I know that the characteristic function $f(t, u) = \mathbb{E}(e^{iuX_t})$ of some random variable $X_t$ depending on $t \geq 0$ has to solve: $$f_t(t, u) = \left(iu - \frac{u^2}{2}\right) f(t, u) + u f_u(t, u) + \frac{u^2}{2}f_{uu}(t, u),$$ with the following boundary conditions: $f(0, u) = e^{iuy}$, $f(t, 0) = 1$ and $|f(t, u)| \leq 1$, for all $u \in \mathbb{R}$, $t \geq 0$.
I would like to obtain an explicit solution of this equation, if possible. It is not possible to use the separation of variables since the boundary conditions are not homogenous. An other idea is to put $f(t, u) = \exp(g(t, u))$, to obtain: $$g_t(t, u) = \left(iu - \frac{u^2}{2}\right) + u g_u(t, u) + \frac{u^2}{2}\left(g_u(t, u)^2 + g_{uu}(t, u)\right),$$ with the following boundary conditions: $g(0, u) = iuy$, $g(t, 0) = 0$ and $g(t, u) \leq 0$, for all $u \in \mathbb{R}$, $t \geq 0$.
Does anyone have an idea how to solve either equation? Thanks a lot!