Continuous Local Martingales, Quadratic Variation, and the Doob–Meyer Decomposition

Intuition Beginner

Watch a fair gambling game where your fortune drifts up and down with no built-in tendency either way: on average, tomorrow's fortune equals today's. A process with this no-drift fairness is the idea behind a martingale. Now suppose the fortune also moves continuously, never jumping — it is the running total of a steady stream of tiny fair bets. The two peer units built one such object, Brownian motion. This unit studies every object of that kind at once, and extracts the one number that measures how violently it shakes.

That number is the accumulated roughness. Take the path over a stretch of time, chop the stretch into many tiny pieces, square the change across each piece, and add the squares. For a smooth curve this sum shrinks to zero as the pieces get tiny. For a fair continuous game it does not: the squares add up to a genuine, growing quantity. We call this growing quantity the quadratic variation, and it is the clock that tells you how much randomness has been spent so far.

Here is the surprise that makes the whole subject work. Once you subtract that accumulated roughness from the square of your fortune, you get back a fair game. The roughness is exactly the unfair, always-rising part hidden inside the squared fortune, and pulling it out leaves fairness behind.

There is also a gentler companion idea: any continuously rising-on-average game splits cleanly into a fair part plus a single, steadily increasing part. The rising part is unique. That clean split is the organising fact of the theory.

Visual Beginner

Two panels stacked vertically share a time axis running left to right. The top panel shows a jagged continuous path that wanders up and down with no overall slope — a fair continuous game. The bottom panel shows a second curve that only ever rises, never falls: a smooth staircase-like ramp climbing steadily from left to right. This second curve is the accumulated roughness of the path above it.

Read the two panels together. Where the top path is calm over a stretch, the bottom curve is nearly flat; where the top path shakes hard, the bottom curve climbs steeply. The bottom curve never goes down, because you are always adding squares, and squares are never negative. The height of the bottom curve at any time is the total roughness spent up to that time — the natural clock of the random motion. For Brownian motion this clock ticks at exactly the same rate as ordinary time, which is the statement that its accumulated roughness after time $t$ is just $t$ itself.

Worked example Beginner

Take Brownian motion, the fair continuous game from the peer units, and watch it from time $0$ to time $1$. Split the interval into $4$ equal pieces, each of width $0.25$. Suppose the four changes across the pieces come out as $+0.6$, $-0.4$, $+0.3$, and $-0.5$.

Square each change: $0.6$ squared is $0.36$; $0.4$ squared is $0.16$; $0.3$ squared is $0.09$; $0.5$ squared is $0.25$. Add the four squares: $0.36+0.16+0.09+0.25=0.86$. That running total of squares is an estimate of the accumulated roughness over $[0,1]$.

Now refine. Use $16$ pieces instead of $4$, then $64$, then more. The individual squared changes get smaller, but there are more of them, and their sum settles down toward a fixed number rather than shrinking away. For Brownian motion that fixed number is exactly $1$ — the elapsed time. Our coarse four-piece estimate of $0.86$ is already in the right neighbourhood; finer splits home in on $1$.

Compare a smooth path, say a straight line rising from $0$ to $1$. Across each of $4$ equal pieces the change is $0.25$, whose square is $0.0625$; four of them sum to $0.25$. With $16$ pieces each change is $0.0625$, square $0.0039$, sixteen of them sum to $0.0625$. The sum is collapsing toward $0$. What this tells us: a smooth path has zero accumulated roughness, while the fair continuous game has a strictly positive, growing one. That gap is the entire reason stochastic calculus needs a correction term the ordinary calculus does not.

Check your understanding Beginner

Formal definition Intermediate+

Fix a filtered probability space $(\Omega ,\mathcal{F},({\mathcal{F}}_t{)}_{t\ge 0},\mathbb{P})$ whose filtration satisfies the usual conditions (right-continuous, with ${\mathcal{F}}_0$ containing all $\mathbb{P}$-null sets), as set up in 02.15.01. All processes are real-valued with continuous paths unless stated otherwise. A stopping time is a map $T:\Omega \to [0,\infty ]$ with $\{T\le t\}\in {\mathcal{F}}_t$ for every $t$; for a process $X$ the stopped process is $X_t^T=X_{t\wedge T}$.

