Itos Formula and Brownian Martingales

Assume the Axiom of Choice and the standing hypothesis (H) of Elementary predictable Brownian integrands: $B$ is a standard Brownian motion on a filtered probability space, adapted to $({\mathcal{F}}_t{)}_{t\ge 0}$, with $B_t-B_s$ independent of ${\mathcal{F}}_s$ of law $N(0,t-s)$ for $0\le s<t$. For this localized-calculus definition the filtration satisfies the usual conditions, as required by Locally square-integrable predictable Brownian integrands.

1. Real continuous Brownian Ito process. A real progressively measurable process $X=(X_t{)}_{t\ge 0}$ is a continuous Brownian Ito process when there are

   • a finite real ${\mathcal{F}}_0$-measurable $X_0$,
   • a real progressively measurable process $b=(b_s{)}_{s\ge 0}$ Progressively measurable and predictable processes with ${\int }_0^t\mathrm{\mid }{b}_s\mathrm{\mid }ds<\mathrm{\infty }$ almost surely for every finite $t\ge 0$, and
   • a real predictable process $\sigma =({\sigma }_s{)}_{s\ge 0}$, locally square-integrable in the sense that ${\int }_0^t{\sigma }_s^{2}ds<\mathrm{\infty }$ almost surely for every finite $t$ Locally square-integrable predictable Brownian integrands,

   such that, up to equality at all times on a measurable probability-one event, $X_t=X_0+{\int }_0^t{b}_{s}ds+{\int }_0^t{\sigma }_{s}d{B}_s,t\ge 0,$ where the first integral is the pathwise Lebesgue integral of $s\mapsto b_s(\omega )$ and the second is the localized Ito integral $\sigma \cdot B$ of Localized Ito integral. One writes ${\displaystyle d{X}_t=b_{t}dt+{\sigma }_{t}d{B}_t}$ for the display, and $b$ is called the drift and $\sigma$ the diffusion coefficient of this representation.

2. Multidimensional Brownian-driven process. Let $d\ge 1$ and $m\ge 1$ be finite integers, let $B=(B^1,\dots ,B^m)$ be a standard $m$-dimensional Brownian motion $d$-dimensional Brownian motion that is adapted to $({\mathcal{F}}_t)$ and whose vector increment $B_t-B_s$ is independent of ${\mathcal{F}}_s$ for $0\le s<t$, let $X_0$ be a finite ${\mathcal{F}}_0$-measurable ${\mathbb{R}}^d$-valued random vector, let $b$ be an ${\mathbb{R}}^d$-valued progressively measurable process with ${\int }_0^t\mathrm{\mid }{b}_s^i\mathrm{\mid }ds<\mathrm{\infty }$ almost surely for every $i$ and finite $t$, and let $\sigma =({\sigma }^{ik})$ be an ${\mathbb{R}}^{d\times m}$-valued predictable process, each entry predictable Progressively measurable and predictable processes, with ${\int }_0^t({\sigma }_s^{ik}{)}^{2}ds<\mathrm{\infty }$ almost surely for every $i,k$ and finite $t$. Then $X_t^i=X_0^i+{\int }_0^t{b}_s^{i}ds+{\sum }_{k=1}^m{\int }_0^t{\sigma }_s^{ik}d{B}_s^k,i=1,\dots ,d,$ defines an ${\mathbb{R}}^d$-valued continuous Brownian Ito process driven by $B$, using the progressive integral versions of Localized Ito integral. Other progressive representatives agreeing on one measurable full event at every time represent the same process. No claim is made that an arbitrary null-path modification remains adapted.

The following well-definedness clauses are part of the definition and are used throughout.

3. The drift integral is a genuine pathwise integral. Progressive measurability gives measurable sections and makes the positive and negative integrals ${\mathcal{F}}_t$-measurable by the parameter-integral part of Tonelli Tonelli's theorem for nonnegative measurable functions on a sigma-finite product, applied to Lebesgue measure on $[0,t]$ and the probability measure on ${\mathcal{F}}_t$. On the event where both extended integrals are finite, their difference is the drift integral; set it to zero otherwise. This is an adapted measurable convention, is finite everywhere, and on one probability-one event is absolutely continuous on every finite interval (take the countable intersection over integer horizons). It is progressive: the map $(t,\omega )\mapsto {\int }_0^t{b}_s(\omega )ds$, with the preceding finite-value convention, is measurable on every $[0,T]\times \mathrm{\Omega }$ by parameter integration, and is adapted. Thus the definition uses the standard almost-sure drift and local-energy classes under the usual conditions.

