What the Construction Already Knows
The previous page built the
Itô integral
\(\int_S^T f(t, \omega)\, dw_t\) for integrands \(f\) in the class
\(\mathcal{V}(S, T)\),
as the \(L^2(\mathbb{P})\) limit of integrals of bounded
elementary processes,
and it closed with a promise. Viewed as a process in its upper limit of integration, the integral
is a martingale and admits a continuous version, and this page keeps that promise.
Throughout, \(w_t\) is the standard one-dimensional Brownian motion of the last two pages, carried
on \((\Omega, \mathcal{F}, \mathbb{P})\), and \(\{\mathcal{F}_t\}_{t \geq 0}\) is the
Brownian filtration,
with its convention that every set of \(\mathbb{P}\)-measure zero belongs to each
\(\mathcal{F}_t\). The norm \(\|f\|_{\mathcal{V}}^2 = \mathbb{E}\bigl[\int_S^T f^2\, dt\bigr]\) is
the one along which elementary processes approximate general integrands.
Before the process view, we collect what the fixed-interval construction already implies. The
four properties below are routinely summarized as "they hold for elementary processes, hence
in the limit". In fact each one crosses the limit by a different mechanism. The first travels
by an exact identity between approximating sums, the second by the linear structure of
\(L^2\) limits, the third by convergence of expectations, and the fourth by the measurability
of almost-everywhere limits.
We cross each bridge in full, because two of them collect a toll. The third needs a norm
comparison between \(L^1\) and \(L^2\), and the fourth needs precisely the null-set convention
quoted above, adopted on the previous page.
Theorem: Properties of the Itô Integral
Let \(0 \leq S \lt U \lt T\), let \(f, g \in \mathcal{V}(S, T)\), and let \(a, b \in \mathbb{R}\).
Then \(f \in \mathcal{V}(S, U)\) and \(f \in \mathcal{V}(U, T)\), so that all integrals below
are defined, and the following hold almost surely.
(i) Additivity over intervals.
\[
\int_S^T f\, dw_t = \int_S^U f\, dw_t + \int_U^T f\, dw_t .
\]
(ii) Linearity.
\[
\int_S^T (a f + b g)\, dw_t = a \int_S^T f\, dw_t + b \int_S^T g\, dw_t .
\]
(iii) Zero mean.
\[
\mathbb{E}\Bigl[ \int_S^T f\, dw_t \Bigr] = 0 .
\]
(iv) Measurability.
Every representative of the class
\(\int_S^T f\, dw_t \in L^2(\mathbb{P})\) is \(\mathcal{F}_T\)-measurable.
Proof
We first verify the opening claim. Conditions (V1) and (V2) defining
\(\mathcal{V}\)
constrain \(f\) as a function on all of \([0, \infty) \times \Omega\) and do not mention the
interval. Only (V3) does, and since \(f^2 \geq 0\), the integral of \(f^2\) over \([S, U]\) or
\([U, T]\) is bounded by the integral over \([S, T]\). Hence membership in
\(\mathcal{V}(S, T)\) implies membership in \(\mathcal{V}(S, U)\) and \(\mathcal{V}(U, T)\).
Throughout the proof, fix a sequence \(\{\phi_n\}\) of bounded elementary processes with
\(\|f - \phi_n\|_{\mathcal{V}} \to 0\) on \([S, T]\). Such a sequence exists by the three
approximation steps of the previous page.
(i).
Refine the partition of each \(\phi_n\) so that it contains the
point \(U\). As noted with the definition of the elementary integral, refinement changes
neither the function nor its integral. Now split each \(\phi_n\) at \(U\):
\[
\begin{align*}
\psi_n &= \phi_n\, \mathbf{1}_{[S, U)} \\\\
\psi_n' &= \phi_n\, \mathbf{1}_{[U, T)} .
\end{align*}
\]
The process \(\psi_n\) is bounded elementary on \([S, U]\). Its partition is the part of the
refined partition up to \(U\), its levels are those of \(\phi_n\), and multiplication by a
deterministic indicator disturbs neither joint measurability nor adaptedness. The same holds
for \(\psi_n'\) on \([U, T]\). Splitting the defining sum of the elementary integral at \(U\)
gives, for every \(\omega\),
\[
\int_S^T \phi_n\, dw_t = \int_S^U \psi_n\, dw_t + \int_U^T \psi_n'\, dw_t .
\]
The sequences are admissible for \(f\) on the subintervals. Since \(\psi_n = \phi_n\) on
\([S, U)\) and the endpoint \(\{U\}\) has Lebesgue measure zero,
\[
\begin{align*}
\mathbb{E}\Bigl[ \int_S^U (f - \psi_n)^2\, dt \Bigr]
&= \mathbb{E}\Bigl[ \int_S^U (f - \phi_n)^2\, dt \Bigr] \\\\
&\leq \|f - \phi_n\|_{\mathcal{V}}^2 \\\\
&\longrightarrow 0 ,
\end{align*}
\]
and likewise on \([U, T]\). By the definition of the integral, the three integrals of
\(\phi_n\), \(\psi_n\), \(\psi_n'\) converge in \(L^2(\mathbb{P})\) to the three integrals of
\(f\) over \([S, T]\), \([S, U]\), \([U, T]\) respectively. Limits in \(L^2\) respect sums,
and two \(L^2\) limits of the same sequence agree almost surely, so the displayed identity
survives the passage, and (i) holds almost surely.
(ii).