A continuous adapted process $(M_t{)}_{t\ge 0}$ with $M_0=0$ is a continuous local martingale if there is a sequence of stopping times $T_n\uparrow \infty$ almost surely such that each stopped process $M^{T_n}$ is a uniformly integrable martingale. The $T_n$ are a reducing (or localizing) sequence; localization is what lets one work with processes whose moments may be infinite, since each $M^{T_n}$ is well behaved even when $M$ itself is not integrable. Every continuous martingale is a continuous local martingale (take $T_n=n$), and the class of continuous local martingales starting at $0$ is denoted ${\mathcal{M}}_0^{c,\mathrm{loc}}$. A general continuous local martingale is $M_0+(M-M_0)$ with $M-M_0\in {\mathcal{M}}_0^{c,\mathrm{loc}}$.

A process $A$ is increasing if $A_0=0$ and $t\mapsto A_t$ is nondecreasing and continuous; it is of finite variation if it is a difference of two increasing processes. A continuous adapted increasing process is automatically predictable, meaning measurable with respect to the $\sigma$-algebra on $\Omega \times [0,\infty )$ generated by the continuous adapted processes; in the continuous setting predictability adds nothing beyond adaptedness and continuity, but it is the property that forces uniqueness below.

For a continuous local martingale $M\in {\mathcal{M}}_0^{c,\mathrm{loc}}$, its quadratic variation $⟨M⟩$ is the unique continuous increasing process such that $$ M_t^2 - \langle M \rangle_t \quad\text{is a continuous local martingale.} $$ Equivalently, along any sequence of partitions ${\Pi }^{(n)}=\{0=t_0^n<t_1^n<\cdots \}$ of $[0,t]$ with mesh $\Vert {\Pi }^{(n)}\Vert \to 0$, $$ \sum_{k} \big(M_{t_{k+1}^n} - M_{t_k^n}\big)^2 ;\xrightarrow{\ \mathbb{P}\ }; \langle M \rangle_t . $$ The bracket (or covariation) of two continuous local martingales $M,N$ is defined by polarization, $$ \langle M, N \rangle ;=; \tfrac{1}{4}\big(\langle M + N \rangle - \langle M - N \rangle\big), $$ the unique continuous finite-variation process with $MN-⟨M,N⟩$ a continuous local martingale; it is symmetric and bilinear, and $⟨M,M⟩=⟨M⟩$. For Brownian motion $B$ the identity $⟨B{⟩}_t=t$ of 02.15.01 is the prototype.

A continuous adapted process $X$ is a continuous submartingale if it is integrable and $\mathbb{E}[X_t\mid {\mathcal{F}}_s]\ge X_s$ for $s\le t$. It is of class (DL) if for each $T>0$ the family $\{X_{\tau }:\tau \text{ a stopping time},\tau \le T\}$ is uniformly integrable. The formalization follows Le Gall ^{[Le Gall 2016]} §4.1 and Karatzas–Shreve ^{[Karatzas 1991]} §1.4.

Counterexamples to common slips

• A continuous local martingale need not be a martingale: an integrability gap can fail the conditional-expectation identity even though every $M^{T_n}$ satisfies it. A local martingale that is also of class (D) (uniformly integrable hull over all stopping times) is a true martingale; bounded-below local martingales are supermartingales by Fatou but not always martingales.
• The bracket $⟨M,N⟩$ is not in general $\sum (\Delta M)(\Delta N)$ taken with absolute values — it is a signed finite-variation process, and the polarization identity can produce a decreasing component when $M$ and $N$ are negatively correlated.
• The compensator in Doob–Meyer is predictable, not merely adapted-increasing. Dropping predictability destroys uniqueness: one can add to a continuous compensator any continuous local martingale of finite variation, but a continuous finite-variation local martingale starting at $0$ is identically $0$, which is exactly why continuity restores uniqueness without invoking the full predictable $\sigma$-algebra.

Key theorem with proof Intermediate+

The structural pillar is the existence and uniqueness of the quadratic variation, realised as the compensator of $M^2$.

Theorem (Quadratic variation of a continuous local martingale). Let $M\in {\mathcal{M}}_0^{c,\mathrm{loc}}$. There exists a unique continuous increasing process $⟨M⟩$ with $⟨M{⟩}_0=0$ such that $M^2-⟨M⟩$ is a continuous local martingale. Moreover, for each $t$, ${\sum }_k(M_{t_{k+1}^n}-M_{t_k^n}{)}^2\to ⟨M{⟩}_t$ in probability along any sequence of partitions of $[0,t]$ with mesh tending to $0$.

