The Ito Integral with Respect to Brownian Motion

Assume the Axiom of Choice The Axiom of Choice. Fix a probability space $(\mathrm{\Omega },\mathcal{F},P)$ with a continuous-time filtration $({\mathcal{F}}_t{)}_{t\ge 0}$ in the sense of Continuous-time filtrations and all-pairs martingales. All processes in this definition are real-valued and indexed by $[0,\mathrm{\infty })$; the filtration is neither assumed complete nor right-continuous. The Axiom of Choice is declared because the conditional-expectation classes used in clause 3 are supplied by the Radon--Nikodym interface of Conditional expectation as an ae class, which assumes it; the countable-choice obligations inherited from that interface are declared as dependencies of this item.

1. Adapted. $X=(X_t{)}_{t\ge 0}$ is adapted to $({\mathcal{F}}_t)$ when $X_t$ is ${\mathcal{F}}_t$-measurable for every $t\ge 0$. This is exactly the notion of Continuous-time filtrations and all-pairs martingales.

2. Stopped process. For a stopping time $\tau$ for $({\mathcal{F}}_t)$ Continuous-time stopping times and stopped sigma-algebras the stopped process is $X_t^{\tau }(\omega ):=X_{t\wedge \tau (\omega )}(\omega ),t\ge 0,$ with the convention $t\wedge \mathrm{\infty }:=t$, so no value $X_{\mathrm{\infty }}$ is ever required and $X_0^{\tau }=X_0$ identically. This formula defines a pathwise family; adaptedness of $X$ alone does not assert measurability or adaptedness of its stopped values. If in addition $\tau \le c$ for a deterministic constant $c$, then $X_t^{\tau }=X_{t\wedge \tau }$, and only the values of $X$ on $[0,c]$ enter. Stopping at a stopping time is not the same as replacing a process by a modification; it is a pathwise operation.

3. Martingale. $M=(M_t{)}_{t\ge 0}$ is a martingale (an all-pairs continuous-time martingale) relative to $({\mathcal{F}}_t)$ when it is adapted, $E\mathrm{\mid }{M}_t\mathrm{\mid }<\mathrm{\infty }$ for every $t\ge 0$, and for all $0\le s\le t$ $E[M_t\mid {\mathcal{F}}_s]=M_s\text{almost surely}\mathrm{.}$ The equality is an equality of the almost-everywhere classes of Conditional expectation as an ae class; equivalently, every version of the conditional expectation on the left equals $M_s$ off one null set. At $s=t$ the identity reduces to the known-variable case. The word "continuous-time" refers to the index set only and does not assert path continuity.

4. Local martingale. $X=(X_t{)}_{t\ge 0}$ is a local martingale relative to $({\mathcal{F}}_t)$ when it is adapted, $E\mathrm{\mid }{X}_0\mathrm{\mid }<\mathrm{\infty }$, and there exist stopping times $({\tau }_n{)}_{n\ge 0}$ with ${\tau }_0\le {\tau }_1\le \cdots$ and ${\tau }_n\uparrow \mathrm{\infty }$ almost surely such that for every $n\ge 0$ the stopped process $X^{{\tau }_n}-X_0=(X_{t\wedge {\tau }_n}-X_0{)}_{t\ge 0}$ is a martingale in the sense of clause 3. The sequence $({\tau }_n)$ is called a localizing sequence. Equivalently, every $X^{{\tau }_n}$ is a martingale: adaptedness makes the integrable variable $X_0$ measurable with respect to every ${\mathcal{F}}_t$, so the constant process with value $X_0$ is a martingale and may be added to, or subtracted from, each stopped process. The centering merely makes every localized process start at $0$; apart from the required integrability of $X_0$, no claim is made that $X$ is integrable at a positive deterministic time, and no claim is made that $X$ has continuous paths.

5. Path and integrability attributes. A process has continuous paths when $t\mapsto X_t(\omega )$ is continuous on $[0,\mathrm{\infty })$ for every $\omega$ in an event of probability one; the usual almost-sure path conventions of Process law, modification, and indistinguishability apply. A martingale $M$ is square-integrable when $E{M}_t^2<\mathrm{\infty }$ for every $t\ge 0$, and $L^2$-bounded when ${\sup }_{t\ge 0}E{M}_t^2<\mathrm{\infty }$. These are properties of the single process under consideration, not of its versions: a modification of a martingale need not be adapted, so every later statement names the adapted versions it uses.