This was established in the course of proving that the
definition of the integral
is legitimate, by approximating \(f\) and \(g\) separately, combining the approximants linearly,
and using linearity of the elementary integral together with linearity of \(L^2\) limits. We
record the statement here for reference and add nothing to that argument.
(iii).
For a bounded elementary process
\(\phi_n = \sum_j e_j\, \mathbf{1}_{[t_j, t_{j+1})}\), expectation is linear over the finite
defining sum by the
properties of expectation, so
\[
\begin{align*}
\mathbb{E}\Bigl[ \int_S^T \phi_n\, dw_t \Bigr]
&= \sum_{j} \mathbb{E}\bigl[ e_j\, \Delta w_j \bigr] , \\\\
\Delta w_j &= w_{t_{j+1}} - w_{t_j} .
\end{align*}
\]
Each term vanishes by the argument that powered the elementary isometry. The level \(e_j\) is
\(\mathcal{F}_{t_j}\)-measurable and bounded, the increment \(\Delta w_j\) is
independent of \(\mathcal{F}_{t_j}\), the
expectation of the product factors,
and increments are centered by (W1) of the
definition of Brownian motion.
Hence \(\mathbb{E}\bigl[ \int_S^T \phi_n\, dw_t \bigr] = 0\) for every \(n\).
For the limit, compare the norms of \(L^1\) and \(L^2\). Write
\(X_n = \int_S^T f\, dw_t - \int_S^T \phi_n\, dw_t\). Since the subtracted term has
expectation zero by the above, the triangle inequality for expectation and the
Cauchy-Schwarz inequality
in the inner product space \(L^2(\mathbb{P})\), applied against the constant function
\(\mathbf{1}\) with \(\|\mathbf{1}\|_{L^2(\mathbb{P})} = 1\), give
\[
\begin{align*}
\Bigl| \mathbb{E}\Bigl[ \int_S^T f\, dw_t \Bigr] \Bigr|
&= \bigl| \mathbb{E}[X_n] \bigr| \\\\
&\leq \mathbb{E}\bigl[ |X_n| \bigr] \\\\
&\leq \| X_n \|_{L^2(\mathbb{P})} \\\\
&\longrightarrow 0 ,
\end{align*}
\]
the convergence being the definition of the integral. The left side does not depend on \(n\),
so it is zero.
(iv).
We first exhibit one \(\mathcal{F}_T\)-measurable representative
and then extend to all of them. Each elementary integral \(\sum_j e_j\, \Delta w_j\) is a
genuine function on \(\Omega\), and it is \(\mathcal{F}_T\)-measurable. Every level \(e_j\) is
\(\mathcal{F}_{t_j}\)-measurable with \(\mathcal{F}_{t_j} \subseteq \mathcal{F}_T\). Every
increment is a difference of values of the path at times at most \(T\), each
\(\mathcal{F}_T\)-measurable by the definition of the
Brownian filtration.
Sums and products of \(\mathcal{F}_T\)-measurable functions are \(\mathcal{F}_T\)-measurable,
the closure already used on the previous page.
Since
\[
\int_S^T \phi_n\, dw_t \to \int_S^T f\, dw_t
\]
in \(L^2(\mathbb{P})\), a
subsequence converges almost surely. We write
\[
Z_k = \int_S^T \phi_{n_k}\, dw_t
\]
for its terms, and let \(\widetilde{X}\) equal \(\lim_k Z_k\) where that limit exists in
\(\mathbb{R}\), and \(0\) elsewhere. Measurability survives countable suprema and infima (for
instance \(\{\sup_k Z_k \leq a\} = \bigcap_k \{Z_k \leq a\}\)), so \(\limsup_k Z_k\) and
\(\liminf_k Z_k\) are \(\mathcal{F}_T\)-measurable maps into \([-\infty, \infty]\), the set
where they agree and are finite lies in \(\mathcal{F}_T\), and \(\widetilde{X}\) is
\(\mathcal{F}_T\)-measurable. Being the almost-sure limit of a subsequence converging in
\(L^2\), \(\widetilde{X}\) is a representative of \(\int_S^T f\, dw_t\).
Now let \(Y\) be any representative. As an element of \(L^2(\mathbb{P})\) it is
\(\mathcal{F}\)-measurable, and \(Y = \widetilde{X}\) off a \(\mathbb{P}\)-null set. For each
\(a \in \mathbb{R}\), the symmetric difference
\(D_a = \{Y \leq a\} \,\triangle\, \{\widetilde{X} \leq a\}\) is an \(\mathcal{F}\)-measurable
subset of \(\{Y \neq \widetilde{X}\}\), hence a set of \(\mathbb{P}\)-measure zero, hence a
member of \(\mathcal{F}_T\). This is the moment the null-set convention adopted with the
Brownian filtration earns its keep. Since \(A \,\triangle\, (A \,\triangle\, B) = B\) for any
sets \(A, B\), we conclude
\[
\{Y \leq a\} = \{\widetilde{X} \leq a\} \,\triangle\, D_a \in \mathcal{F}_T,
\]
and \(Y\) is \(\mathcal{F}_T\)-measurable by the
half-line criterion
applied to \(\mathcal{F}_T\).
Property (i) is the license to let the upper endpoint move. For \(f \in \mathcal{V}(0, T)\) and
\(t \in (0, T]\), the integral \(\int_0^t f\, dw_s\) is defined by the opening claim of the
theorem, and additivity makes the family of integrals over nested intervals consistent. The
integral up to \(t\) plus the increment from \(t\) to \(t'\) is the integral up to \(t'\). Writing
\[
M_t(\omega) = \int_0^t f(s, \omega)\, dw_s ,
\quad 0 \leq t \leq T ,
\]
with the convention \(M_0 = 0\), we thus obtain not a single random variable but
a stochastic process.