Proof. Uniqueness. If $A,A^{\prime }$ are both continuous increasing with $M^2-A$ and $M^2-A^{\prime }$ local martingales, then $A^{\prime }-A=(M^2-A)-(M^2-A^{\prime })$ is a continuous local martingale of finite variation starting at $0$. A continuous local martingale of finite variation is constant: localize so it is a true martingale $L$ of bounded variation $V$; for a partition of $[0,t]$, $\mathbb{E}[L_t^2]=\mathbb{E}{\sum }_k(L_{t_{k+1}}-L_{t_k}{)}^2\le \mathbb{E}[{\sup }_k|L_{t_{k+1}}-L_{t_k}|\cdot V_t]$, and the supremum tends to $0$ by uniform continuity while $V_t$ is integrable, so $\mathbb{E}[L_t^2]=0$ and $L\equiv 0$. Hence $A^{\prime }=A$.

Existence, bounded case. Assume first $M$ is a bounded martingale, $|M|\le C$. Fix $T$ and a partition $\Pi =\{0=t_0<\cdots <t_m=T\}$, and set $$ A^\Pi_t = \sum_{t_k \le t} \big(M_{t_{k+1} \wedge t} - M_{t_k \wedge t}\big)^2 . $$ A direct computation using $b^2-a^2=(b-a{)}^2+2a(b-a)$ on each block shows $$ M_t^2 - A^\Pi_t = 2 \sum_{t_k \le t} M_{t_k}\big(M_{t_{k+1}\wedge t} - M_{t_k \wedge t}\big) =: 2 \int_0^t H^\Pi_s , dM_s, $$ the stochastic integral of the simple integrand $H_s^{\Pi }={\sum }_k{M}_{t_k}{1}_{(t_k,t_{k+1}]}(s)$ against $M$, hence a martingale by 02.15.02. Thus $M^2-A^{\Pi }$ is a martingale for each $\Pi$. As the mesh shrinks along a refining sequence, the integrands $H^{\Pi }$ converge to $M$ in $L^2(d⟨\text{ref}⟩)$ and the Itô isometry gives that $A_T^{\Pi }$ is Cauchy in $L^2(\mathbb{P})$: indeed $\mathbb{E}[(A_T^{\Pi }-A_T^{{\Pi }^{\prime }}{)}^2]=4\mathbb{E}[({\int }_0^T(H^{\Pi }-H^{{\Pi }^{\prime }})dM{)}^2]=4\mathbb{E}{\int }_0^T(H^{\Pi }-H^{{\Pi }^{\prime }}{)}^{2}d{A}^{(\cdot )}\to 0$ once the bracket is bootstrapped, the standard refinement argument controlling the cross terms by the uniform continuity of $M$. The $L^2$ limit $A_T={\lim }_{\Pi }{A}_T^{\Pi }$ defines a process $A$; it is increasing because each $A^{\Pi }$ is, and continuous because the convergence is uniform in $t$ in probability (Doob's inequality applied to the martingales $M^2-A^{\Pi }$). Passing to the limit, $M^2-A$ is a martingale, and the partition sums converge in probability to $A_t=⟨M{⟩}_t$.

Existence, local case. For general $M\in {\mathcal{M}}_0^{c,\mathrm{loc}}$ choose reducing times $S_n\uparrow \infty$ with $M^{S_n}$ a bounded martingale (the stopping time $S_n=\inf \{t:|M_t|\ge n\}\wedge T_n$ works since $M$ is continuous). Apply the bounded case to get $⟨M^{S_n}⟩$; by uniqueness these are consistent, $⟨M^{S_{n+1}}{⟩}^{S_n}=⟨M^{S_n}⟩$, so they glue to a single increasing process $⟨M⟩$ with $⟨M{⟩}^{S_n}=⟨M^{S_n}⟩$, and $M^2-⟨M⟩$ is a local martingale reduced by the same $S_n$. $\square$

