Differentiating Stochastic Integral I was wondering how to write the following integral in differential form: $$\int^t_0 f(s,t)dW_s$$ where $W_s$ is a standard Brownian Motion. In my understanding, if $f(s,t)$ can be written as $f(s)g(t)$, we can take $g(t)$ out of the integral and apply Ito's Lemma. However, if we are unable to separate the functional form of $f(s,t)$, how can I differentiate the integral? Thanks for your help!
 A: We look at the integral at time $t+dt$:
\begin{equation}
\begin{split}
\int_0^{t+dt}f(s,t+dt)dW_s &= \int_0^{t+dt}\left[f(s,t)+\partial_t f(s,t)dt \right]dW_s \\
&= \int_0^{t}\left[f(s,t)+\partial_t f(s,t)dt \right]dW_s+\left[f(t,t)+\partial_t f(t,t)dt \right]dW_t
\end{split}
\end{equation}
In the first line we expanded the integrand around $t$ in the second argument of $f$. In the second line we approximated the integral from $t$ to $t+dt$ by its integrand evaluated at $t$. Now we may use the rules of stochastic calculus to drop the term $dt\, dW_t$ and write
\begin{equation}
\begin{split}
\int_0^{t+dt}f(s,t+dt)dW_s &= \int_0^{t}f(s,t)dW_s + dt\cdot \int_0^{t}\partial_tf(s,t)dW_s + f(t,t)dW_t.
\end{split}
\end{equation}
Thus, if we define 
\begin{equation}
\begin{split}
X^{(0)}_t=\int_0^{t}f(s,t)dW_s, \quad X^{(n)}_t=\int_0^{t}f^{(0,n)}dW_s,
\end{split}
\end{equation}
where $f^{(0,n)}$ is the $n$'th derivative of $f$ in the second argument, $X^{(0)}_t$ satisfies the Ito SDE
\begin{equation}
\begin{split}
dX^{(0)}_t=X^{(1)}_t dt+f(t,t)dW_t.
\end{split}
\end{equation}
One remark: In practice this SDE is not very helpful on its own, since $X^{(1)}$ is of the same form as $X^{(0)}$ and its SDE depends on $X^{(2)}$, and so on. This generates a closure problem, requiring us to always know one of the integrals beforehand. However, if $f^{(0,n)}\equiv 0$ for some positive integer $n$, we get a closed system of SDEs
\begin{align}
dX^{(0)}_t&=X^{(1)}_t dt+f(t,t)dW_t,\\
dX^{(1)}_t&=X^{(2)}_t dt+f^{(0,1)}(t,t)dW_t,\\
\vdots\quad & \qquad\qquad\vdots\\
dX^{(n-2)}_t&=X^{(n-1)}_t dt+f^{(0,n-2)}(t,t)dW_t,\\
dX^{(n-1)}_t&=f^{(0,n-1)}(t,t)dW_t.
\end{align}