The following remarks specify what follows directly from this vocabulary.

1. A martingale is a local martingale. If $M$ is a martingale, the constant sequence ${\tau }_n:=n$, $n\ge 0$, localizes it: the stopped process $M^n-M_0$ is again a martingale by the martingale identity applied at the deterministic times $s\wedge n\le t\wedge n$. The converse fails; a local martingale need not be a martingale, and no such implication is used in this development.
2. Localization after a separately justified stopping operation. Suppose $({\tau }_n)$ localizes $X$, $\sigma$ is a stopping time, $X^{\sigma }$ is adapted, and each $(X^{{\tau }_n}-X_0{)}^{\sigma }$ is a martingale. Then $({\tau }_n)$ localizes $X^{\sigma }$. Indeed it still increases to infinity, $(X^{\sigma }{)}_0=X_0$, and the pathwise identity $(X^{\sigma }{)}^{{\tau }_n}-(X^{\sigma }{)}_0=X^{\sigma \wedge {\tau }_n}-X_0=(X^{{\tau }_n}-X_0{)}^{\sigma }$ verifies exactly clause 4. The stopped-piece martingale assertion and adaptedness are hypotheses here, not consequences of the unrestricted all-pairs definition. They must be established in each application. The sequence $(\sigma \wedge {\tau }_n)$ is not a substitute for $({\tau }_n)$: its almost-sure limit is $\sigma$, which need not be infinity.

No path continuity, no right continuity of the filtration, and no completeness of the underlying probability space is imposed by this definition. Choice enters only through the conditional-expectation interface named above, whose countable-choice obligations are declared as dependencies of this item.

Assume the Axiom of Choice The Axiom of Choice. Let $(\mathrm{\Omega },\mathcal{F},P)$ be a probability space with a continuous-time filtration $({\mathcal{F}}_t{)}_{t\ge 0}$ Continuous-time filtrations and all-pairs martingales. All processes here are real-valued and indexed by $[0,\mathrm{\infty })$. AC is inherited from the cited continuous-time-filtration interface, whose separate martingale clause uses conditional expectation. Once the filtered probability space and stopping times are given, the progressive and predictable constructions below make no additional choice: they use only generated sigma-algebras and product rectangles.

1. Progressively measurable. A process $X=(X_t{)}_{t\ge 0}$ is progressively measurable relative to $({\mathcal{F}}_t)$ when for every $t\ge 0$ the restricted map $[0,t]\times \mathrm{\Omega }\to \mathbb{R}$, $(s,\omega )\mapsto X_s(\omega )$, is measurable for the product sigma-algebra $\mathcal{B}([0,t])\otimes {\mathcal{F}}_t$ The product sigma-algebra and its finite iterates.

2. Time-zero and interval generators. On the product space $[0,\mathrm{\infty })\times \mathrm{\Omega }$, let $\mathcal{P}$ be the sigma-algebra generated by the family $\mathcal{G}:=\{(s,u]\times A:0\le s<u<\mathrm{\infty },A\in {\mathcal{F}}_s\}\cup \{\{0\}\times A:A\in {\mathcal{F}}_0\}\mathrm{.}$ The interval generators with $s=0$ are included, so $(0,u]\times A$ with $A\in {\mathcal{F}}_0$ is a generator. The sigma-algebra $\mathcal{P}$ exists by Nonempty intersections of sigma-algebras are sigma-algebras, so the generated sigma-algebra exists and is minimal and is called the predictable sigma-algebra. A process $H=(H_t{)}_{t\ge 0}$ is predictable when the map $(s,\omega )\mapsto H_s(\omega )$ is $\mathcal{P}$-measurable. For a finite horizon $T>0$ we write ${\mathcal{P}}_T$ for the sigma-algebra on $[0,T]\times \mathrm{\Omega }$ generated by the same family restricted to $u\le T$ together with the time-zero generators; this is the trace of $\mathcal{P}$ on $[0,T]\times \mathrm{\Omega }$. Indeed, the trace of an interval generator is empty when $s\ge T$, and otherwise is $(s,\min (u,T)]\times A$, while time-zero generators are unchanged. Conversely every listed finite-horizon generator is such a trace.