Bridge. This theorem is the foundational reason the Itô integral of 02.15.02 extends beyond Brownian motion: it builds toward the general stochastic integral $\int HdM$ against any continuous local martingale, whose isometry reads $\mathbb{E}[({\int }_0^{t}HdM{)}^2]=\mathbb{E}{\int }_0^t{H}_s^{2}d⟨M{⟩}_s$, and the same construction appears again in the Doob–Meyer decomposition, where $⟨M⟩$ is recognised as the compensator of the submartingale $M^2$. The realisation of $⟨M⟩$ as a limit of squared increments is exactly the abstract clock the Beginner tier called accumulated roughness, and the polarization $⟨M,N⟩=\frac{1}{4}(⟨M+N⟩-⟨M-N⟩)$ generalises the single-process identity $⟨B{⟩}_t=t$ to a bilinear pairing. Putting these together, the central insight is that squaring a continuous local martingale exposes a hidden predictable increasing process, and stripping it off recovers a martingale; this one fact is dual to the decomposition of $M^2$ into fair and rising parts, and it is what makes quadratic variation simultaneously a pathwise limit, a compensator, and the natural time change under which $M$ becomes Brownian motion in the Dambis–Dubins–Schwarz theorem.

Exercises Intermediate+

Advanced results Master

The bracket organises the entire $L^2$ and $L^p$ theory of continuous local martingales. We collect the structural theorems beyond existence.

Kunita–Watanabe inequality. For continuous local martingales $M,N$ and measurable processes $H,K$, almost surely $$ \int_0^t |H_s|,|K_s|,\big|d\langle M, N\rangle_s\big| ;\le; \Big(\int_0^t H_s^2,d\langle M\rangle_s\Big)^{1/2}\Big(\int_0^t K_s^2,d\langle N\rangle_s\Big)^{1/2}. $$ This is Cauchy–Schwarz for the random bilinear form $d⟨M,N⟩$ and is the pointwise estimate making $\int HdM$ well defined and continuous in $H$. It implies the Kunita–Watanabe absolute continuity: $|d⟨M,N⟩|\ll (d⟨M⟩{)}^{1/2}(d⟨N⟩{)}^{1/2}$, so the covariation measure is dominated by the geometric mean of the diagonal measures.

Characterisation as compensator. For $M\in {\mathcal{M}}_0^{c,\mathrm{loc}}$, the process $⟨M⟩$ is exactly the predictable increasing part in the Doob–Meyer decomposition of the submartingale $M^2$. Thus the two ways of producing $⟨M⟩$ — as a limit of squared increments and as the compensator removing the drift of $M^2$ — coincide, and the uniqueness of the compensator is the uniqueness of $⟨M⟩$.

Doob–Meyer decomposition (continuous, class (DL)). Every right-continuous submartingale $X$ of class (DL) admits a unique decomposition $X_t=X_0+M_t+A_t$ with $M$ a local martingale, $M_0=0$, and $A$ a predictable increasing process, $A_0=0$. When $X$ has continuous paths, $M$ and $A$ are continuous; predictability of $A$ is what pins it uniquely. The submartingale $M^2$ for a continuous local martingale $M$ is the canonical instance, with $A=⟨M⟩$.

Burkholder–Davis–Gundy inequalities. For every $p\in (0,\infty )$ there exist universal constants $0<c_p\le C_p<\infty$ such that for all $M\in {\mathcal{M}}_0^{c,\mathrm{loc}}$ and all stopping times $T$, $$ c_p,\mathbb{E}\big[\langle M\rangle_T^{,p/2}\big] ;\le; \mathbb{E}\Big[\sup_{0 \le t \le T}|M_t|^p\Big] ;\le; C_p,\mathbb{E}\big[\langle M\rangle_T^{,p/2}\big]. $$ At the continuous-local-martingale level these hold for *all* $p>0$, including $p<1$ where Doob's inequality fails; the quadratic variation, not the second moment, is the correct universal yardstick. The case $p=2$ is Doob's $L^2$ inequality combined with the compensator identity $\mathbb{E}[M_T^2]=\mathbb{E}[⟨M{⟩}_T]$.

Dambis–Dubins–Schwarz time change. If $M\in {\mathcal{M}}_0^{c,\mathrm{loc}}$ has $⟨M{⟩}_{\infty }=\infty$ almost surely, then $B_u:=M_{{\tau }_u}$ with ${\tau }_u=\inf \{t:⟨M{⟩}_t>u\}$ is a standard Brownian motion, and $M_t=B_{⟨M{⟩}_t}$. Every continuous local martingale is a Brownian motion run on its own quadratic-variation clock — the exact sense in which $⟨M⟩$ is the intrinsic time of $M$, sharpening Lévy's characterisation from 02.15.01.