Property (iv), applied on the interval \([0, t]\), says that \(M_t\) is
\(\mathcal{F}_t\)-measurable for every \(t\), so the process \(\{M_t\}\) is
\(\mathcal{F}_t\)-adapted and inherits the causality of its integrand. Property (iii) says that
each \(M_t\) is centered. The next section names the structure that these facts point to and
proves that \(\{M_t\}\) has it.
Processes That Forecast Themselves
The process \(\{M_t\}\) assembled at the end of the last section is adapted, centered, and driven
by increments that the present cannot foresee. There is a classical name for processes with this
character. The defining property behind that name is a statement about prediction. Given
everything observable now, the expected value of the process at any future time is its value now.
Gains and losses are possible, but neither is favored. The process models a fair game.
The definition is stated relative to an arbitrary
filtration,
not only the Brownian one, because the notion belongs to any setting where information accumulates
over time. The prediction itself is the
conditional expectation
given the current \(\sigma\)-algebra.
Definition: Martingale
Let \(\{\mathcal{M}_t\}_{t \geq 0}\) be a filtration on \((\Omega, \mathcal{F}, \mathbb{P})\).
A real-valued stochastic process \(\{X_t\}_{t \geq 0}\) is a martingale with
respect to \(\{\mathcal{M}_t\}\) (and with respect to \(\mathbb{P}\)) if:
(M1)
The process \(\{X_t\}\) is
\(\mathcal{M}_t\)-adapted.
(M2)
\(\mathbb{E}\bigl[ |X_t| \bigr] \lt \infty\) for every \(t \geq 0\).
(M3)
\(\mathbb{E}\bigl[ X_s \mid \mathcal{M}_t \bigr] = X_t\) almost surely,
for all \(0 \leq t \leq s\).
Condition (M2) makes the conditional expectation in (M3) meaningful. The definition applies
verbatim when the time index ranges over a bounded interval \([0, T]\), the case our integrals
inhabit.
We note two immediate readings. First, condition (M3) and the averaging identity of conditional
expectation, applied with \(A = \Omega\), give
\[
\mathbb{E}[X_s] = \mathbb{E}\bigl[ \mathbb{E}[X_s \mid \mathcal{M}_t] \bigr] = \mathbb{E}[X_t],
\]
so a martingale has constant expectation. Second, and stronger, (M3) pins down not just the
average over all of \(\Omega\) but the average over every event observable at time \(t\). On each
such event, the future is expected to go nowhere. Constancy of expectation is what remains of the
martingale property when the information structure is thrown away.
The prototype is Brownian motion itself, with respect to its own filtration. The proof is a
dividend of machinery already in place. Every step is a citation.
Theorem: Brownian Motion Is a Martingale
The process \(\{w_t\}_{t \geq 0}\) is a martingale with respect to the Brownian filtration
\(\{\mathcal{F}_t\}_{t \geq 0}\).
Proof
(M1).
The event \(\{w_t \in F\}\), for \(F\) Borel, is one of the generating
sets of the
Brownian filtration
at time \(t\), so \(w_t\) is \(\mathcal{F}_t\)-measurable.
(M2).
By the
Cauchy-Schwarz inequality
in \(L^2(\mathbb{P})\) against the constant function \(\mathbf{1}\), the same \(L^1\)-\(L^2\)
comparison as in the previous section,
\[
\mathbb{E}[|w_t|] \leq \bigl( \mathbb{E}[w_t^2] \bigr)^{1/2} = \sqrt{t} \lt \infty ,
\]
the variance being \(t\) by (W1) of the
definition of Brownian motion.
(M3).
Let \(0 \leq t \leq s\) and write \(w_s = (w_s - w_t) + w_t\), both summands integrable
(\(w_t\) by the bound above, the increment by the triangle inequality). By
linearity of conditional expectation,
\[
\mathbb{E}[w_s \mid \mathcal{F}_t]
= \mathbb{E}[w_s - w_t \mid \mathcal{F}_t] + \mathbb{E}[w_t \mid \mathcal{F}_t] .
\]
For the first term, if \(t = s\) the increment is \(0\) and its conditional expectation
vanishes trivially. For \(t \lt s\) the increment is
independent of \(\mathcal{F}_t\).
The lemma states precisely that every event \(\{w_s - w_t \in B\}\) decouples from every event
in \(\mathcal{F}_t\), and such events constitute all of \(\sigma(w_s - w_t)\), so the
independence collapse
applies and yields
\[
\mathbb{E}[w_s - w_t \mid \mathcal{F}_t] = \mathbb{E}[w_s - w_t] = 0 ,
\]
increments being centered by (W1). For the second term, \(w_t\) is
\(\mathcal{F}_t\)-measurable and integrable, so the
take-out property
with the constant factor \(X = \mathbf{1}\) gives
\[
\mathbb{E}[w_t \mid \mathcal{F}_t] = w_t \cdot \mathbb{E}[\mathbf{1} \mid \mathcal{F}_t] = w_t .
\]
Hence, almost surely,
\[
\mathbb{E}[w_s \mid \mathcal{F}_t] = w_t .
