I am currently taking a course on Stochastic Analysis, and in the chapter entitled It$\hat{o}$'s Calculus, I am introduced to this integral: $$ \int^{t}_{0}Y_s\,dW_s$$ as a stochastic process, where $Y_s$ is a simple process. I have understood up to now what a Brownian Motion is, what a Martingale, Filtration, probability space, etc… is. But this, I am really stumped on. The notion of $dW_s$ is confusing me, and why this integral is so important in the grand scheme of things? What is the significance of this integral, and what exactly is $Y_s$? I feel that I can't move on with this course until I understand the very essentials of what Ito calculus is and why we are concerned with it. A answers that shed some light on this topic would be vastly appreciated.
It$\hat{o}$’s stochastic integral
brownian motionstochastic-processes
Related Solutions
There are two notions that are getting mixed up here.
When people speak of a Brownian motion on the real line (or more generally of a martingale on the real line) they usually refer to a martingale that is indexed by $\mathbb{R}$ with $\lim_{t \to - \infty} X_{t} = 0$. In that case the fact that you index your process with $[0,\infty)$ or $\{-\infty\} \cup \mathbb{R}$ does not change anything and all the formula you know for martingales or Brownian motion stay valid in this setting. This is what Peccati is referring to in his paper you mentioned.
On the other hand, a two-sided Brownian motion cannot be a martingale nor even a local martingale in any filtration and you therefore cannot perform stochastic integration with respect to it. There was some research in that direction in this paper if you are interested.
The reason for using Ito's formula is that we know stochastic integrals are martingales (up to some finite variance assumptions that can usually be ignored when you're first learning the material).
As an example, let $M_t = e^{B_t - \frac 12 t}$ where $B$ is a Brownian motion. To use Ito's formula, we can write $f(t,x) = e^{x - \frac 12 t}$ and $M_t = f(t,B_t)$. We compute \begin{align*} \partial_t f(t,x) &= -\frac 12 f(t,x) \\ \partial_x f(t,x) &= f(t,x) \\ \partial_{xx} f(t,x) &= f(t,x). \end{align*}
Then Ito's formula says \begin{align*}dM_t &= \partial_t f(t,B_t) dt + \partial_x f(t,B_t)dB_t + \frac 12 \partial_{xx}f(t,B_t)dB_tdB_t \\ &= -\frac 12 f(t,B_t) dt + f(t,B_t)dB_t + \frac 12 f(t,B_t)dt \\ &= f(t,B_t)dB_t. \end{align*}
Since the differential (not the derivative) of $M_t$ contains no $dt$ terms and only contains a $dB_t$ term, we conclude that $M_t$ is a martingale.
This was a very simple example and we could've checked that $M$ is a martingale without using Ito's formula, but for more complicated functions it is usually the easiest way to check. As for resources for learning it, I would recommend Steven Shreve's Stochastic Calculus for Finance II. Even if you aren't interested in financial mathematics, it contains a very approachable introduction to Ito calculus.
Best Answer
You can think of the Itô integral much like the Riemann-Stieltjes integral: you are integrating with respect to a Wiener process. In general, you need $Y$ to be a locally square-integrable process which is adapted to the filtration generated by $W$. It seems you know what $W$ is, but for the sake of completeness: $W$ is usually a Brownian motion, but in general it can be a semimartingale. Itô integrals are important for giving a rigorous meaning to stochastic differential equations, which are extremely useful for modeling all kinds of things. For example, SDE are incredibly important in mathematical finance.
In defining the Itô integral, one generally starts by defining it for simple processes (much like defining the Lebesgue integral for simple functions) and then shows that you can approximate other processes by limits of these simple processes.
Edit: A simple process is like a simple function, it is of the form $$X_t = \sum_j x_j \chi_{E_j}(t),$$ where $E_j$ is a measurable set and $\chi_E$ denotes the characteristic function of the set $E$. For example, one might take $E_j = [t_{j-1},t_j)$. I generally think of a simple functions and simple processes as those with finite image.