# Stochastic Integration

In this below, some results on stochastic integration are given here based on [Karatzas and Shreve 1998]([KS98]).

Fix a probability space ${(\Omega, \mathcal{F}, \mathbb{P}, \{\mathcal{F}_t\})}$ which satisfies the usual conditions.

Definition 1 [KS98, 1.5.1] Let ${X = \{X_t, \mathcal{F}_t; 0\le t<\infty\}}$ be a right-continuous martingale. We say that ${X}$ is square-integrable if ${\mathbb{E}[X_t^2] < \infty}$ for every ${t\ge 0}$. If, in addition, ${X_0 = 0}$ a.s., we write ${X\in \mathcal{M}_2}$ (or ${X\in \mathcal{M}_2^c}$, if ${X}$ is also continuous).

For any ${X \in \mathcal{M}_2}$, ${X^2}$ is a nonnegative submartingale, hence of class DL, and so ${X^2}$ has a unique Doob-Mayer decomposition

$\displaystyle X_t^2 = M_t + A_t; \quad 0\le t<\infty.$

Now, ${\langle X\rangle_t := A_t}$ is referred as quadratic variation of ${X}$. ([KS98, 1.5.3]) The cross-variation process ${\langle X, Y \rangle}$, for any ${X, Y \in \mathcal{M}_2}$, is defined by

$\displaystyle \langle X, Y\rangle_t := \frac 1 4 [\langle X+Y \rangle_t - \langle X-Y \rangle_t]; \quad 0\le t <\infty.$

Following statements explains name of quadratic variation. Let ${\Pi}$ be a partition of ${[0,t]}$, and ${V_t^{(p)} (\Pi)}$ is ${p}$th variation over ${\Pi}$. Then, we have, by [KS98, 1.5.8]

$\displaystyle \lim_{\|\Pi\|\rightarrow 0} V_t^{(2)} (\Pi) = \langle X \rangle_t, \quad \hbox{in probability}.$

A metric structure is imposed on the space ${\mathcal{M}_2}$ and its subspace ${\mathcal{M}_2^c}$ as follows, under which ${\mathcal{M}_2}$ is a complete metric space, and ${\mathcal{M}_2^c}$ is a closed subspace of ${\mathcal{M}_2}$.

Definition 2 [KS98, 1.5.22] For any ${X\in \mathcal{M}_2}$ and ${0\le t <\infty}$, we define

$\displaystyle \|X\|_t := \sqrt{\mathbb{E}[X_t^2]}.$

We also set

$\displaystyle \|X\| = \sum_{n=1}^\infty \frac{\|X\|_n \wedge 1}{2^n}.$

Next, local martingale is defined.

Definition 3 [KS98,1.5.15] Let ${X= \{X_t, \mathcal{F}_t; 0\le t<\infty\}}$ be a (continuous) process. If there exists a nondecreasing sequence ${\{T_n\}_{n=1}^\infty}$ of stopping times of ${\{\mathcal{F}_t\}}$, such that ${\{X_t^{(n)} := X_{t\wedge T_n}, \mathcal{F}_t; 0\le t <\infty\}}$ is a martingale for each ${n\ge 1}$ and ${\mathbb{P}\{\lim_{n\rightarrow \infty} T_n = \infty \} = 1}$, then we say that ${X}$ is a (continuous) local martingale; if, in addition, ${X_0 = 0}$ a.s., we write ${X \in \mathcal{M}^{loc} }$ (respectively, ${X\in \mathcal{M}^{c,loc}}$ if ${X}$ is continuous).

For measurable, ${\mathcal{F}_t}$-adapted process ${X}$, we define, for ${M\in \mathcal{M}_2^c}$

$\displaystyle [X]_T^2 := \mathbb{E} \Big[ \int_0^T X_t^2 d \langle M \rangle_t \Big].$

Definition 4 [KS98, 3.2.1] Let ${M \in \mathcal{M}_2^c}$. Let ${\mathcal{L}}$ denote the set of equivalence classes of all measurable, ${\{\mathcal{F}_t\}}$-adapted processes ${X}$, for which ${[X]_T < \infty}$ for all ${T>0}$. We define a metric on ${\mathcal{L}}$ by ${[X - Y]}$, where

$\displaystyle [X] := \sum_{n=1}^\infty 2^{-n} (1\wedge [X]_n).$

Let ${\mathcal{L}^*}$ denote the set of equivalence classes of progressively measurable processes satisfying ${[X]_T <\infty}$ for all ${T>0}$, and define a metric on ${\mathcal{L}^*}$ in the same way.

${\mathcal{L}^*}$ consists of those processes in ${\mathcal{L}}$ which are progressively measurable. When we wish to indicate the dependence on ${M}$, we write ${\mathcal{L}(M)}$ and ${\mathcal{L}^*(M)}$.

For ${M\in \mathcal{M}_2^c}$, ${I(X)}$ is well defined for every ${X\in \mathcal{L}^*}$, and ${I(X) \in \mathcal{M}_2^c}$. In addition, if the sample paths ${t\rightarrow \langle M \rangle_t(\omega)}$ of the quadratic variation process ${\langle M \rangle}$ are absolutely continuous functions of ${t}$ for ${\mathbb{P}}$-a.e. ${\omega}$, then ${I(X)}$ is well defined for all ${X\in \mathcal{L}}$, and all the properties of Proposition 6 hold in this case. (Remark 3.2.11 of [KS98]).

