Why do we unavoidably (or not) use Riemann integral to define Itô integral?

Solution 1:

First of all, note that the limit

$$\int_0^t H(s) \, dB_s = \lim_{n \to \infty} \sum_{i=1}^n H_{t_i} (B_{t_i}-B_{t_{i-1}}) \tag{1}$$

is a limit in $L^2$ (or, alternatively, in probability); the limit does, in general, not exist in a pointwise sense. This means, in particular, that the existence of this limit does not imply that the function $t \mapsto H_t(\omega)$ is Riemann-integrable for each $\omega \in \Omega$. This is really something you should be careful about when working with such limits: in which sense does the limit exist?

Let's turn to the question why we cannot introduce the Itô integral in a pointwise sense. There is the following general statement which is a direct consequence of the Banach-Steinhaus theorem (see below for a detailed proof):

Theorem Let $\alpha:[a,b] \to \mathbb{R}$ be a mapping. If the Riemann-Stieltjes integral $$I(f) := \int_a^b f(t) \, d\alpha(t)$$ exists for all continuous functions $f:[a,b] \to \mathbb{R}$, then $\alpha$ is of bounded variation.

This means that if we define a stochastic integral with respect to a stochastic process $(X_t)_{t \geq 0}$ in a pointwise sense

$$\int_0^t f(s) \, dX_s(\omega), \qquad \omega \in \Omega$$

as a Riemann-Stieltjes integral, then this integral is only well-defined for all continuous functions if $t \mapsto X(t,\omega)$ is of bounded variation (on compacts) for all $\omega \in \Omega$. However, it is well-known that the sample paths of a Brownian motion are almost surely of unbounded variation, and therefore the definition of a stochastic integral in a pointwise sense is not a good idea: the class of functions which we can integrate would not even include the continuous functions.

Proof of the above theorem: The idea for this proof is taken from R. Schilling & L. Partzsch: Brownian Motion - An Introduction to Stochastic Processes, Corollary A.41.

Consider the normed spaces $$(X,\|\cdot\|_X) := (C[a,b],\|\cdot\|_{\infty}) \qquad \text{and} \qquad (Y,\|\cdot\|_Y) := (\mathbb{R},|\cdot|).$$ For a partition $\Pi = \{a=t_0<\ldots<t_n=b\}$ of the interval $[a,b]$ define $I^{\Pi}:X \to Y$ by

$$I^{\Pi}(f) := \sum_{t_j \in \Pi} f(t_{j-1}) (\alpha(t_j)-\alpha(t_{j-1})).$$

Since, by assumption, $I^{\Pi}(f) \to I(f)$ as $|\Pi| \to 0$ for all $f \in C[a,b]$, we have

$$\sup_{\Pi} |I^{\Pi}(f)| \leq c_f < \infty$$

for all $f \in C[a,b]$. Applying the Banach Steinhaus theorem we find

$$\sup_{f \in C[a,b], \|f\|_{\infty} \leq 1} \sup_{\Pi} I^{\Pi}(f) < \infty$$

which is equivalent to

$$\sup_{\Pi} \sup_{f \in C[a,b],\|f\|_{\infty} \leq 1} |I^{\Pi}(f)| < \infty. \tag{2}$$

For a partition $\Pi$ let $f_{\Pi}:[a,b] \to \mathbb{R}$ be a piecewise linear continuous function such that

$$f_{\Pi}(t_{j-1}) = \text{sgn} \, (\alpha(t_j)-\alpha(t_{j-1}))$$

for all $j=1,\ldots,n$. By the very choice of $f_{\Pi}$, we have

$$I^{\Pi}(f_{\Pi}) = \sum_{t_j \in \Pi} |\alpha(t_j)-\alpha(t_{j-1}))| \leq \sup_{f \in X, \|f\|_X \leq 1} |I^{\Pi}(f)|. \tag{3}$$

Combining $(2)$ and $(3)$, we conclude

$$\text{VAR}_1(\alpha,[a,b]) = \sup_{\Pi} I^{\Pi}(f_{\pi}) \leq \sup_{\Pi} \sup_{f \in C[a,b],\|f\|_{\infty} \leq 1} |I^{\Pi}(f)| < \infty.$$