Radial second moment of multidimensional Brownian motion

Write $B_t=(B_t^1,\dots ,B_t^d)$. By [F1], each $B_t^{\alpha }$ has law $N(0,t)$, so [F6] gives $E[(B_t^{\alpha }{)}^2]=t$. The coordinate-square sum in [F5] and finite $L^1$ linearity in [F7] therefore give $E\parallel B_t{\parallel }_2^2={\sum }_{\alpha =1}^{d}E[(B_t^{\alpha }{)}^2]=dt\mathrm{.}$ Thus $\parallel B_t{\parallel }_2^2$ and $M_t$ are integrable. The map $x\mapsto \parallel x{\parallel }_2^2$ is continuous, hence Borel by [F5]; since $B_t$ is an observation generating ${\mathcal{F}}_t$, both $\parallel B_t{\parallel }_2^2$ and $M_t$ are ${\mathcal{F}}_t$-measurable. Consequently $M$ is adapted. At $t=0$, the same calculation gives expectation zero although $B_0=0$ is only an almost-sure identity.

Fix $0\le s<t$ and put $\mathrm{\Delta }=B_t-B_s$. Let ${\mathrm{\Pi }}_s$ consist of $\mathrm{\Omega }$ and all finite intersections ${\bigcap }_{j=1}^m\{B_{u_j}\in A_j\},0\le u_j\le s,A_j\in \mathcal{B}({\mathbb{R}}^d)\mathrm{.}$ It is a pi-system and its generated sigma-algebra is ${\mathcal{F}}_s$ by [F2]. Given one such cylinder, sort its distinct positive observation times and adjoin $0,s,t$. On the probability-one event $\{B_0=0\}$, every displayed past observation is a finite cumulative sum of the vector increments ending no later than $s$. Those increments and $\mathrm{\Delta }$ are disjoint groups of the mutually independent increment family in [F1]; [F3] makes their grouped vectors, and then the past observation tuple and $\mathrm{\Delta }$, independent. Replacing the tuple by its almost-surely equal cumulative-sum expression changes the cylinder event by a null set, so for every Borel $C\subseteq {\mathbb{R}}^d$ and every $A\in {\mathrm{\Pi }}_s$, $P(A\cap \{\mathrm{\Delta }\in C\})=P(A)P(\mathrm{\Delta }\in C)\mathrm{.}$ Empty cylinders give $\mathrm{\Omega }$. If $s=0$, every finite past tuple is almost surely the constant zero tuple, and the same null-set argument gives the identity.

Fix a Borel $C\subseteq {\mathbb{R}}^d$ and let ${\mathrm{\Lambda }}_C$ be the events $A\in {\mathcal{F}}_s$ satisfying the factorization in step 1.2. The class contains $\mathrm{\Omega }$; finite differences of nested members follow by subtracting the two finite probability identities, and increasing countable unions follow from probability continuity in [F4]. Thus ${\mathrm{\Lambda }}_C$ is a lambda-system containing ${\mathrm{\Pi }}_s$. By [F4], it contains $\sigma ({\mathrm{\Pi }}_s)={\mathcal{F}}_s$. Since $C$ was arbitrary, $\mathrm{\Delta }$ is independent of ${\mathcal{F}}_s$. For $s=t$, $\mathrm{\Delta }=0$ identically and the same independence conclusion is immediate.

Let $h=t-s\ge 0$. By [F1], the coordinates of $\mathrm{\Delta }$ have laws $N(0,h)$; [F6] and [F7] give $E[{\mathrm{\Delta }}^{\alpha }\mid {\mathcal{F}}_s]=0,E[\parallel \mathrm{\Delta }{\parallel }_2^2\mid {\mathcal{F}}_s]=E\parallel \mathrm{\Delta }{\parallel }_2^2=dh\mathrm{.}$ Indeed each scalar coordinate and the squared norm are Borel functions of the independent vector from step 2.1, and the squared norm is integrable by [F5]--[F7]. Also $B_s^{\alpha }$ and ${\mathrm{\Delta }}^{\alpha }$ are square-integrable by [F1] and [F6], so [F8] makes their product integrable. Because $B_s^{\alpha }$ is ${\mathcal{F}}_s$-measurable, taking out what is known yields $E[B_s^{\alpha }{\mathrm{\Delta }}^{\alpha }\mid {\mathcal{F}}_s]=B_s^{\alpha }E[{\mathrm{\Delta }}^{\alpha }\mid {\mathcal{F}}_s]=0.$ The formulas also hold when $h=0$, where $\mathrm{\Delta }=0$ pointwise.

The pathwise Euclidean identity $\parallel B_t{\parallel }_2^2=\parallel B_s{\parallel }_2^2+2{\sum }_{\alpha =1}^d{B}_s^{\alpha }{\mathrm{\Delta }}^{\alpha }+\parallel \mathrm{\Delta }{\parallel }_2^2$ and conditional linearity now give $E[\parallel B_t{\parallel }_2^2\mid {\mathcal{F}}_s]=\parallel B_s{\parallel }_2^2+d(t-s)\mathrm{.}$ Subtracting the deterministic $dt$ proves $E[M_t\mid {\mathcal{F}}_s]=\parallel B_s{\parallel }_2^2+d(t-s)-dt=M_s\mathrm{.}$ With the adaptation and integrability from step 1.1, this is exactly the all-pairs martingale definition in [F2].

Steps 1.1--4.1 prove both displayed claims. The case $d=1$ is Durrett's scalar square martingale; $d=0$ is excluded. Time zero, $s=0$, $s=t$, and $t=0$ are all covered, and no completed or right-continuously augmented filtration has been substituted for the stated natural filtration. There is no biconditional. AC is used only through [F1], [F6], and [F7]; the finite grid, pi-lambda promotion, and coordinate sum make no additional choices.