📂Stochastic Differential Equations

m2 Space

Definition ^{1 2}

Given that there is a probability space $( \Omega , \mathcal{F} , P)$,

1. A sequence of sub sigma fields $\left\{ \mathcal{F}_{t} \right\}_{t \ge 0}$ of $\mathcal{F}$ is called a Filtration if it satisfies the following: $$ \forall s < t, \mathcal{F}_{s} \subset \mathcal{F}_{t} $$
2. A stochastic process $g(t,\omega) : [0,\infty) \times \Omega \to \mathbb{R}^{n}$ is said to be $\mathcal{F}_{t}$-Adapted if for all $t \ge 0$, $\omega \mapsto g (t,\omega)$ is $\mathcal{F}_{t}$-measurable.
3. For an interval $I := [a,b]$, the set of functions $f$ that satisfy the following three conditions is denoted as $m^{2} = m^{2} [a,b]$. This $I$, in particular, is called the Natural Domain of the Ito Integral.
   • (i): $\mathcal{B}$ is $\mathcal{B} \times \mathcal{F}$-measurable with respect to the Borel sigma field of $[0, \infty)$.
   • (ii): $f (t,\omega)$ is $\mathcal{F}_{t}$-Adapted.
   • (iii): It has the structure of a Hilbert space, i.e., $$ \left\| f \right\|_{2}^{2} \left( [a,b] \right) = E \left( \int_{a}^{b} \left| f(t,\omega) \right|^{2} dt \right) < \infty $$

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• For $\mathcal{F}_{t}$ to be a sub sigma field of $\mathcal{F}$ means both are sigma fields of $\Omega$, but $\mathcal{F}_{t} \subset \mathcal{F}$.
• A function $f$ is $\mathcal{F}_{t}$-measurable means for all Borel set $B \in \mathcal{B}([0,\infty))$, $f^{-1} (B) \in \mathcal{F}_{t}$.

Explanation

Given a filtration, saying a stochastic process $f$ is $\mathcal{F}_{t}$-measurable means it encompasses the history or information up to time $t$. Since a filtration is a sequence of increasingly larger sub sigma fields, it aligns with the notion that information grows over time.

Naturally, the naming of $m^{2}$ space comes from the $L^{2}$ space as seen in condition (iii).