Definition 5 [KS98, 3.2.22] Let ${M \in \mathcal{M}^{c,loc}}$. We denote by ${\mathcal{P}}$ the collection of equivalence classes of all measurable, adapted processes ${X = \{X_t, \mathcal{F}_t; 0\le t < \infty\}}$ satisfying

$\displaystyle \mathbb{P}\Big[ \int_0^T X_t^2 d \langle M \rangle_t <\infty \Big] = 1 \quad \hbox{ for every } T\in [0,\infty).$

We denote by ${\mathcal{P}^*}$ the collection of equivalence classes of all progressively measurable processes satisfying this condition.

If ${M\in \mathcal{M}_2^c}$, then both ${\mathcal{L}}$ and ${\mathcal{L}^*}$ are well defined, and we have ${\mathcal{L} \subset \mathcal{P}}$ and ${\mathcal{L}^* \subset \mathcal{P}^*}$.

For ${M \in \mathcal{M}^{c,loc}}$ and ${X\in \mathcal{P}^*}$, the stochastic integral of ${X}$ with respect to ${M}$, denoted by ${I(X)}$, is well defined, and ${I(X) \in \mathcal{M}^{c,loc}}$.

If a.e. path ${t\rightarrow \langle M \rangle_t(\omega)}$ of the quadratic variation process ${\langle M \rangle}$ is an absolutely continuous function, we can choose integrands from wider class ${\mathcal{P}}$, instead of ${\mathcal{P}^*}$.

When ${M \in \mathcal{M}^{c,loc}}$ and ${X\in \mathcal{P}^*}$, the integral ${I(X)}$ will not in general satisfy conditions involved with expectations, but only satisfy (1), (6), (7), and (11).

Proposition 6 [KS98, 3.2.10] For ${M \in \mathcal{M}_2^c}$ and ${X\in \mathcal{L}^*}$, the stochastic integral ${I(X) = \{I_t(X), \mathcal{F}_t: 0\le t< \infty\}}$ of ${X}$ with respect to ${M}$ satisfies following elementary properties: For any numbers of ${0\le s < t <\infty}$

$\displaystyle I_0(X) = 0, \hbox{ a.s. } \mathbb{P} \ \ \ \ \ (1)$

$\displaystyle \mathbb{E}[I_t(X)|\mathcal{F}_s] = I_s(X), \hbox{ a.s. } \mathbb{P} \ \ \ \ \ (2)$

$\displaystyle \mathbb{E}[(I_t(X))^2] = \mathbb{E} \int_0^t X_u^2 d \langle M \rangle_u \ \ \ \ \ (3)$

$\displaystyle \|I(X)\| = [X] \ \ \ \ \ (4)$

$\displaystyle \mathbb{E}[(I_t(X) - I_s(X))^2| \mathcal{F}_s] = \mathbb{E} \Big[ \int_s^t X_u^2 d\langle M \rangle_u \Big | \mathcal{F}_s \Big], \hbox{ a.s. } \mathbb{P} \ \ \ \ \ (5)$

and

$\displaystyle I( \alpha X + \beta Y) = \alpha I(X) + \beta I(Y); \ \forall \alpha, \beta \in \mathbb{R}, \ Y\in \mathcal{L}^*. \ \ \ \ \ (6)$

Also, its quadratic variation is given by

$\displaystyle \langle I(X)\rangle_t = \int_0^t X_u^2 d \langle M \rangle_u. \ \ \ \ \ (7)$

Further more, for any two stopping times ${S \le T}$ of the filtration ${\{\mathcal{F}_t\}}$ and any number ${t>0}$, we have

$\displaystyle \mathbb{E} [ I_{t\wedge T} (X) | \mathcal{F}_S] = I_{t\wedge S} (X), \ \hbox{ a.s. } \mathbb{P}. \ \ \ \ \ (8)$

With ${X, Y \in \mathcal{L}^*}$ we have, a.s. ${\mathbb{P}}$:

$\displaystyle \mathbb{E}[(I_{t\wedge T}(X) - I_{t\wedge S}(X)) (I_{t\wedge T}(Y) - I_{t\wedge S}(Y)) | \mathcal{F}_S] = \mathbb{E} \Big[ \int_{t\wedge S}^{t\wedge T} X_u Y_u d \langle M \rangle_u \Big | \mathcal{F}_S \Big], \ \ \ \ \ (9)$

and in particular, for any number ${s}$ in ${[0,t]}$,

$\displaystyle \mathbb{E}[(I_{t}(X) - I_{s}(X)) (I_{t}(Y) - I_{s}(Y)) | \mathcal{F}_s] = \mathbb{E} \Big[ \int_{s}^{t} X_u Y_u d \langle M \rangle_u \Big | \mathcal{F}_s \Big], \ \ \ \ \ (10)$

Finally,

$\displaystyle I_{t\wedge T}(X) = I_t(\tilde X) \ \hbox{a.s.}, \ \ \ \ \ (11)$

where ${\tilde X_t(\omega) := X_t(\omega) 1_{\{t\le T(\omega)\}}.}$