4. The stochastic integral exists, is unique and is continuous. By Locally square-integrable predictable Brownian integrands and Localized Ito integral the localized integral $\sigma \cdot B$ is a well-defined adapted process with continuous paths, unique up to indistinguishability, and each of its finite-energy pieces is a square-integrable martingale. The displayed integral representative is progressive; $X$ is adapted by its required progressive measurability and has continuous paths, $X_t$ is finite almost surely for each $t$, and $M_t^i:={\sum }_k{\int }_0^t{\sigma }_s^{ik}d{B}_s^k$ is a continuous local martingale relative to $({\mathcal{F}}_t)$ in the sense of Continuous-time adapted processes and martingales for every $i$, with the common energy localizers described in clause 6.

5. The coefficients are not part of the process. A continuous Brownian Ito process may admit several representations $X=X_0+b\cdot \mathrm{L}\mathrm{e}\mathrm{b}+\sigma \cdot B$ with different pointwise representatives $(b,\sigma )$ (for example, altering a coefficient only at the single time zero), even for the same Brownian motion and filtration; this does not assert nonuniqueness of their equivalence classes modulo $\mathrm{d}t\otimes P$; conversely, two processes with the same displayed integral representation and the same $X_0$ are indistinguishable. Every statement on this page that mentions a decomposition uses only properties common to all representations of the given process. The separate quadratic-covariation definition that follows this item on its owning page is formulated from $X$ itself and not from $(b,\sigma )$; no quadratic-covariation definition is made inside this item.

6. The class is closed under stopping and under linear combinations. If $\tau$ is a stopping time, the literal stopped process $X^{\tau }$ is progressive. Indeed on $[0,T]\times \mathrm{\Omega }$ the map $(t,\omega )\mapsto (t\wedge \tau (\omega ),\omega )$ is measurable into $\mathcal{B}([0,T])\otimes {\mathcal{F}}_T$: the truncated time $\tau \wedge T$ is ${\mathcal{F}}_T$-measurable, minimum is continuous, and rectangle inverse images give the assertion. Compose with the progressive restriction of $X$. Its drift is $b{1}_{[0,\tau ]}$ and its diffusion is $\sigma 1_{[0,\tau ]}$; these retain almost-sure integrability and the required progressive/predictable measurability. The drift identity is pathwise. For the stochastic identity, apply the finite-energy stopping identity of Localized Ito integral, clause 4, on the canonical energy intervals of $\sigma$. The truncated integrand has finite energy on these same intervals. Comparing its canonical intervals with them at their pairwise minima by clause 4 identifies the localized integrals there. Countably many full-event agreements and exhaustion then give the identity at all times on one full event. Thus stopping closure uses localization, not a direct application of clause 4 to a possibly infinite-energy integrand. For finitely many coefficients, use the common continuous energy $E_t={\sum }_{i,k}{\int }_0^t({\sigma }_s^{ik}{)}^{2}ds$ and times ${\kappa }_n=\inf \{t:E_t\ge n\}\wedge n$, $n\ge 1$. The same continuous-energy test as in the local-integrability definition makes these stopping times increasing to infinity almost surely, and bounds every stopped coefficient's expected energy by $n$. The preceding pairwise-minimum comparison shows that every integral stopped there is its finite-energy integral. Finite sums of these square-integrable martingales are martingales: finite sums preserve adaptation and integrability, and summing the integral test over any $A\in {\mathcal{F}}_s$ proves the conditional-expectation identity; hence their sums are local martingales with this common sequence. For fixed real constants, finite linear combinations of processes driven by the same vector Brownian motion have the combined coefficients. Their integrability follows from the triangle inequality and $({\sum }_{j=1}^r{u}_j{)}^2\le r{\sum }_j{u}_j^2$.

7. No pathwise integral against Brownian motion is defined. The display defines the stochastic integral only as the localized limit of Localized Ito integral; no integral ${\int }_0^t{H}_s(\omega )d{B}_s(\omega )$ along the individual path is asserted, and the page's examples show that the pathwise Riemann--Stieltjes route is unavailable.

No choice beyond the declared AC of the ambient interfaces enters the construction: the coefficients $b,\sigma$ and the process are given data, and the canonically localized integral is constructed in Localized Ito integral. The Axiom of Choice is declared because the conditional-expectation and $L^2$ interfaces used downstream assume it, and the countable-choice obligations inherited from those interfaces are declared as dependencies of this item.