\]
Brownian motion is the raw material, and the Itô integral is machined from it. The next
theorem says the machining preserves the fair-game property, and the reason is the arrow of time
built into the class \(\mathcal{V}\). Because the integrand is adapted, every increment of the
integral pairs a level readable from the past with a piece of noise invisible to it, and the
conditional expectation of such a pairing vanishes.
Theorem: The Itô Integral Is a Martingale
Let \(f \in \mathcal{V}(0, T)\). Then the process
\[
M_t(\omega) = \int_0^t f(s, \omega)\, dw_s ,
\quad 0 \leq t \leq T ,
\]
with the convention \(M_0 = 0\), is a martingale with respect to
\(\{\mathcal{F}_t\}_{0 \leq t \leq T}\).
Proof
(M1).
This is part (iv) of the
properties theorem
applied on the interval \([0, t]\), and is trivial at \(t = 0\).
(M2).
This is quantitative. By the
Itô isometry
on \([0, t]\) and the \(L^1\)-\(L^2\) comparison,
\[
\begin{align*}
\mathbb{E}\bigl[ |M_t| \bigr]
&\leq \bigl( \mathbb{E}[M_t^2] \bigr)^{1/2} \\\\
&= \Bigl( \mathbb{E}\Bigl[ \int_0^t f^2\, ds \Bigr] \Bigr)^{1/2} \\\\
&\leq \|f\|_{\mathcal{V}} \lt \infty ,
\end{align*}
\]
the last bound because \(f^2 \geq 0\) makes the integral monotone in the interval.
(M3).
Fix \(0 \leq t \leq s \leq T\). If \(t = s\) the claim is
\(\mathbb{E}[M_t \mid \mathcal{F}_t] = M_t\), which is the take-out property as in the
previous proof. Assume now \(t \lt s\). The membership \(f \in \mathcal{V}(0, s)\) holds by
the same monotonicity of (V3), so for \(t \gt 0\), additivity (part (i) of the properties
theorem, applied with the triple \((0, t, s)\)) gives, almost surely,
\[
M_s = M_t + \int_t^s f\, dw_u ,
\]
and this identity holds trivially at \(t = 0\), where \(M_t = 0\) by convention and the
remaining integral is \(M_s\) itself.
Conditional expectations of almost-surely equal variables agree almost surely, by the
uniqueness clause in their definition. By linearity of conditional expectation and, once more,
the take-out property, \(\mathbb{E}[M_t \mid \mathcal{F}_t] = M_t\), so everything reduces to
a conditional zero mean. Almost surely,
\[
\mathbb{E}\Bigl[ \int_t^s f\, dw_u \,\Bigm|\, \mathcal{F}_t \Bigr] = 0 .
\]
Step 1: bounded elementary integrands.
Membership \(f \in \mathcal{V}(t, s)\) holds because (V1) and (V2) are interval-free and (V3)
is monotone in the interval, the argument that opened the properties theorem. The
approximation procedure of the previous page, applied on \([t, s]\), therefore supplies bounded
elementary processes
\(\phi_n\) with \(\mathbb{E}\bigl[ \int_t^s (f - \phi_n)^2\, du \bigr] \to 0\). Fix one such
\(\phi = \sum_j e_j \mathbf{1}_{[u_j, u_{j+1})}\) with partition
\(t = u_0 \lt u_1 \lt \cdots \lt u_m = s\) and bounded levels \(e_j\), each
\(\mathcal{F}_{u_j}\)-measurable by adaptedness. Write \(\Delta w_j = w_{u_{j+1}} - w_{u_j}\)
for the matching increments, so that \(\int_t^s \phi\, dw_u = \sum_j e_j\, \Delta w_j\).
For a single term, since \(\mathcal{F}_t \subseteq \mathcal{F}_{u_j}\) and
\(e_j\, \Delta w_j \in L^1(\mathbb{P})\), the
tower property
lets us condition first on the larger \(\sigma\)-algebra:
\[
\begin{align*}
\mathbb{E}\bigl[ e_j\, \Delta w_j \mid \mathcal{F}_t \bigr]
&= \mathbb{E}\Bigl[\, \mathbb{E}\bigl[ e_j\, \Delta w_j \mid \mathcal{F}_{u_j} \bigr] \,\Bigm|\, \mathcal{F}_t \Bigr] \\\\
&= \mathbb{E}\Bigl[\, e_j\, \mathbb{E}\bigl[ \Delta w_j \mid \mathcal{F}_{u_j} \bigr] \,\Bigm|\, \mathcal{F}_t \Bigr] \\\\
&= 0 .
\end{align*}
\]
Here the middle equality is the take-out property, since the factor \(e_j\) is
\(\mathcal{F}_{u_j}\)-measurable and bounded and the increment is integrable. The last
equality holds because
\(\mathbb{E}[\Delta w_j \mid \mathcal{F}_{u_j}] = \mathbb{E}[\Delta w_j] = 0\) by increment
independence, the independence collapse, and centeredness, the same chain as for Brownian
motion above. Summing over \(j\) by linearity, we obtain
\[
\mathbb{E}\bigl[ \int_t^s \phi\, dw_u \mid \mathcal{F}_t \bigr] = 0 .
\]
The elementary integral over the future is invisible to the present.
Step 2: passage to the limit.
