Since by Ito's lemma you have for $\phi(t,x) \in \mathcal C ^{1,2}$

$$ \phi(t, X_t) = \phi(o,X_0) + \int _0 ^t (\partial_t +b\partial_x +\frac{\sigma^2}{2}\partial^2_{xx})\phi(s,X_s) ~ ds + \int_0^t\partial_x\phi(s,X_s) ~dB_s$$
 if $X$ is a Ito's diffusion

$$ X_t = X_0 + \int _0 ^t b~ ds + \int_0^t\sigma ~dB_s$$

you must search for the simples $\phi$ such that $\partial_x \phi =x $ with $X =B$ ( so $b=0$ and $\sigma =1$) for the first integral and  $\partial_x \phi =f $ for the second one also with $X =B$ .

Indeed, 
$$ \int_0^T B_s ~dB_s=\frac{1}{2}(B^2_T-B^2_0 -T)$$
and 
$$ \int_0^T f(B_s) ~dB_s=F(B_T)-F(B_0) -\frac{1}{2}\int_0^T\partial_x f (B_s) ~ds$$

where $F$ is a primitive of $f$.

How to solve this stochastic integrals?
At the mean-field level, the dynamics of a polariton condensate can be described by a type of nonlinear Schrodinger equation (Gross-Pitaevskii-type), for a classical (complex-number) wavefunction $\psi_{LP}$. Its form in momentum space reads:

\begin{multline}
i \frac{d}{dt}\psi_{LP}(k) =\left[\epsilon(k) 
-i\frac{\gamma(k)}{2}\right] \psi_{LP}(k) 
+F_{p}(k)\,\, e^{-i\omega_{p}t}  \\
+  \sum_{q_1,q_2} g_{k,q_1,q_2}\, \psi^{\star}_{LP}(q_1+q_2-k)
\, \psi_{LP}(q_1)\, \psi_{LP}(q_2).
\end{multline}

The function $\epsilon(k)$ is the dispersion of the particles (polaritons). The polaritons are a non-equilibrium system, due to their finite lifetime (damping rate $\gamma$). Therefore, they need continuous pumping with amplitude $F_p$ at energy $\omega_p$.
Finally, there exists a momentum-dependent nonlinear interaction $g_{k,q_1,q_2}$ that depends of the so-called Hopfield coefficients $X$ (simple functions of momentum) as:

\begin{equation}
g_{k,q_1,q_2}=g\, X^{\star}(k)\, X^{\star}(q_1+q_2-k)\, X(q_1)\, X(q_2)
\end{equation}

How can one transform the equation for $\psi$ to real-space?