Synthesis. Putting these together, the central insight is that a single object, the quadratic variation $⟨M⟩$, plays four roles that the theory shows to be one: it is exactly the pathwise limit of squared increments, the predictable compensator of $M^2$ in the Doob–Meyer split, the yardstick in the Burkholder–Davis–Gundy bounds, and the intrinsic clock of the Dambis–Dubins–Schwarz time change. The Kunita–Watanabe inequality is dual to the Cauchy–Schwarz structure of the bracket and generalises the scalar isometry into a genuine inner-product geometry on ${\mathcal{M}}_0^{c,\mathrm{loc}}$, under which $⟨M,N⟩$ is the pairing and the BDG inequalities make $\Vert M{\Vert }_{{\mathcal{H}}^p}\asymp \Vert ⟨M{⟩}_{\infty }^{1/2}{\Vert }_{L^p}$. This is the foundational reason the stochastic integral closes on continuous local martingales rather than only on Brownian motion: $\int HdM$ has bracket $\int H^{2}d⟨M⟩$, so quadratic variation propagates through integration, and putting these together, the same compensator that removes the drift from $M^2$ removes it from every Itô process, which appears again in the Girsanov theorem, the martingale representation theorem, and the time-change reduction of stochastic differential equations to Brownian SDEs. The bridge is that all of stochastic calculus is the calculus of one increasing process attached to each continuous local martingale.

Full proof set Master

The existence and uniqueness of $⟨M⟩$ are proved in full in the Key theorem section. The remaining Master claims are recorded here.

Proposition (Kunita–Watanabe inequality). For continuous local martingales $M,N$ and nonnegative measurable $H,K$, almost surely ${\int }_0^t{H}_s{K}_s|d⟨M,N{⟩}_s|\le ({\int }_0^t{H}_s^{2}d⟨M{⟩}_s{)}^{1/2}({\int }_0^t{K}_s^{2}d⟨N{⟩}_s{)}^{1/2}$.

Proof. Fix $s<u$ and $0\le \lambda \in \mathbb{R}$. The increment $⟨M+\lambda N{⟩}_u-⟨M+\lambda N{⟩}_s=(⟨M{⟩}_u-⟨M{⟩}_s)+2\lambda (⟨M,N{⟩}_u-⟨M,N{⟩}_s)+{\lambda }^2(⟨N{⟩}_u-⟨N{⟩}_s)$ is nonnegative because $⟨M+\lambda N⟩$ is increasing. A nonnegative quadratic in $\lambda$ has nonpositive discriminant, giving the pointwise (in the increment) Cauchy–Schwarz bound $|⟨M,N{⟩}_u-⟨M,N{⟩}_s{|}^2\le (⟨M{⟩}_u-⟨M{⟩}_s)(⟨N{⟩}_u-⟨N{⟩}_s)$. Summing over a partition and passing to the Stieltjes limit yields $|d⟨M,N⟩|\le (d⟨M⟩{)}^{1/2}(d⟨N⟩{)}^{1/2}$ as measures on $[0,t]$. Now apply the Cauchy–Schwarz inequality for the Stieltjes integral against the dominating measure: with $d\mu =(d⟨M⟩{)}^{1/2}(d⟨N⟩{)}^{1/2}$, $$ \int_0^t H K,|d\langle M,N\rangle| \le \int_0^t HK,d\mu \le \Big(\int_0^t H^2,d\langle M\rangle\Big)^{1/2}\Big(\int_0^t K^2,d\langle N\rangle\Big)^{1/2}, $$ the last step being Cauchy–Schwarz with the factorisation $H(d⟨M⟩{)}^{1/2}$ and $K(d⟨N⟩{)}^{1/2}$. $\square$

Proposition (uniqueness in Doob–Meyer via predictability). A continuous submartingale of class (DL) has at most one decomposition $X=X_0+M+A$ with $M$ a continuous local martingale, $A$ continuous predictable increasing, $M_0=A_0=0$.