3. Sections, adaptedness, and null sets. Two structural conventions are used repeatedly and are part of the definition. (a) Every generator is a measurable rectangle of $\mathcal{B}([0,\mathrm{\infty }))\otimes \mathcal{F}$, hence $\mathcal{P}\subseteq \mathcal{B}([0,\mathrm{\infty }))\otimes \mathcal{F}$; the time-zero generator $\{0\}\times A$ has $dt\otimes P$-measure zero for every $A$. A set in $\mathcal{P}$ has its section at each fixed time $s$ in ${\mathcal{F}}_s$: sets with that section property form a sigma-algebra, and each generator has the property by the increasing-filtration condition. Its section at each fixed $\omega$ is Borel in time, by the same sigma-algebra argument. (b) If a process is predictable then it is progressively measurable: for $s\le t$ the section $\omega \mapsto H_s(\omega )$ is ${\mathcal{F}}_s$-measurable $\subseteq {\mathcal{F}}_t$-measurable, and the progressive-measurability claim is the restriction of the joint map to $[0,t]\times \mathrm{\Omega }$; the restriction of a $\mathcal{P}$-measurable map to $[0,t]\times \mathrm{\Omega }$ is measurable for $\mathcal{B}([0,t])\otimes {\mathcal{F}}_t$ because every generator with $u\le t$ lies in that product sigma-algebra and the generators with $u>t$ have intersection with $[0,t]\times \mathrm{\Omega }$ of the form $(s,t]\times A$ with $s<t$ and $A\in {\mathcal{F}}_s$, or the empty set when $s\ge t$. The time-zero generators also lie in $\mathcal{B}([0,t])\otimes {\mathcal{F}}_t$.

The family $\mathcal{G}$ need not itself contain the whole space, but a finite intersection of generators is empty or again a generator, or a time-zero generator, because $(s,u]\cap (s^{\mathrm{\prime }},u^{\mathrm{\prime }}]=(s\vee s^{\mathrm{\prime }},u\wedge u^{\mathrm{\prime }}]$ and $A\cap A^{\mathrm{\prime }}\in {\mathcal{F}}_{s\vee s^{\mathrm{\prime }}}$ whenever $A\in {\mathcal{F}}_s$, $A^{\mathrm{\prime }}\in {\mathcal{F}}_{s^{\mathrm{\prime }}}$, while $\{0\}\times A$ is disjoint from every interval generator, including $(0,u]\times A^{\mathrm{\prime }}$. Two time-zero generators intersect in $\{0\}\times (A\cap A^{\mathrm{\prime }})$ with $A\cap A^{\mathrm{\prime }}\in {\mathcal{F}}_0$. Thus the family consisting of the whole space together with finite intersections of generators is a pi-system generating $\mathcal{P}$; this is the structural fact used by the density theorem below.

Because predictability is a measurability requirement for the joint map, it is preserved by pointwise limits, products and linear combinations of predictable processes. A deterministic function $h$ of time defines a predictable process exactly when $h$ is Borel measurable: the sets $C$ for which $C\times \mathrm{\Omega }\in \mathcal{P}$ form a sigma-algebra containing the intervals $(s,u]$ and $\{0\}$, hence all Borel sets. Conversely, take any fixed $\omega \in \mathrm{\Omega }$ (a probability space is nonempty) and use the Borel time-section property from 3(a) on every inverse image of an open set. This single choice uses no choice axiom. In particular the indicators $1_{[0,u]}$ and $1_{(0,u]}$ are predictable for every deterministic $u\ge 0$, and the process $s\mapsto 1_{s\le \tau (\omega )}$ for a stopping time $\tau$ Continuous-time stopping times and stopped sigma-algebras is predictable, since $\{(s,\omega ):s>\tau (\omega )\}={\bigcup }_{q\in {\mathbb{Q}}_{>0}}(q,\mathrm{\infty })\times \{\tau <q\}={\bigcup }_{q\in {\mathbb{Q}}_{>0}}{\bigcup }_{n>q}(q,n]\times \{\tau <q\}$ is a countable union of generators: for rational $q>0$ one has $\{\tau <q\}={\bigcup }_{q^{\mathrm{\prime }}<q,q^{\mathrm{\prime }}\in \mathbb{Q}}\{\tau \le q^{\mathrm{\prime }}\}\in {\mathcal{F}}_q$. The complement of that set inside $[0,\mathrm{\infty })\times \mathrm{\Omega }$ is the event of the indicator, so $1_{[0,\tau ]}$ is predictable; its left-continuity in the time variable is the visual form of the same computation.

