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)\}}.}

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