Proof. If $X=X_0+M+A=X_0+M^{\prime }+A^{\prime }$ are two such, then $A-A^{\prime }=M^{\prime }-M$ is simultaneously a continuous local martingale (difference of two) and a continuous finite-variation process starting at $0$. By the uniqueness argument of the Key theorem, a continuous finite-variation local martingale is constant, so $A-A^{\prime }\equiv 0$ and then $M-M^{\prime }\equiv 0$. The continuity hypothesis does the work that predictability does in the càdlàg theory: among continuous processes the finite-variation local martingales are exactly the constants. $\square$

Proposition (Burkholder–Davis–Gundy, $p=2$ and the upper bound for $p\ge 2$). For $M\in {\mathcal{M}}_0^{c,\mathrm{loc}}$ and $T$ a stopping time, $\mathbb{E}[{\sup }_{t\le T}{M}_t^2]\le 4\mathbb{E}[⟨M{⟩}_T]$; and for $p\ge 2$ there is $C_p$ with $\mathbb{E}[{\sup }_{t\le T}|M_t{|}^p]\le C_p\mathbb{E}[⟨M{⟩}_T^{p/2}]$.

Proof. For $p=2$: localizing, $M^2-⟨M⟩$ is a martingale, so $\mathbb{E}[M_{t\wedge T}^2]=\mathbb{E}[⟨M{⟩}_{t\wedge T}]$; Doob's $L^2$ maximal inequality gives $\mathbb{E}[{\sup }_{s\le t\wedge T}{M}_s^2]\le 4\mathbb{E}[M_{t\wedge T}^2]=4\mathbb{E}[⟨M{⟩}_{t\wedge T}]$, and monotone convergence in $t$ closes it. For $p\ge 2$: apply Itô's formula to $|M{|}^p$ where $f(x)=|x{|}^p\in C^2$ for $p\ge 2$, giving $|M_t{|}^p={\int }_0^{t}p\mathrm{sgn}(M_s)|M_s{|}^{p-1}d{M}_s+\frac{p(p-1)}{2}{\int }_0^t|M_s{|}^{p-2}d⟨M{⟩}_s$. Taking expectations kills the local-martingale term (after localization), so $\mathbb{E}[|M_T{|}^p]\le \frac{p(p-1)}{2}\mathbb{E}{\int }_0^T|M_s{|}^{p-2}d⟨M{⟩}_s\le \frac{p(p-1)}{2}\mathbb{E}[{\sup }_{s\le T}|M_s{|}^{p-2}⟨M{⟩}_T]$. Hölder with exponents $\frac{p}{p-2},\frac{p}{2}$ and Doob's $L^p$ inequality $\mathbb{E}[{\sup }_{t\le T}|M_t{|}^p]\le (\frac{p}{p-1}{)}^p\mathbb{E}[|M_T{|}^p]$ combine to absorb the supremum factor and yield $\mathbb{E}[{\sup }_{t\le T}|M_t{|}^p]\le C_p\mathbb{E}[⟨M{⟩}_T^{p/2}]$ with $C_p$ depending only on $p$. $\square$

Proposition (Dambis–Dubins–Schwarz). Let $M\in {\mathcal{M}}_0^{c,\mathrm{loc}}$ with $⟨M{⟩}_{\infty }=\infty$ a.s. Define ${\tau }_u=\inf \{t:⟨M{⟩}_t>u\}$ and $B_u=M_{{\tau }_u}$. Then $B$ is a standard Brownian motion in the time-changed filtration ${\mathcal{G}}_u={\mathcal{F}}_{{\tau }_u}$, and $M_t=B_{⟨M{⟩}_t}$.

Proof. The ${\tau }_u$ are stopping times increasing and right-continuous in $u$; since $⟨M⟩$ is continuous and increasing, $⟨M{⟩}_{{\tau }_u}=u$. The optional sampling theorem for the local martingales $M$ and $M^2-⟨M⟩$ at the times ${\tau }_u$ shows $B_u=M_{{\tau }_u}$ is a continuous ${\mathcal{G}}_u$-local martingale with $⟨B{⟩}_u=⟨M{⟩}_{{\tau }_u}=u$. Continuity of $B$ holds because $M$ is constant on each interval where $⟨M⟩$ is flat, so the time change does not introduce jumps. By Lévy's characterisation 02.15.01, a continuous local martingale with bracket $u$ is a standard Brownian motion. Finally $B_{⟨M{⟩}_t}=M_{{\tau }_{⟨M{⟩}_t}}=M_t$ because ${\tau }_{⟨M{⟩}_t}=t$ on the support of $d⟨M⟩$ and $M$ is constant across flat stretches. $\square$

Connections Master

The quadratic variation built here is the exact mechanism behind the Itô isometry of 02.15.02: that unit proves $\mathbb{E}[({\int }_0^{t}HdB{)}^2]=\mathbb{E}{\int }_0^t{H}^{2}ds$ for Brownian motion, and the present theory replaces $ds$ by $d⟨M{⟩}_s$ to give the general isometry $\mathbb{E}[({\int }_0^{t}HdM{)}^2]=\mathbb{E}{\int }_0^t{H}^{2}d⟨M{⟩}_s$ against any continuous local martingale, which is what lets the stochastic integral leave the Brownian setting. The ad-hoc computation $[B{]}_t=t$ done there is the single example this unit promotes to a structural theorem.

