We know from Itô's lemma that: $$\int_0^T w_t \, dw_t=\frac{w_T^2}{2}-\frac{T}{2},$$ where $w$ is a Wiener process. I don't know how we can interpret the equality!? It is equal for all $\omega\in\Omega$? Or in distribution? Or how? I would say it is true for (almost?) every $\omega$, because the left side of the equation is $$\int_0^T w_t \, dw_t \overset{\circ}{=} \lim_{n\rightarrow\infty}^p \sum_k w_{t_k} \left[w_{t_{k+1}}-w_{t_k}\right],$$ by definition, where $0=t_0\leq t_1\leq t_2\leq\ldots\leq t_n=T$, and in this definition we use the same Wiener process, just like in the right side of the first equation.
So is it true if $\omega=\bar{\omega}$ is fixed, then $$\left(\int_0^T w_t \, dw_t \right) \left(\bar{\omega}\right) = \frac{w_T^2 \left(\bar{\omega}\right)}{2}-\frac{T}{2}\text{?}$$
There are some definitions about the equality of stochastic processes, what can we say about theme in this case? It is just an example, but I am rather interested in how we can interpret the equality in Itô's lemma!? (I hope I ask the same question and it is a relevant example.)
Another train of thought motivated the previous question: Let the following series of random variables be:$$\xi_1,\xi_2,\xi_3,\ldots$$ which is convergent in stochastic convergence. It also means that there exists a $\xi$ random variable, where $$\mathbf{P}\left(\omega:\left|\xi_k-\xi \right| > \varepsilon \right) \longrightarrow 0$$ as $k\rightarrow\infty$. But what is that $\xi$? Of course, if we have a conjecture about $\xi$ and we want to know that $\xi$ is a proper limit: $\lim_{k\rightarrow\infty}^p \xi_k = \xi$, then we should check if the previous $\mathbf{P}\left(\omega:\left|\xi_k-\xi\right| > \varepsilon\right) \longrightarrow 0$ property holds. But if we only know that $(\xi_k)_k$ is convergent in the stochastic way, then how can we "construct" this $\xi$? What can we say if $\xi$ is a random process? The convergence should hold for every $t\in[0,T]$ where the process is defined?
Sorry for the lot of questions, but I think they belong to the same topic.