By the definition of the integral, \(\int_t^s \phi_n\, dw_u \to \int_t^s f\, dw_u\) in
\(L^2(\mathbb{P})\). Conditional expectation given \(\mathcal{F}_t\) is linear and, by the
\(L^p\) contraction
with \(p = 2\), a contraction of \(L^2(\mathbb{P})\). Hence, using Step 1 for each
\(n\), we find
\[
\begin{align*}
\Bigl\| \mathbb{E}\Bigl[ \int_t^s f\, dw_u \Bigm| \mathcal{F}_t \Bigr] \Bigr\|_{L^2(\mathbb{P})}
&= \Bigl\| \mathbb{E}\Bigl[ \int_t^s (f - \phi_n)\, dw_u \Bigm| \mathcal{F}_t \Bigr] \Bigr\|_{L^2(\mathbb{P})} \\\\
&\leq \Bigl\| \int_t^s (f - \phi_n)\, dw_u \Bigr\|_{L^2(\mathbb{P})} \\\\
&\longrightarrow 0 ,
\end{align*}
\]
the first equality also using linearity of the integral and of conditional expectation. A
vector of norm zero in \(L^2(\mathbb{P})\) vanishes almost surely. This establishes the
conditional zero mean, and by the reduction at the start of this proof, condition (M3) of the
martingale definition follows.
Fair Noise Is What Stochastic Gradient Descent Actually Needs
A stochastic optimization step
\[
\boldsymbol{\theta}_{k+1} = \boldsymbol{\theta}_k - \eta \bigl( \nabla L(\boldsymbol{\theta}_k) + \boldsymbol{\xi}_k \bigr)
\]
is driven by gradient noise \(\boldsymbol{\xi}_k\). Convergence analyses need not assume that
the \(\boldsymbol{\xi}_k\) are independent. For minibatch gradient noise, independence across
steps typically fails, since the noise at step \(k\) depends on the current iterate
\(\boldsymbol{\theta}_k\), itself a function of all earlier noise. What many analyses assume
instead is that \(\mathbb{E}[\boldsymbol{\xi}_k \mid \mathcal{M}_k] = 0\), where
\(\mathcal{M}_k\) is the \(\sigma\)-algebra generated by the run so far. This conditional
centering is exactly the discrete-time shadow of the conditional zero mean proved in Step 1.
Each noise term, conditioned on the accumulated history, contributes nothing.
Sequences with this property are called martingale differences, because their partial sums
form a discrete-time martingale by the same tower-property computation as above. The maximal
inequalities of the next section are among the standard tools for controlling their cumulative
effect over an entire run rather than one step at a time.
The martingale property, for all its strength, is a statement about two time points. An inequality
due to Doob removes that limitation, and we prove it next.
Doob's Maximal Inequality
Everything proved so far constrains the integral at fixed times, or at pairs of times. What
remains outstanding, a continuous version and with it control of a whole trajectory,
concerns uncountably many times at once, and no union bound survives that passage. Summing a
per-time estimate that does not shrink from one time to the next over even countably many
times destroys it.
What saves the situation is the fair-game structure itself. If a martingale crosses a high level
\(\lambda\) at some moment and yet ends small, the path must lose ground systematically after the
crossing, and (M3) says the process expects to lose no ground on any event observable at the
crossing time. Made precise, this reasoning bounds the probability that the running maximum ever
reaches \(\lambda\) by an expectation involving the final value alone. The argument runs
most naturally not for martingales but for processes allowed to drift upward, and the inequality
is inherited by \(|M_t|^p\) from the martingale \(M_t\) precisely through that one-sided slack.
Definition: Submartingale
Let \(\{\mathcal{M}_t\}_{t \geq 0}\) be a filtration on \((\Omega, \mathcal{F}, \mathbb{P})\).
A real-valued process \(\{X_t\}_{t \geq 0}\) is a submartingale with respect
to \(\{\mathcal{M}_t\}\) if it satisfies (M1) and (M2) of the definition of a
martingale, together with
(M3')
\(\mathbb{E}\bigl[ X_s \mid \mathcal{M}_t \bigr] \geq X_t\) almost
surely, for all \(0 \leq t \leq s\).
A submartingale is a game tilted in the player's favor. The forecast of the future, given the
present, never falls below the present. Reversing the inequality defines a
supermartingale, which our proofs will not need. Averaging (M3') over \(A = \Omega\)
shows that the expectation \(t \mapsto \mathbb{E}[X_t]\) of a submartingale is nondecreasing, and
every martingale is in particular a submartingale. The maximal inequality is proved first for
finitely many time points, because this is where all the probabilistic content lives. The passage
to the continuum afterwards needs only continuity of measure and path continuity.
Lemma: Maximal Inequality over Finitely Many Times
Let \(\{X_t\}\) be a nonnegative submartingale with respect to
\(\{\mathcal{M}_t\}\), let \(0 \leq t_0 \lt t_1 \lt \cdots \lt t_n\) be times in its
index set, and let \(\lambda \gt 0\). Then
\[
\begin{align*}
\lambda\, \mathbb{P}\Bigl[ \max_{0 \leq k \leq n} X_{t_k} \geq \lambda \Bigr]
&\leq \mathbb{E}\Bigl[ X_{t_n}\, \mathbf{1}_{\{\max_k X_{t_k} \geq \lambda\}} \Bigr] \\\\
&\leq \mathbb{E}\bigl[ X_{t_n} \bigr] .
\end{align*}
\]
Proof
Decompose the event \(A = \{\max_k X_{t_k} \geq \lambda\}\) by the first time the
level is reached. For \(0 \leq k \leq n\), let
\[
A_k = \bigl\{ X_{t_0} \lt \lambda, \ldots, X_{t_{k-1}} \lt \lambda, X_{t_k} \geq \lambda \bigr\} .
