The Brownian square martingale

Given: AC, (H), the usual conditions, the ${\mathcal{F}}_0$-normalized everywhere-continuous adapted representative of a standard Brownian motion $B$ with $B_0=0$ identically, and a finite horizon $T>0$.

$B$ is a continuous Brownian Ito process. Under the usual conditions the full event on which the Brownian paths are continuous and start at $0$ belongs to ${\mathcal{F}}_0$; setting the process to $0$ off that event preserves adaptedness, finite-dimensional laws, and the increment-independence hypothesis. The resulting everywhere-continuous adapted process is predictable and is a continuous Brownian Ito process with drift $0$ and diffusion coefficient $1$. Continuous Brownian Ito processes Brownian motion

Elementary and localized integral of the constant integrand class. The one-block elementary process $1_{(0,T]}$ represents the same $L^2(\mathrm{d}t\otimes P)$ class as the constant process $1$, and its elementary integral is ${\int }_0^t{1}_{(0,T]}dB=B_t-B_0=B_t$. The $L^2$ integral depends only on that class, and the localized integral of the locally square-integrable constant representative is therefore $B$ up to indistinguishability. Elementary predictable Brownian integrands Ito integral of an elementary predictable process Ito integral for square-integrable predictable processes Localized Ito integral Locally square-integrable predictable Brownian integrands

Ito formula for the class. For $f\in C^{1,2}([0,\mathrm{\infty })\times \mathbb{R})$ the one-dimensional Ito formula of One-dimensional Ito formula gives $f(t,X_t)=f(0,X_0)+{\int }_0^t({\mathrm{\partial }}_{t}f+b{\mathrm{\partial }}_{x}f+\frac{1}{2}{\sigma }^2{\mathrm{\partial }}_x^{2}f)(s,X_s)ds+{\int }_0^t{\sigma }_s{\mathrm{\partial }}_{x}f(s,X_s)d{B}_s$ for every continuous Brownian Ito process $X=X_0+\int b+\int \sigma dB$, up to indistinguishability.

Gaussian moments and Tonelli. $B_s$ has law $N(0,s)$ with density ${\varphi }_s(x)=(2\pi s{)}^{-1\mathrm{/}2}{e}^{-x^2\mathrm{/}(2s)}$ for $s>0$, whence $E{B}_s^2=s$; the function $(s,\omega )\mapsto B_s(\omega {)}^2$ is nonnegative and product measurable, so Tonelli gives $E{\int }_0^t{B}_s^{2}ds={\int }_0^{t}sds=t^2\mathrm{/}2$. Standard normal and normal laws Brownian motion The standard normal density has total mass one Tonelli's theorem for nonnegative measurable functions on a sigma-finite product Convergence in probability

True martingales from finite energy. A finite-energy integral $\int HdB$ has a continuous version that is a square-integrable martingale with $E{\int }_0^{t}HdB=0$ and $E({\int }_0^{t}HdB{)}^2=E{\int }_0^t{H}^{2}ds$. The Ito integral process has a continuous martingale version Ito isometry and linearity in predictable L2 Continuous-time adapted processes and martingales

AC bookkeeping. Choice is declared for the ambient conditional-expectation and completeness interfaces. The Axiom of Choice