No completeness of the filtration and no right continuity of $({\mathcal{F}}_t)$ is used or assumed. Apart from the inherited ambient assumption just recorded, the constructions in this definition are choice-free.

Let $(\mathrm{\Omega },\mathcal{F},P)$ be a probability space with a continuous-time filtration $({\mathcal{F}}_t{)}_{t\ge 0}$, and let $X=(X_t{)}_{t\ge 0}$ be a real process that satisfies $X_t:\mathrm{\Omega }\to \mathbb{R}$ is ${\mathcal{F}}_t$-measurable for every $t\ge 0$, and has continuous paths, in the strong sense that $s\mapsto X_s(\omega )$ is continuous on $[0,\mathrm{\infty })$ for every $\omega \in \mathrm{\Omega }$. Then $X$ is progressively measurable and predictable relative to $({\mathcal{F}}_t)$ in the sense of Progressively measurable and predictable processes, including at time zero under the generator convention $\{0\}\times A$, $A\in {\mathcal{F}}_0$. No choice principle is used. Here a filtration means an increasing family of sub-sigma-algebras of $\mathcal{F}$. The fixed-time measurability hypothesis is the adaptedness terminology of clause 1 of Continuous-time adapted processes and martingales; only that clause is used, not its conditional-expectation or martingale interface and its Choice assumption.

If instead the paths are continuous only on an event $A\in \mathcal{F}$ with $P(A)=1$, the conclusion holds for the modification that is set equal to $0$ off $A$ provided $A\in {\mathcal{F}}_0$; without such a measurability assumption on the continuity event no predictability claim is made, because predictability is a property of the given joint map.

The approximation is the standard dyadic-step argument of van der Vaart, Section 5.1; continuous adapted processes generate the predictable sigma-algebra, and the left-continuous staircase approximants are predictable by construction. The time-zero section uses exactly the generators $\{0\}\times A$, $A\in {\mathcal{F}}_0$. The proof explicitly supplies the fixed-time measurable maps and does not invoke any conditional-expectation existence theorem.

Assume the Axiom of Choice The Axiom of Choice, and fix a probability space $(\mathrm{\Omega },\mathcal{F},P)$ with a continuous-time filtration $({\mathcal{F}}_t{)}_{t\ge 0}$ Continuous-time filtrations and all-pairs martingales and a standard Brownian motion $B$ Brownian motion on it. Standing hypothesis (H). $B$ is adapted to $({\mathcal{F}}_t)$ and for all $0\le s<t$ the increment $B_t-B_s$ is independent of ${\mathcal{F}}_s$ and has law $N(0,t-s)$. The raw natural filtration and the usual augmented natural filtration Natural and usual augmented Brownian filtrations both satisfy (H) for a standard Brownian motion, by Future-path Markov property; for a general filtration, (H) is part of the data and is not automatic. Every statement in this development names (H) when it is used.

Throughout, fix a finite horizon $T>0$. An elementary predictable Brownian integrand on $[0,T]$ is a process of the form $H_s(\omega )={\sum }_{k=0}^{m-1}{\xi }_k(\omega )1_{(t_k,t_{k+1}]}(s),s\in [0,T],$ where $0=t_0<t_1<\cdots <t_m=T$ is a finite partition of $[0,T]$, each ${\xi }_k$ is a bounded real ${\mathcal{F}}_{t_k}$-measurable random variable, and the values at the partition points are irrelevant because the intervals are left-open and right-closed. The mesh of the representation is ${\max }_k(t_{k+1}-t_k)$, and the supremum norm of the representation is ${\max }_k\mathrm{\parallel }{\xi }_k{\mathrm{\parallel }}_{\mathrm{\infty }}$. The integrand itself is the process $H$; a second list of the same name with different coefficients is the same elementary integrand only if the two processes coincide in the almost-everywhere sense made precise below.

