Almost sure definition of Ito's lemma but integral terms are defined in $L^2$

Question

I am aware that this question is related to this but I thought it would be useful to clarify a few things.

I am reading this book to get a quick overview of stochastic calculus. It introduces the Ito's lemma in an almost sure sense,

However, the term $\int_0^T F'_x(t,W(t)) dW(t)$ is defined in the book as a limit of sums which converge in $L^2$.

In the proof of Ito's Lemma, the book always mentions that if the convergence is $L^2$, there is a subsequence that converges almost surely.

With this in mind, is it OK then to define stochastic integrals $\int_0^T f(t,W(t)) \,dW(t)$ in the almost sense? I find this confusing since this integral may not even be defined for an abritrary continuous function $f$ especially if $f$ and $W$ don't simultaneously satisfy certain variation conditions. OR, should I interpret it in this manner: Define the stochastic integral in $L^2$ and for those which converge in $L^2$, we can also define the a.s. convergence using the subsequence argument?

Hoping for clarification on the issue. Thanks!

Answer 1

I'm not sure whether I understand your question correctly... so let me know if I got it all wrong.

First of all, notice that Itô's lemma is an identity of the form

$$X=Y+Z \quad \text{a.s.} \tag{1}$$

where $X$, $Y$ and $Z$ are random variables. Here, $X=F(T,W(T))-F(0,W(0))$, $Y$ corresponds to the Riemann integral and $Z$ to the stochastic integral. In order to make sense of $(1)$, it doesn't matter how the integrals are defined - it's really just that: an almost sure equality of random variables.

When proving Itô's lemma, one typically constructs sequences of random variables $(Y_n)_{n \in \mathbb{N}}$ and $(Z_n)_{n \in \mathbb{N}}$ with

• $F(T,W(T))-F(0,W(0)) = Y_n + Z_n$
• $Y_n \to Y$ almost surely (where $Y$ stands for the Riemann integral on the right-hand side of Itô's lemma)
• $Z_n \to Z$ in $L^2$ (where $Z$ stands for the stocahastic integral on the right-hand side of Itô's lemma)

To conclude that $F(T,W(T))-F(0,W(0)) = Y+Z$ one chooses a subsequence $(Z_{n(k)})_{k \in \mathbb{N}}$ of $(Z_n)_{n \in \mathbb{N}}$ such that $Z_{n(k)} \to Z$ almost surely. From

$$F(T,W(T))-F(0,W(0)) = \underbrace{Y_{n(k)}}_{\to Y \, \, \text{a.s.}} + \underbrace{Z_{n(k)}}_{\to Z \, \, \text{a.s.}}$$

one now concludes that $F(T,W(T))-F(0,W(0))=Y+Z$.

Note that the existence of an almost surely convergent subsequence does not imply that we can define the stochastic integral in a pointwise sense, see e.g. this question or this question.