For a simulation in Python, I'd like to numerically solve a PDE of the following form:
$$ du(t,x) = \mu \partial_x u(t,x)dt + \sigma \partial_{xx} u(t,x) dt + \alpha \partial_x u(t,x) dw(t),\quad \text {with}\quad u_t(0)=0 $$
where $\alpha, \sigma, \mu$ are constant, and a Brownian path $w(t)$. The idea behind this is, that for every time change $\bigtriangleup t=t_{i+1}-t_{i}$, the change in $u(t,x)$ is $$\bigtriangleup t(\mu\partial_x u(t,x) + \sigma \partial_{xx} u(t,x)) + (\alpha \partial_x u(t,x))(w(t_{i+1})-w(t_i)).$$
My original approach was applying Fast Fourier Transformation, but I'm only able to get an approximation to the solution without the $w(t)$ part. How is it possible to transform this PDE to a nice form for numerical approximation?