I am studying different approaches to solving stochastic differential equations. I keep seeing Ito Calculus being described as a probabilistic approach to solving SDEs, as opposed to being a pathwise approach. Similarly, the Theory of Rough Paths is being described as a deterministic pathwise approach.
I am confused as to what it means for an approach to be pathwise when integrating a stochastic process driven by Brownian Motion. How is it possible to just ignore/remove the probabilistic/random qualities of an SDE. Does this mean that we a fix a random path/trajectory of the Brownian Motion and integrate simply with respect to that?
Thanks
It is a good idea to simulate pathwise the (strong) solution of the SDE $$dX_t=X_t\,dW_t$$ by discretizing using, say, the Euler scheme: $$ X_{t_i}=X_{t_{i-1}}+X_{t_{i-1}}\sqrt{t_i-t_{i-1}}\,\varepsilon_k $$ where you draw $\varepsilon_k$ independently from $N(0,1)$ every time you need one such $\varepsilon_k$ (you will need $nm$ such draws when your time interval has $n$ points and you simulate $m$ paths).
That SDE has the advantage of having a closed form solution $$ X_t=X_0\,e^{W_t-t/2} $$ which you can compare with the simulation.
In short: strong solutions of SDEs can be understood as pathwise solutions: every path $\omega$ of $t\mapsto W_t(\omega)$ of the given Brownian motion leads to a path $t\mapsto X_t(\omega)$ of the solution.
Rough paths seems to be a topic for specialists in control theory. I know nothing about that.