The following properties are part of the definition and are used at once.

1. Predictability. $H$ is predictable Progressively measurable and predictable processes. Indeed, for a Borel set $\mathrm{\Gamma }\subseteq \mathbb{R}$, $\{H\in \mathrm{\Gamma }\}=Z_{\mathrm{\Gamma }}\cup {\bigcup }_{k=0}^{m-1}((t_k,t_{k+1}]\times \{{\xi }_k\in \mathrm{\Gamma }\}),Z_{\mathrm{\Gamma }}:=\{\begin{array}{ll}\{0\}\times \mathrm{\Omega },& 0\in \mathrm{\Gamma },\\ \mathrm{\varnothing },& 0\mathrm{\notin }\mathrm{\Gamma },\end{array}$ a finite union of generators of the predictable sigma-algebra because $\{{\xi }_k\in \mathrm{\Gamma }\}\in {\mathcal{F}}_{t_k}$ and $\{0\}\times \mathrm{\Omega }$ is a time-zero generator. Consequently $H$ is progressively measurable and measurable for the product sigma-algebra $\mathcal{B}([0,T])\otimes \mathcal{F}$, and $H$ belongs to $L^2([0,T]\times \mathrm{\Omega },\mathrm{d}t\otimes P)$: with $c:={\max }_k\mathrm{\parallel }{\xi }_k{\mathrm{\parallel }}_{\mathrm{\infty }}<\mathrm{\infty }$ one has $E{\int }_0^T{H}_s^{2}ds\le c^{2}T<\mathrm{\infty }$.

2. Endpoint and null-set conventions. Replacing the intervals by $(t_k,t_{k+1}]$ makes the value at $s=0$ zero for every elementary integrand. More generally, if two elementary integrands agree for all $s\in [0,T]$ except at finitely many deterministic times, then they agree $(\mathrm{d}t\otimes P)$-almost everywhere, since a finite set of times is Lebesgue-null and Tonelli computes $\mathrm{d}t\otimes P(\{u\}\times \mathrm{\Omega })=0$. All integrands and all integrals below are therefore elements of the quotient spaces of $L^2(\mathrm{d}t\otimes P)$ and $L^2(P)$; a claim about a process is a claim about its almost-everywhere class unless a representative is explicitly named, and path statements name the continuous representative.

3. Deterministic coefficients. If every ${\xi }_k$ is a deterministic real number, $H$ is a deterministic step function on $[0,T]$, so the elementary integrands include all step functions with deterministic coefficients. These are the integrands for which the integral is a Gaussian variable below.

The Axiom of Choice is declared because the Brownian construction and the conditional-expectation interface used in items 6, 7 and 13 assume it; the definition itself, including the predictability computation of clause 1, uses no choice. The countable-choice obligations inherited from that interface are declared as dependencies of this item.

Assume the Axiom of Choice and work under the standing hypothesis (H) of Elementary predictable Brownian integrands: a filtered probability space with a standard Brownian motion $B$ adapted to the filtration, with increments $B_t-B_s$ independent of ${\mathcal{F}}_s$ of law $N(0,t-s)$. Fix $T>0$ and an elementary predictable integrand $H_s={\sum }_{k=0}^{m-1}{\xi }_k{1}_{(t_k,t_{k+1}]}(s),0=t_0<\cdots <t_m=T,$ with bounded ${\mathcal{F}}_{t_k}$-measurable coefficients ${\xi }_k$. The Ito integral of $H$ against $B$ is the process $I_t(H):={\sum }_{k=0}^{m-1}{\xi }_k(B_{t\wedge t_{k+1}}-B_{t\wedge t_k}),0\le t\le T,$ also written ${\int }_0^t{H}_{s}d{B}_s$ or $(H\cdot B{)}_t$. The sum is finite and is evaluated with the Brownian path of the given representative; on the event where $t\mapsto B_t$ is continuous it is continuous in $t$, and $I_0(H)=0$ because $0\wedge t_k=0$ for every $k$.

Three conventions are part of the definition.

