# Ito Formula for Stochastic Integral

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?

• Are you missing a $\partial_t y_t$? – Chinny84 Feb 8 '16 at 7:30
• @Kenneth Chen : well as long as your process S is well defined (i.e. the sde has a solution) your process $y_t$ is correctly defined and the differential form is simply : $dy_t=S_tdt+S_tdW_t$ no more, no less. Your formulas are certainly all wrong. Best regards – TheBridge Feb 8 '16 at 22:25
• @TheBridge Thanks! I've noted that. – Kenneth Chen Feb 8 '16 at 22:27