Fix processes $X=(X_t{)}_{t\ge 0}$ and $Y=(Y_t{)}_{t\ge 0}$ of real random variables on one probability space $(\mathrm{\Omega },\mathcal{F},P)$: for every $\omega$ in a measurable event of probability one the paths $t\mapsto X_t$ and $t\mapsto Y_t$ are continuous on $[0,\mathrm{\infty })$ Continuity of $f:A\to \mathbb{R}$ at a point of $A$ and on $A$: the $\epsilon$-$\delta$ condition, its agreement with ${\lim }_{x\to c}f(x)=f(c)$ at a limit point, and continuity at an isolated point. Fix $T>0$ and a deterministic partition sequence $({\pi }_n)$ of $[0,T]$ with mesh tending to $0$ Quadratic variation along a partition sequence. For each $n$ write ${\pi }_n=(m_n,s^{(n)})$ and form the two families of cross-increment partial sums $S_n^{\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{p}}(t):={\sum }_{\begin{array}{c}1\le k\le m_n\\ s_k^{(n)}\le t\end{array}}(X_{s_k^{(n)}}-X_{s_{k-1}^{(n)}})(Y_{s_k^{(n)}}-Y_{s_{k-1}^{(n)}}),$ $S_n^{\mathrm{p}\mathrm{a}\mathrm{r}\mathrm{t}}(t):=S_n^{\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{p}}(t)+(X_t-X_{s_{k(t)}^{(n)}})(Y_t-Y_{s_{k(t)}^{(n)}})(k(t)<m_n),$ with $S_n^{\mathrm{p}\mathrm{a}\mathrm{r}\mathrm{t}}(T):=S_n^{\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{p}}(T)$ and $k(t)$ the largest index in $\{0,\dots ,m_n\}$ with $s_{k(t)}^{(n)}\le t$; these are the cross sums corresponding to the two conventions of Quadratic variation along a partition sequence specialized to the pair $(X,Y)$, and both are $0$ at $t=0$.

All suprema in probability statements below use measurable versions. For any finite collection of the processes involved, including a candidate limit, intersect their measurable probability-one continuity events and set all of them to zero off that intersection. These representatives have everywhere continuous paths and retain measurable fixed-time values. The partial-sum paths are continuous; the step-sum paths are right-continuous on $[0,T)$ with the specified value at $T$. Their uniform distances from a continuous candidate therefore equal suprema over $(\mathbb{Q}\cap [0,T])\cup \{T\}$, which are finite measurable random variables. Another such normalization agrees on a measurable probability-one event and gives the same probability limits. Here indistinguishability means agreement at every time on a measurable probability-one event. No completeness or adaptedness is needed for this convention.

1. Existence and value of the covariation. We say that the quadratic covariation $[X,Y]$ exists on $[0,T]$ when there is a real process $t\mapsto [X,Y{]}_t$ of real random variables with almost-sure continuous paths on $[0,T]$ such that for every deterministic partition sequence $({\pi }_n)$ of $[0,T]$ with mesh tending to $0$ both families converge to it uniformly in probability on $[0,T]$: ${\sup }_{0\le t\le T}\mid S_n^{\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{p}}(t)-[X,Y{]}_t\mid ⟶0,{\sup }_{0\le t\le T}\mid S_n^{\mathrm{p}\mathrm{a}\mathrm{r}\mathrm{t}}(t)-[X,Y{]}_t\mid ⟶0,$ the convergence being convergence in probability Convergence in probability. The two displayed requirements are part of one condition: the same process $[X,Y]$ must arise for every admissible sequence and for both conventions. When the condition holds we call $[X,Y{]}_t$ the quadratic covariation of $X$ and $Y$ at time $t$, and we write $[X]:=[X,X]$ and call it the quadratic variation of $X$. For two candidate limits $U,V$, fix a deterministic dyadic partition sequence. The triangle inequality bounds ${\sup }_t\mathrm{\mid }{U}_t-V_t\mathrm{\mid }$ by the sum of their uniform errors against its partial sums. For each $\epsilon >0$ the probability that this supremum exceeds $\epsilon$ is at most the sum of the two error probabilities at $\epsilon \mathrm{/}2$, and hence is zero. Taking $\epsilon =1\mathrm{/}j$, $j\ge 1$, proves uniqueness on one measurable probability-one event; the definition is applied separately on each finite horizon, and when the covariations on all horizons are compatible we write the resulting process on $[0,\mathrm{\infty })$ again as $[X,Y]$.

