Solving the SDE for GBM [closed]

Let's assume that we have the following stochastic differential equation:

$$dX_t = \mu X_t dt + \sigma X_tdW_t$$

and that we have to prove that this is its solution:

$$X_t = X_0 \exp\left(\left(\mu -{\frac {\sigma ^{2}}{2}}\right)t+\sigma W_t\right)$$

I can solve this thanks to the application of Ito's lemma on this deterministic regular function defined in two variables (time and process),

$$f(t, X_t) = \ln(X_t)$$

The point is that I can't understand why the partial derivative of that function with respect to t is equal to zero. Why do we use a function defined in t if it is not differentiable in t?

$$f$$ as you've described it is differentiable in $$t$$ -- the derivative is just equal to zero.
The larger point is that Ito's lemma applies to a broader class of functions $$f(t, X_t)$$ -- the one we've chosen happens to have $$\frac{df}{dt} = 0$$ everywhere.
• Yes. So for example if we choose $f(t,X_t)=ln(X_t)+7t$ then there would be a non-zero derivative with respect to t. But the f you have chosen does not have this explicit dependence on t. The subscript on X does not count. If I rewrite your f function as $f(arg1,arg2)= ln(arg2)$ it is clear there is no dependence on arg1, which you call t, it is what is called in programming an unused input to a function. Oct 20, 2023 at 17:40