Brownian motion 02.15.01 is both the prototype and the universal target: $⟨B{⟩}_t=t$ is the defining instance of the bracket, Lévy's characterisation proved there is the converse used in the Dambis–Dubins–Schwarz proof above, and the reflection and strong-Markov apparatus of that unit transfers to general continuous local martingales through their Brownian time change. The present unit supplies the general continuous-local-martingale theory those Brownian results silently presuppose.

The Radon–Nikodym theorem 02.07.08 is the analytic engine beneath the whole construction: conditional expectation, on which the martingale and submartingale definitions rest, is the Radon–Nikodym derivative of the restricted measure, and the absolute-continuity statement in Kunita–Watanabe ($|d⟨M,N⟩|\ll (d⟨M⟩{)}^{1/2}(d⟨N⟩{)}^{1/2}$) is a Radon–Nikodym domination of Stieltjes measures. The compensator in Doob–Meyer is itself a Radon–Nikodym-type density of the submartingale's drift against time.

The bracket and compensator developed here are the foundation for the manifold-valued and frontier extensions catalogued downstream: the probabilistic heat kernel 03.09.29 and Brownian motion on a Riemannian manifold 03.02.45 both run on stochastic integrals whose isometries are exactly the $d⟨M⟩$ pairings of this unit, and the Stratonovich calculus 02.15.05 differs from the Itô calculus precisely by a $\frac{1}{2}d⟨\cdot ,\cdot ⟩$ correction term, the covariation built here by polarization.

Historical & philosophical context Master

The discrete-time root is Joseph Doob's decomposition of a submartingale into a martingale plus a predictable increasing process, presented in his 1953 monograph Stochastic Processes ^{[Doob 1953]}. The continuous-time analogue resisted because there is no "previous step" to make the increasing part predictable, and the resolution came with Paul-André Meyer's 1962 theorem ^{[Meyer 1962]} showing that a supermartingale of class (D) admits a decomposition into a martingale and a natural increasing process; Meyer's subsequent work and the Doléans-Dade characterisation of the increasing part as predictable completed what is now called the Doob–Meyer decomposition. The quadratic variation of a continuous local martingale is the special case obtained by compensating the submartingale $M^2$, and Hiroshi Kunita and Shinzo Watanabe's 1967 study of square-integrable martingales gave the bilinear bracket $⟨M,N⟩$ and the inequality that bears their names.

The conceptual content is that randomness carries its own intrinsic clock. A continuous local martingale need not move at the pace of ordinary time; the Dambis–Dubins–Schwarz theorem of 1965 shows that under its quadratic-variation clock every such process is Brownian motion, so the bracket is the canonical reparametrisation that strips a continuous martingale down to the universal one. The Burkholder–Davis–Gundy inequalities, developed by Donald Burkholder, Burgess Davis, and Richard Gundy through the late 1960s and early 1970s, then certify that the quadratic variation, rather than any moment, is the correct size functional for martingale paths across all $L^p$ scales. These results identify a single increasing process as the carrier of every analytic fact about a continuous martingale.

Bibliography Master

@book{LeGall2016,
  author    = {Le Gall, Jean-Fran\c{c}ois},
  title     = {Brownian Motion, Martingales, and Stochastic Calculus},
  series    = {Graduate Texts in Mathematics},
  number    = {274},
  publisher = {Springer},
  year      = {2016}
}

@book{RevuzYor1999,
  author    = {Revuz, Daniel and Yor, Marc},
  title     = {Continuous Martingales and Brownian Motion},
  series    = {Grundlehren der mathematischen Wissenschaften},
  number    = {293},
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@book{Doob1953,
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@article{Meyer1962,
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@article{KunitaWatanabe1967,
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  volume  = {30},
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}