Ito Calculus
Buildup
Before discussing stochastic integrals, it is crucial to define an essential probabilistic process called the Elementary Process. Elementary processes play a similar role to simple functions, which were necessary for defining the Lebesgue integral in [Measure Theory](../../categories/Measure Theory).
$$ a = t_{0} < t_{1} < \cdots < t_{k} = b $$ Considering such a partition in the Natural Domain $[a,b]$, an Elementary Process is defined as follows for indicator functions $\chi$ and $\mathcal{F}_{t_{j}}$-measurable functions (random variables) $e_{j}$: $$ \phi (t,\omega) := \sum_{j=0}^{k-1} e_{j} (\omega) \chi_{ \left[ t_{j} , t_{j+1} \right] } (t) $$ Integrating this function with the Wiener Process $W(t)$ can be thought of in the same vein as the Riemann Sum: $$ \int_{a}^{b} \phi (t,\omega) d W_{t} (\omega) = \sum_{j=0}^{k-1} e_{j} (\omega) \left[ W_{t_{j+1}} - W_{t_{j}} \right] ( \omega ) $$ Based on this, the following Stochastic Integral is defined.
Definition 1
The Itô Integral is defined as follows $f \in m^{2} [a,b]$. $$ \int_{a}^{b} f (t,\omega) d W_{t} (\omega) := \lim_{n \to \infty} \int_{a}^{b} \phi_{n} (t,\omega) d W_{t} (\omega) $$ Here, the sequence $\left\{ \phi_{n} \right\}_{n \in \mathbb{N}}$ is a sequence of elementary processes that satisfy the following: $$ \lim_{n \to \infty} E \left[ \int_{a}^{b} \left( f (t,\omega) - \phi_{n} (t,\omega) \right)^{2} dt \right] = 0 $$
Explanation
In the definition, $\left\{ \phi_{n} \right\}_{n \in \mathbb{N}}$ can be specifically chosen in any way as long as it satisfies condition $E \int [f-\phi_{n}]^{2} dt \to 0$.
Basic Properties 2
Let us assume $f, g \in m^{2} [a,b]$ and a given filtration $\left\{ \mathcal{F}_{t} \right\}_{t \ge 0}$.
- [1] Measurability: $\displaystyle \int_{a}^{b} f d W_{t} = \left( \int_{a}^{b} f d W_{t} \right) (\omega)$ is $\mathcal{F}_{b}$-measurable.
- [2] Linearity: For a constant $c$, $$ \int_{a}^{b} \left( c f + g \right) d W_{t} = \int_{a}^{b} c f d W_{t} + \int_{a}^{b} g d W_{t} $$
- [3] Additivity: For $a < c < b$, $$ \int_{a}^{b} f d W_{t} = \int_{a}^{c} f d W_{t} + \int_{c}^{b} f d W_{t} $$
- [4] Normality: If $f$ is independent with $\omega \in \Omega$, in other words, if $f$ is deterministic $$ \int_{a}^{b} f d W_{t} \sim N \left( 0, \int_{a}^{b} \left( f \right)^{2} dt \right) $$
- [5] Bounded Random Variables: If $Z$ is $\mathcal{F}_{b}$-measurable, then $Z f \in m^{2}[a,b]$ and the following holds: $$ \int_{a}^{b} Z f (t) d W_{t} = Z \int_{a}^{b} f (t) d W_{t} $$
- [6] Expectation: For the sub-sigma field $\mathcal{F}_{a}$ $$ E \left[ \int_{a}^{b} f d W_{t} \right] = E \left[ \int_{a}^{b} f d W_{t} | \mathcal{F}_{a} \right] = 0 $$ and, regarding $f,g$, the following holds: $$ E \left( \int_{a}^{b} f(t) d W_{t} \int_{a}^{b} g(t) d W_{t} \right) = E \left( \int_{a}^{b} f(t) g(t) d W_{t} \right) $$
- [7] Itô Isometry Equality: $$ E \left( \left| \int_{a}^{b} f W_{t} \right|^{2} \right) = E \left( \int_{a}^{b} \left| f \right|^{2} W_{t} \right) $$ This also applies to Conditional Expectations, satisfying the following: $$ E \left( \left| \int_{a}^{b} f d W_{t} \right|^{2} | \mathcal{F}_{a} \right) = E \left( \int_{a}^{b} \left| f \right|^{2} d W_{t} | \mathcal{F}_{a} \right) = \int_{a}^{b} E \left( \left| f \right|^{2} | \mathcal{F}_{a} \right) d W_{t} $$
- [8] Let’s consider $f \in m^2$ and $\left\{ f_{n} \right\}_{n \in \mathbb{N}} \subset m^{2}$. If $n \to \infty$, then $$ E \left[ \int_{a}^{b} \left( f_{n} - f \right)^{2} dt \right] \to 0 $$ it converges as per $\mathcal{L}_{2}$ Convergence when $n \to \infty$. $$ \int_{a}^{b} f_{n} W_{t} \to \int_{a}^{b} f W_{t} $$
- $\mathcal{F}_{t}$ being a sub-sigma field of $\mathcal{F}$ means both are Sigma Fields of $\Omega$, but $\mathcal{F}_{t} \subset \mathcal{F}$ applies.
- $f$ being a $\mathcal{F}_{t}$-measurable function means that for all Borel Sets $B \in \mathcal{B}([0,\infty))$, $f^{-1} (B) \in \mathcal{F}_{t}$ applies.