\]
The \(A_k\) are pairwise disjoint with union \(A\), and \(A_k \in \mathcal{M}_{t_k}\). Each
event \(\{X_{t_j} \lt \lambda\}\) with \(j \lt k\) lies in
\(\mathcal{M}_{t_j} \subseteq \mathcal{M}_{t_k}\), and \(\{X_{t_k} \geq \lambda\}\) lies in
\(\mathcal{M}_{t_k}\), both by adaptedness (M1).
On \(A_k\) the process has reached the level, so
\(\lambda\, \mathbf{1}_{A_k} \leq X_{t_k}\, \mathbf{1}_{A_k}\) pointwise, and by monotonicity
of expectation, among the
properties of expectation,
\[
\lambda\, \mathbb{P}(A_k) \leq \mathbb{E}\bigl[ X_{t_k}\, \mathbf{1}_{A_k} \bigr] .
\]
Now trade the time-\(t_k\) value for the terminal one. The submartingale inequality (M3')
gives \(X_{t_k} \leq \mathbb{E}[X_{t_n} \mid \mathcal{M}_{t_k}]\) almost surely. Multiplying
by the nonnegative \(\mathbf{1}_{A_k}\) and taking expectations gives
\[
\begin{align*}
\mathbb{E}\bigl[ X_{t_k}\, \mathbf{1}_{A_k} \bigr]
&\leq \mathbb{E}\bigl[\, \mathbb{E}[X_{t_n} \mid \mathcal{M}_{t_k}]\, \mathbf{1}_{A_k} \bigr] \\\\
&= \int_{A_k} \mathbb{E}[X_{t_n} \mid \mathcal{M}_{t_k}]\, d\mathbb{P} \\\\
&= \int_{A_k} X_{t_n}\, d\mathbb{P} ,
\end{align*}
\]
the last step being the averaging identity that
defines conditional expectation,
legitimate because \(A_k \in \mathcal{M}_{t_k}\). This is the fair-game mechanism in one line.
On an event decided by time \(t_k\), the terminal value integrates to at least as much as the
value at the crossing.
Summing over \(k\) and using disjointness, we arrive at
\[
\begin{align*}
\lambda\, \mathbb{P}(A)
&= \sum_{k=0}^{n} \lambda\, \mathbb{P}(A_k) \\\\
&\leq \sum_{k=0}^{n} \mathbb{E}\bigl[ X_{t_n}\, \mathbf{1}_{A_k} \bigr] \\\\
&= \mathbb{E}\bigl[ X_{t_n}\, \mathbf{1}_{A} \bigr] \\\\
&\leq \mathbb{E}\bigl[ X_{t_n} \bigr] ,
\end{align*}
\]
the final inequality because \(X_{t_n} \geq 0\).
To apply the lemma to a martingale \(M_t\), we feed it \(|M_t|^p\). The observation that convex
images of martingales drift upward is worth isolating, since it is the standard device for converting
two-sided processes into nonnegative ones without losing the filtration structure.
Lemma: Convex Images of Martingales Are Submartingales
Let \(\{X_t\}\) be a martingale with respect to \(\{\mathcal{M}_t\}\), and let
\(\varphi : \mathbb{R} \to \mathbb{R}\) be a
convex function
with \(\varphi(X_t) \in L^1(\mathbb{P})\) for every \(t\). Then \(\{\varphi(X_t)\}\) is a
submartingale with respect to \(\{\mathcal{M}_t\}\).
Proof
A convex function on \(\mathbb{R}\) is continuous, one of the two standard facts that the
supporting-line proof of conditional Jensen takes without proof and that we likewise take for
granted here. Hence \(\varphi\) is Borel measurable, \(\varphi(X_t)\) is
\(\mathcal{M}_t\)-measurable, and (M1) holds. (M2) is the integrability hypothesis. For (M3'),
let \(0 \leq t \leq s\). Since \(X_s \in L^1(\mathbb{P})\) by (M2) for the martingale and
\(\varphi(X_s) \in L^1(\mathbb{P})\) by hypothesis,
Jensen's inequality for conditional expectation
applies, and almost surely
\[
\begin{align*}
\varphi(X_t)
&= \varphi\bigl( \mathbb{E}[X_s \mid \mathcal{M}_t] \bigr) \\\\
&\leq \mathbb{E}\bigl[ \varphi(X_s) \mid \mathcal{M}_t \bigr] ,
\end{align*}
\]
the first equality being the martingale property (M3) inside the continuous \(\varphi\).
One further reading convention is needed before the main theorem can be stated. A supremum of
\(|M_t|\) over the uncountable set \([0, T]\) is a supremum of uncountably many random variables
and need not be measurable, so the probability appearing below must be read with care. Given the
horizon \(T\), we take the supremum along the countable set \(D = \bigcup_N D_N\) of dyadic times
\[
D_N = \{ k T 2^{-N} : k = 0, 1, \ldots, 2^N \} ,
\]
which is dense in \([0, T]\) and contains \(0\) and \(T\). A countable supremum of measurable
functions is measurable, as in part (iv) of the properties theorem.
For almost every \(\omega\) the path \(t \mapsto M_t(\omega)\) is continuous by the hypothesis
of the theorem below, so \(t \mapsto |M_t(\omega)|\) is continuous and its supremum over the
dense set \(D\) equals its supremum over all of \([0, T]\). The reading convention costs nothing
where it matters.
