint as to how I can't any further finding a solution?

Kind Regards