I have an equation like $dX(t)/dt = f(X(t),t)+\int_{0}^t dW_s$. I was wondering if there is a way to solve it (even in the simple case like $f(X(t),t) = g(X(t))h(t)$). Any hint will be much appreciated!
EDIT: I found this paper Second order stochastic differential equations with Dirichlet boundary conditions. But I would like to have a Cauchy boundary condition
\begin{align} \frac{dX_t}{dt} &= f(X_t,t) + \int_0^tdW_s \\ &:= f(X_t,t) + dW_t \\ &\Longleftrightarrow \\ dX_t &= f(X_t,t)dt + dW_tdt. \end{align} Now since $dW_tdt = 0$ we have that the stochastic part vanishes, so
\begin{align} dX_t &= f(X_t,t)dt \\ &\Longleftrightarrow \\ \frac{dX_t}{dt} &- f(X_t,t) = 0, \\ \end{align} which is a differential equation of first order. The solution depends on the form of $f$
If $f$ is separable as $f(X_t,t)=g(X_t)h(t)$, that leads to \begin{align} \frac{1}{g(X)}dX_t = h(t)dt, \end{align} which can be solved following separation of variables.