Stochastic differential equation of a Brownian Motion

Question

I have two questions about Ito's Lemma with respect to calculating SDEs. The examples are simple enough, but I haven't found an answer yet.

Take $W_t$ as a standard Brownian motion and $g(s)$ as some function of $s$. Assume that all regularities etc. are fulfilled and take $F$ as some function. I know that if $F = \int_0^tg(s)dW_s$, then the corresponding SDE is $dF = g(t)dW_t$. However, applying Ito's Lemma, I'm not sure how this SDE is derived. I am unsure about the next part:

• $dF = \underbrace{\frac{\partial F}{\partial t}}_{=g(t)dW_t}dt + \underbrace{\frac{\partial F}{\partial W_t}}_{=g(t)}dW_t + \underbrace{\frac{1}{2}\frac{\partial^2 F}{\partial W_t^2}}_{=0}dt = g(t)dW_tdt + g(t)dW_t = g(t)dW_t$

Question 1: is $\frac{\partial F}{\partial t} = g(t)dW_t$ correct? Or should this be zero?

Now take $F=\int_0^tW_s^2dW_s$. My approach would be:

• $dF = \underbrace{\frac{\partial F}{\partial t}}_{=W_t^2dW_t}dt + \underbrace{\frac{\partial F}{\partial W_t}}_{=W_t^2}dW_t + \underbrace{\frac{1}{2}\frac{\partial^2 F}{\partial W_t^2}}_{=W_t}dt = W_t^2dW_t+W_tdt$

Question 2: Are the partial derivatives in the above example correct?

Answer 1

In stochastic calculus, only stochastic integrals are defined. The differential form is just a notation. That is, $$dF=g(t)dW_t$$ is just another expression for the integral $$F=\int_0^t g(s) dW_s.$$ See, for example, in this book or this book, all Ito's lemmas are expressed in integral forms.

For your question, note that $F$ is not a function of $t$ and $W_t$, that is, it is not of the form $F(t, W_t)$. In fact, it depends on the whole path of $W_s$ from $0$ to $t$. Then Ito's lemma can not be applied to $F$. The application for both of your questions are incorrect.