Theorem: Doob's Martingale Inequality
Let \(T \geq 0\) and let \(\{M_t\}_{0 \leq t \leq T}\) be a martingale with respect to a
filtration \(\{\mathcal{M}_t\}\), such that \(t \mapsto M_t(\omega)\) is continuous for almost
every \(\omega\). Then for every \(p \geq 1\) and \(\lambda \gt 0\),
\[
\mathbb{P}\Bigl[ \sup_{0 \leq t \leq T} |M_t| \geq \lambda \Bigr]
\leq \frac{1}{\lambda^p}\, \mathbb{E}\bigl[ |M_T|^p \bigr] ,
\]
the supremum being taken over the dyadic times \(k T 2^{-N}\), with \(N \in \mathbb{N}\) and
\(k = 0, 1, \ldots, 2^N\), and for almost every \(\omega\) coinciding with the supremum over
all of \([0, T]\).
Proof
If \(\mathbb{E}[|M_T|^p] = \infty\) there is nothing to prove, so assume it finite. The map
\(x \mapsto |x|^p\) is convex for \(p \geq 1\). On \([0, \infty)\) the power
\(u \mapsto u^p\) is convex and nondecreasing, its derivative \(p\, u^{p-1}\) being
nonnegative and nondecreasing, and composing a nondecreasing convex function with the convex
\(x \mapsto |x|\) preserves convexity. Moreover \(|M_t|^p\) is integrable for every
\(t \in [0, T]\). The martingale property gives \(M_t = \mathbb{E}[M_T \mid \mathcal{M}_t]\)
almost surely, so the
\(L^p\) contraction
yields
\[
\|M_t\|_{L^p(\mathbb{P})} \leq \|M_T\|_{L^p(\mathbb{P})} \lt \infty .
\]
By the
convex-image lemma,
\(X_t = |M_t|^p\) is therefore a nonnegative submartingale on \([0, T]\).
Fix \(0 \lt \mu \lt \lambda\) and \(N \in \mathbb{N}\), and apply the
maximal inequality
to \(X\) at the finitely many times of \(D_N\), whose largest element is \(T\), with
threshold \(\mu^p\). Since \(u \mapsto u^p\) is strictly increasing on \([0, \infty)\),
the events \(\{\max_{t \in D_N} |M_t| \geq \mu\}\) and
\(\{\max_{t \in D_N} X_t \geq \mu^p\}\) coincide, so
\[
\begin{align*}
\mathbb{P}\Bigl[ \max_{t \in D_N} |M_t| \gt \mu \Bigr]
&\leq \mathbb{P}\Bigl[ \max_{t \in D_N} |M_t| \geq \mu \Bigr] \\\\
&\leq \frac{1}{\mu^p}\, \mathbb{E}\bigl[ |M_T|^p \bigr].
\end{align*}
\]
The sets \(D_N\) increase with \(N\), so the events \(\{\max_{t \in D_N} |M_t| \gt \mu\}\)
increase, and their union over \(N\) is \(\{\sup_{t \in D} |M_t| \gt \mu\}\), since a supremum
exceeds \(\mu\) strictly exactly when some member does. By
continuity of measure from below,
\[
\begin{align*}
\mathbb{P}\Bigl[ \sup_{t \in D} |M_t| \gt \mu \Bigr]
&= \lim_{N \to \infty} \mathbb{P}\Bigl[ \max_{t \in D_N} |M_t| \gt \mu \Bigr] \\\\
&\leq \frac{1}{\mu^p}\, \mathbb{E}\bigl[ |M_T|^p \bigr] .
\end{align*}
\]
Finally,
\[
\{\sup_{t \in D} |M_t| \geq \lambda\} \subseteq \{\sup_{t \in D} |M_t| \gt \mu\}
\]
because \(\mu \lt \lambda\), so the left side of the theorem is bounded by
\(\mu^{-p}\, \mathbb{E}[|M_T|^p]\) for every \(\mu \in (0, \lambda)\). Letting
\(\mu \uparrow \lambda\) gives the claim.
Note where each hypothesis worked. The strict inequality \(\gt \mu\) is what makes the events
increase to the union (with \(\geq\) the identity fails, since a supremum can reach a level no
single term attains), and the reserve \(\mu \lt \lambda\) repays that debt at the end. Path
continuity entered only through the reading convention. It is what entitles a countable set of
times to speak for the continuum.
Watching the Whole Run at Once
A confidence bound valid at one prespecified time becomes invalid the moment one monitors
continuously and stops on a favorable fluctuation. This is the peeking problem of sequential
testing, the statistical face of taking a supremum over times.
Doob's inequality is the template for the repair. It prices the running maximum of an entire
trajectory at the cost of a single terminal moment, with no union bound over times and no
widening factor growing with the horizon. Anytime-valid inference (confidence sequences that
a practitioner may check after every observation, stopping whenever they like) is built on
maximal inequalities of the same kind, chiefly the infinite-horizon bound for nonnegative
supermartingales known as Ville's inequality, applied to likelihood ratios and to exponential
transforms of the error processes of the experiment.
Doob's inequality, in the form proved here, speaks for the whole trajectory only when the
paths are continuous, and there lies the last gap. Nothing so far ties the uncountably many
\(L^2\) classes \(M_t\) of the
previous section into a
single continuous path. Producing one is the task of the next section, and Doob's inequality,
applied to the differences of approximating integrals, is the tool that accomplishes it.