1. Adaptedness and continuity. For fixed $t$, each summand ${\xi }_k(B_{t\wedge t_{k+1}}-B_{t\wedge t_k})$ is ${\mathcal{F}}_t$-measurable: if $t<t_k$, the Brownian difference and hence the summand are zero; if $t\ge t_k$, then ${\mathcal{F}}_{t_k}\subseteq {\mathcal{F}}_t$, while $B_{t\wedge u}$ is ${\mathcal{F}}_t$-measurable for every $u$ because $t\wedge u\le t$. Thus $I(H)$ is adapted. For each fixed $\omega$ the map $t\mapsto B_{t\wedge t_{k+1}}(\omega )-B_{t\wedge t_k}(\omega )$ is continuous outside the single exceptional null set of (H) on which the Brownian path is discontinuous; the finite sum is therefore continuous on the same event.

2. Linearity on a common refinement. If $H$ and $K$ are elementary integrands and a partition refines both representations, then the defining sums of $H$, $K$, $H+K$ and $aH$ are taken over that common partition, and the finite sums give $I_t(H+K)=I_t(H)+I_t(K)$ and $I_t(aH)=a{I}_t(H)$ for real $a$ identically. In particular $I_t(H)-I_t(K)=I_t(H-K)$ for the elementary integrand $H-K$ represented on that refinement. This is a rearrangement of finitely many terms, not a limiting statement.

3. Dependence on the representation is temporary. The definition attaches $I_t(H)$ to a chosen elementary representation. Elementary Ito integrals do not depend on step representation ↗ proves that two representations that agree $(\mathrm{d}t\otimes P)$-almost everywhere produce the same random variables almost surely at each fixed time, so that $I_t(H)$ is a function of the $(\mathrm{d}t\otimes P)$-class of $H$ alone. Until then, every statement about an elementary integrand names the representation it uses.

The Axiom of Choice is declared because the standing hypothesis (H) is part of the Brownian interface of Elementary predictable Brownian integrands, which assumes it; the definition of the finite sum uses no choice. The countable-choice obligations inherited from that interface are declared as dependencies of this item.

Assume the Axiom of Choice and the standing hypothesis (H) of Elementary predictable Brownian integrands. Let $H$ and $H^{\mathrm{\prime }}$ be two elementary predictable integrands on $[0,T]$ whose values agree $(\mathrm{d}t\otimes P)$-almost everywhere, and let $I_t(H)$, $I_t(H^{\mathrm{\prime }})$ be their defining sums Ito integral of an elementary predictable process for the chosen representations. Then $I_T(H)=I_T(H^{\mathrm{\prime }})$ almost surely, and in fact $I_t(H)=I_t(H^{\mathrm{\prime }})$ almost surely for every deterministic $t\in [0,T]$. In particular, replacing a representation by a deterministic refinement of its partition, or by any other elementary representation of the same $(\mathrm{d}t\otimes P)$-class, does not change the sums; the elementary integral is a function of that class.

Van der Vaart, Definition 5.20 and Lemma 5.22, first defines the integral on step processes and then checks that the definition does not depend on the representation; the isometry computation for the difference is the same conditional-centering expansion used here. The nearly-sure statement at every fixed time is what makes the notation ${\int }_0^{t}HdB$ well defined before the completion step of item 10.

Assume the Axiom of Choice and the standing hypothesis (H) of Elementary predictable Brownian integrands. Let $H$ be an elementary predictable integrand on $[0,T]$ with representation $H_s={\sum }_{k=0}^{m-1}{\xi }_k{1}_{(t_k,t_{k+1}]}(s)$ and defining sums $I_t(H)$ Ito integral of an elementary predictable process. Extend those sums to $[0,\mathrm{\infty })$ by setting $I_t(H):=I_T(H)$ for $t\ge T$. Then $t\mapsto I_t(H)$ is a continuous square-integrable martingale relative to $({\mathcal{F}}_t)$ Continuous-time adapted processes and martingales, and for every $t\in [0,T]$ $E[I_t(H{)}^2]=E{\int }_0^t{H}_s^{2}ds={\sum }_{k=0}^{m-1}E[{\xi }_k^2](t\wedge t_{k+1}-t\wedge t_k)\mathrm{.}$ By the representation independence of Elementary Ito integrals do not depend on step representation both sides depend only on the $(\mathrm{d}t\otimes P)$-class of $H$.