Continuous Brownian Ito processes

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.