# Using approximations of approximations to define the Itô integral

In Oksendal's book of Stochastic Differential Equations, the author does a reasoning similar to what's presented below.

The follwing three points are proved:

1. I can find $$\{\phi_n\}$$ such that $$E(\int_S^T(g-\phi_n)^2dt)\rightarrow 0$$, as $$n\rightarrow \infty$$.
2. I can find $$\{g_n\}$$ (each $$g_n$$ is a $$g$$ from point 1) such that $$E(\int_S^T(h-g_n)^2dt)\rightarrow 0$$, as $$n\rightarrow \infty$$.
3. I can find $$\{h_n\}$$ (each $$h_n$$ is an $$h$$ from point 2) such that $$E(\int_S^T(f-h_n)^2dt)\rightarrow 0$$, as $$n\rightarrow \infty$$.

Then the author states that by the three points above, for any $$f$$ there is a sequence of $$\{\phi_n\}$$ such that $$E(\int_S^T(f-\phi_n)^2dt)\rightarrow 0$$, as $$n\rightarrow \infty$$.

How is he able to state that? I think he's probably using some simple relationship/inequality which allows him to conclude that, and I'm not getting it.

All of these functions have their own properties, stated in the book, which I didn't write, because in my opinion they would make the question bigger, without improving it.

He is essentially using the triangle inequality.

Fix some function $$f$$. Then by, 3., there is some $$h_n$$ with $$\sqrt{\mathbb{E}\int_S^T (f-h_n)^2 \, dt} \leq \frac{1}{3n}$$. Now, by 2., there is some $$g_n$$ with $$\sqrt{\mathbb{E}\int_S^T (h_n-g_n)^2 \, dt} \leq \frac{1}{3n}$$. Finally, by 1., there is $$\phi_n$$ with $$\sqrt{\mathbb{E}\int_S^T (g_n-\phi_n)^2 \, dt} \leq \frac{1}{3n}$$. Since $$\sqrt{\mathbb{E}\int_S^T f(t)^2 \, dt}$$ is a norm (the $$L^2$$-norm with respect to the product measure $$\mathbb{P} \otimes \lambda$$), it satisfies the triangle inequality. Writing $$f-\phi_n = (f-h_n)+(h_n-g_n)+(g_n-\phi_n)$$ we thus get

\begin{align*} \sqrt{\mathbb{E}\int_S^T (f-\phi_n)^2 \, dt} &\leq \sqrt{\mathbb{E}\int_S^T (f-h_n)^2 \, dt} + \sqrt{\mathbb{E}\int_S^T (h_n-g_n)^2 \, dt} \\ &\quad + \sqrt{\mathbb{E}\int_S^T (g_n-\phi_n)^2 \, dt} \\ &\leq 3 \frac{1}{3n}\end{align*}

and so $$\sqrt{\mathbb{E}\int_S^T (f-\phi_n)^2 \, dt} \to0$$.

Remark: You can also read the statements a different (more analytic) way. Say, you have a normed space $$X$$ and two subsets $$C$$ and $$D$$. If $$C$$ is dense in $$D$$ and $$D$$ is dense in $$X$$, then $$C$$ is dense in $$X$$. Of course, you can iterate this (i.e. take another set which is dense in $$C$$ and so on). That's exactly Oksendal uses in his book. Statement 3. says that the functions $$h$$ (with certain properties) are dense (w.r.t. $$L^2$$-norm) in your original space of functions $$f$$. Statement 2. says that the functions $$g$$ (with certain properties) are dense (w.r.t. $$L^2$$-norm) in the set of functions $$h$$, and so on.

• Saz, thanks for the answer. I really appreciate it. I still have some questions though... 1) - The 2-norm as is defined in the book is $E(|X|^2)^{1/2}$. How does one go about to see that yours is also a norm? If we could use the Itô isometry, than I understand. However, at that point in the book, we only have isometry for simple functions. 2) - In the approximations that you use, shouldn't we use subsequences, i.e., for each $h_n$, we have $g_{n_i}$, and for each $g_{n_i}$ we can approximate it by $\phi_{n_{j_i}}$ 3) - I think there's a typo in the formula following "there's a $g_n$" – An old man in the sea. Jun 30 at 17:26
• @Anoldmaninthesea. 1) This has nothing to do with Ito^'s isometry. For any measure $\mu$ the mapping $f \mapsto \sqrt{\int f^2 \, d\mu}$ defines a norm. Here, in our case, $\mu$ is the product measure of the probability measure $\mathbb{P}$ and Lebesgue measure $\lambda$. 2) No, we do not use subsequences here ... we do the whole argumentation for fixed $n$. E.g. after we have chosen $h_n$, we need to find some function $g$ "(with the properties of the "g-functions", which you did not state explitily) such that $\sqrt{\mathbb{E}\int |h_n-g|^2 \, dt} \leq \frac{1}{3n}$. This is possible because – saz Jun 30 at 18:30
• of 2. (If you prefer think of $h$ instead of $h_n$... by 2. you can approximate this by some sequence of "g-functions"... in particular there is some $g$ (...or rather $g_n$) which is close to $h$ (i.e. $h_n$) in $L^2$ norm). Did you think about my remark? From my point of view, it's much easier to read the statements as denseness of certain subsets. 3)Yeah, right – saz Jun 30 at 18:32
• Many thanks! Now I understand! ;) Btw, I did read your remark, and found it very intuitive. I just wasn't sure how it should be interpreted at the light of your answer. Now, I know. Thanks ;) +1 – An old man in the sea. Jun 30 at 19:14