Finding an SDE satisfied by the stochastic process $X_t=\int_{0}^{t}f(u,t)dW_{u}$

Question

I am wondering about stochastic processes of the form

$$X_t=\int_{0}^{t}f(u,t)dW_{u}$$

I can't figure out how to convert this into a stochastic differential equation if there were no t dependence on f I could have written $$dX_t=f(t) dW_{t} $$ But I don't know how to do it for f(u,t). What is the SDE satisfied by this process?

Answer 1

First starting from functions of the form

$$f(u, t) =a(u) b(t) $$

$$X_t=b(t) \int_{0}^{t}a(u)dW_{u}$$

Using Ito's product rule ,we get

$$dX_t= (\int_{0}^{t}a(u)b'(t) dW_{u}) dt+(a(t) b(t)) dW_t$$

Equivalently

$$dX_t= (\int_{0}^{t}\frac{\partial f} {\partial t} dW_{u}) dt+f(t, t) dW_t$$

I understand this much but I am not aware of any method for getting an SDE if the function is not separable.