2. Symmetry and polarization. Exchanging the two factors does not change the definition, so $[X,Y]=[Y,X]$ whenever either side exists. The identities $[X+Y]=[X]+2[X,Y]+[Y],[X-Y]=[X]-2[X,Y]+[Y]$ hold on the domain where all covariations appearing in them exist: the cross-increment sums are bilinear in the pair, so the displayed identities are exact at the level of partial sums for every partition, and probability limits pass through finite algebraic identities. In particular $[X,Y]=\frac{1}{2}([X+Y]-[X]-[Y])$ and $4[X,Y]=[X+Y]-[X-Y]$ on that domain.

3. Bilinearity and insensitivity to constants. If the covariations $[X_1,Y]$, $[X_2,Y]$ and $[X_1+X_2,Y]$ exist on $[0,T]$, then $[X_1+X_2,Y]=[X_1,Y]+[X_2,Y]$ there, and $[cX,Y]=c[X,Y]$ for real $c$; moreover $[X+c,Y]=[X,Y]$ for every real constant $c$, because the increments of a constant process are zero. These are again exact identities of partial sums plus uniqueness of limits.

4. Zero covariation with a continuous finite-variation process. Let $A=(A_t)$ be continuous on a full-measure event and suppose, for every $\omega$ in that event, that $t\mapsto A_t(\omega )$ has bounded variation on every finite interval in the sense of Bounded variation and total variation on an interval. Then for each $T$ and each admissible partition sequence, the cross sums of $A$ against any continuous process $Y$ satisfy $\mid {\sum }_k(A_{s_k}-A_{s_{k-1}})(Y_{s_k}-Y_{s_{k-1}})\mid \le ({\max }_k{\sup }_{u,v\in [s_{k-1},s_k]}\mathrm{\mid }{Y}_u-Y_v\mathrm{\mid }){\mathrm{Var}}_{[0,T]}(A),$ because every sum of absolute $A$-increments is bounded by the path's total variation on $[0,T]$; the maximum tends to $0$ along vanishing meshes by uniform continuity of the continuous path $Y$ on the compact interval $[0,T]$. Hence $[A,Y]$ exists and equals the zero process for every such $A$ and every continuous $Y$, and in particular $[A,A^{\mathrm{\prime }}]=0$ for two such processes. In the notation of the page this says that continuous finite-variation parts contribute nothing to quadratic covariation.

5. Scope and choice. The inputs and candidate limits have measurable fixed-time values and almost-sure continuous paths. The normalization above makes the uniform errors real random variables, as required by Convergence in probability. No filtration or adaptedness is used. The algebra in clauses 2--3 passes to limits by the uniform triangle inequality and the union bound. In clause 4 the same estimate holds for every partial sum, including its terminal partial increment, because its intervals are disjoint; it thus proves uniform convergence to zero on the common continuity event. For each epsilon, the measurable events that some error after index n exceeds epsilon decrease to a null event; continuity from above of probability gives convergence in probability. Uniform continuity is justified by the choice-free finite-cover argument in Quadratic variation along a partition sequence. The partitions are given and normalization uses a finite intersection of supplied full-measure events, so no choice axiom is used. Existence for Brownian Ito processes is a separate theorem, not a presupposition of this definition.

Assume the Axiom of Choice and the standing hypothesis (H) of Elementary predictable Brownian integrands. Let $B=(B^1,\dots ,B^m)$ be a standard $m$-dimensional Brownian motion and let $X=(X^1,\dots ,X^d)$ be an ${\mathbb{R}}^d$-valued continuous Brownian Ito process driven by $B$, $d{X}_t^i=b_t^{i}dt+{\sum }_{k=1}^m{\sigma }_t^{ik}d{B}_t^k,i=1,\dots ,d,$ in the sense of Continuous Brownian Ito processes, including its progressive representative, usual-filtration, almost-sure coefficient-integrability and vector-increment independence hypotheses. Indistinguishability and measurable suprema use the full-event normalization convention of Quadratic covariation of Brownian Ito processes. All coefficient path integrals below use zero outside the common measurable probability-one event on which every drift is locally absolutely integrable and every diffusion entry is locally square-integrable. Such an event is obtained by intersecting the finitely many coefficient conditions at integer horizons; its complement belongs to ${\mathcal{F}}_0$ under the usual conditions. On this event, Cauchy–Schwarz makes every product ${\sigma }^{ik}{\sigma }^{jk}$ locally integrable. The normalized covariance integrals therefore have finite continuous paths and measurable time sections.

