Suppose I have $$dS_t = \mu(S_t,t) dt + \sigma(S_t,t)dW_t$$ What would be the process satisfying the following process of $y_t$? $$y_t = \int_0^t S_u du + \int_0^t S_u dW_u$$
I'm not quite sure about differentiating $y_t$. The following is what I did $$\frac{\partial y_t}{\partial S_t} =dt +dW_t $$ and $$\frac{\partial^2 dy_t}{\partial dS_t^2} = 0$$
Are these right? Then Ito's Formula gives $$dy_t = (dt+dW_t)dS_t = \sigma(S_t,t)dt $$
But this feels wrong. Can anyone helps me with it please?