1. Existence and value. For every pair $i,j$ the quadratic covariation $[X^i,X^j]$ exists on every finite horizon in the sense of Quadratic covariation of Brownian Ito processes, and for every $t\ge 0$ $[X^i,X^j{]}_t={\sum }_{k=1}^m{\int }_0^t{\sigma }_s^{ik}{\sigma }_s^{jk}ds={\int }_0^t(\sigma {\sigma }^{\mathsf{T}}{)}_s^{ij}ds$ up to indistinguishability, where $(\sigma {\sigma }^{\mathsf{T}}{)}^{ij}={\sum }_k{\sigma }^{ik}{\sigma }^{jk}$. In particular the real continuous Brownian Ito process $X_t=X_0+{\int }_0^t{b}_{s}ds+{\int }_0^t{\sigma }_{s}d{B}_s$ has $[X{]}_t={\int }_0^t{\sigma }_s^{2}ds$.

2. Finite-variation parts contribute nothing. If $A$ is a continuous process whose paths are absolutely continuous, $A_t={\int }_0^t{a}_{s}ds$, and $Y$ is any continuous process, then the covariation of $A$ with $Y$ exists and is the zero process. Consequently in the decomposition $X^i=X_0^i+{\int }_0^t{b}_s^{i}ds+{\sum }_k{\int }_0^t{\sigma }_s^{ik}d{B}_s^k$ the drift part contributes zero cross sums against every continuous process, including against the driving Brownian coordinates.

Van der Vaart, Theorem 5.64, identifies covariation by partition limits; Lemma 5.77 gives the stochastic-integral covariation rule for a locally bounded predictable integrand. The arbitrary-partition, arbitrary locally square-integrable Brownian case here is proved directly by the conditional vector-increment estimate, explicit refinement error, finite-energy approximation and common localization. No independence of the general stochastic integrals is assumed.

Assume the Axiom of Choice and the standing hypothesis (H) of Elementary predictable Brownian integrands. Let $X=X_0+{\int }_0^{\cdot }{b}_s^{X}ds+{\int }_0^{\cdot }{\xi }_{s}d{B}_s$ and $Y=Y_0+{\int }_0^{\cdot }{b}_s^{Y}ds+{\int }_0^{\cdot }{\eta }_{s}d{B}_s$ be real continuous Brownian Ito processes over the same Brownian motion $B$ and filtration Continuous Brownian Ito processes, including its usual-filtration conditions. For the displayed integrands use simultaneous everywhere-continuous representatives of $X,Y$: intersect their measurable full events of continuity and decomposition with the coefficient-integrability events at integer horizons, and set the processes, initial values and coefficients to zero on the complement. This complement is an ${\mathcal{F}}_0$-null event. All conclusions concern the original processes up to indistinguishability and do not depend on this choice. Define the differential integrals ${\int }_0^t{X}_{s}d{Y}_s:={\int }_0^t{X}_s{b}_s^{Y}ds+{\int }_0^t{X}_s{\eta }_{s}d{B}_s,{\int }_0^t{Y}_{s}d{X}_s:={\int }_0^t{Y}_s{b}_s^{X}ds+{\int }_0^t{Y}_s{\xi }_{s}d{B}_s,$ where the stochastic terms are the localized Ito integrals of the predictable locally square-integrable integrands $X\eta$ and $Y\xi$ Localized Ito integral and the Lebesgue terms are pathwise integrals. Then, up to indistinguishability, for every $t\ge 0$ $X_t{Y}_t=X_0{Y}_0+{\int }_0^t{X}_{s}d{Y}_s+{\int }_0^t{Y}_{s}d{X}_s+[X,Y{]}_t,$ with $[X,Y]$ the quadratic covariation of Quadratic covariation of Brownian Ito processes. In particular, when $b^X=b^Y=0$, the centered process $X_t{Y}_t-[X,Y{]}_t-X_0{Y}_0$ is a continuous local martingale. If additionally $E\mathrm{\mid }{X}_0{Y}_0\mathrm{\mid }<\mathrm{\infty }$, then $X_t{Y}_t-[X,Y{]}_t$ is itself a continuous local martingale. In this zero-drift case, if in addition $X\eta$ and $Y\xi$ have finite energy on $[0,t]$, $X_0{Y}_0$ is integrable, and $E{\int }_0^t\mathrm{\mid }{\xi }_s{\eta }_s\mathrm{\mid }ds<\mathrm{\infty }$, then $E[X_t{Y}_t]=E[X_0{Y}_0]+E[X,Y{]}_t$.