A Continuous Version
Here is the gap left by the construction, stated plainly. For each fixed \(t\), the integral
\(\int_0^t f\, dw_s\) is an element of \(L^2(\mathbb{P})\), an equivalence class pinned down
only up to a null set. A path \(t \mapsto M_t(\omega)\) requires choosing one representative for
every \(t\), and there are uncountably many \(t\). The null sets on which the choices misbehave
can accumulate into a set of full measure, and no property of the individual classes prevents it.
Continuity is a property of the joint selection, not of the separate values, and it must be
engineered.
The plan is to realize the approximating elementary integrals as genuinely continuous paths, to
force a subsequence of them to converge uniformly in \(t\) on a single almost-sure event,
and to take the limit path by path. Doob's inequality carries the load in the middle step. We
first fix some vocabulary and prove one classical lemma.
Definition: Version of a Process
Let \(\{X_t\}_{t \in I}\) and \(\{Y_t\}_{t \in I}\) be stochastic processes on
\((\Omega, \mathcal{F}, \mathbb{P})\) with a common index set \(I\). The process \(Y\) is a
version (or modification) of \(X\) if, for every
\(t \in I\),
\[
\mathbb{P}\bigl[ X_t = Y_t \bigr] = 1 .
\]
The relation is symmetric, and versions agree almost surely at each finite collection of times,
so every fixed-time statement of the preceding sections (martingale property, isometry, zero
mean) transfers freely between versions, adaptedness included because every null set belongs to
every \(\mathcal{F}_t\). Path properties do not transfer. Modifying a process at a single random
time \(\tau\) with \(\mathbb{P}[\tau = t] = 0\) for every \(t\) produces a version with jumps.
The saving grace is that continuity, once achieved, is essentially unique. If two versions of the
same process, indexed by an interval, both have almost surely continuous paths, their difference
vanishes almost surely at each time in a countable dense subset of the interval. Countably many
null sets have a null union, so the difference vanishes at all of these times at once, and
continuity propagates the equality to every \(t\). Two continuous versions are thus equal for all
\(t\) simultaneously, almost surely, and one may speak of the continuous version.
Lemma: Borel-Cantelli
Let \(\{A_k\}_{k \in \mathbb{N}} \subseteq \mathcal{F}\) satisfy
\(\sum_{k=1}^{\infty} \mathbb{P}(A_k) \lt \infty\). Then
\[
\mathbb{P}\Bigl[ \bigcap_{N \geq 1} \bigcup_{k \geq N} A_k \Bigr] = 0 ,
\]
that is, almost every \(\omega\) belongs to only finitely many of the \(A_k\).
Proof
We begin with countable subadditivity. For any sequence \(\{B_k\} \subseteq \mathcal{F}\), the
sets \(C_k = B_k \setminus \bigcup_{j \lt k} B_j\) are disjoint, lie in \(\mathcal{F}\),
satisfy \(C_k \subseteq B_k\), and have the same union as the \(B_k\). By countable
additivity, which is part of the
definition of a measure,
and by monotonicity (itself immediate from additivity applied to
\(B_k = C_k \cup (B_k \setminus C_k)\)),
\[
\begin{align*}
\mathbb{P}\Bigl[ \bigcup_{k} B_k \Bigr]
&= \sum_{k} \mathbb{P}(C_k) \\\\
&\leq \sum_{k} \mathbb{P}(B_k) .
\end{align*}
\]
Now let \(A\) denote the intersection in the statement. For every \(N\), monotonicity
and subadditivity give
\[
\begin{align*}
\mathbb{P}(A)
&\leq \mathbb{P}\Bigl[ \bigcup_{k \geq N} A_k \Bigr] \\\\
&\leq \sum_{k \geq N} \mathbb{P}(A_k) ,
\end{align*}
\]
and the right side is the tail of a convergent series, hence tends to \(0\) as
\(N \to \infty\). The left side does not depend on \(N\), so \(\mathbb{P}(A) = 0\).
The lemma converts a summable sequence of failure probabilities into almost-sure eventual
success, and it is exactly the shape of conclusion we need. Doob's inequality will price
each failure event, the isometry, along a fast subsequence, will make the prices
summable, and Borel-Cantelli will collect the winnings. Here is the theorem that keeps
the previous page's promise.
Theorem: The Itô Integral Admits a Continuous Version
Let \(f \in \mathcal{V}(0, T)\). There exists a stochastic process \(\{J_t\}_{0 \leq t \leq T}\)
on \((\Omega, \mathcal{F}, \mathbb{P})\) such that \(t \mapsto J_t(\omega)\) is continuous
for almost every \(\omega\), and, for every \(t \in [0, T]\),
\[
\mathbb{P}\Bigl[ J_t = \int_0^t f\, dw_s \Bigr] = 1 .
\]
Proof
Step 1: elementary integrals have continuous realizations.
Let \(\phi = \sum_{j} e_j\, \mathbf{1}_{[t_j, t_{j+1})}\) be bounded elementary on \([0, T]\)
and define, pointwise in \(\omega\) and for every \(t \in [0, T]\),
\[
I^{\phi}(t, \omega)
= \sum_{j} e_j(\omega) \bigl( w_{t \wedge t_{j+1}}(\omega) - w_{t \wedge t_j}(\omega) \bigr) ,
\]
where \(a \wedge b\) denotes \(\min(a, b)\). For \(t \in [t_i, t_{i+1})\) the terms with
\(j \lt i\) contribute full increments, the term \(j = i\) contributes
\(e_i (w_t - w_{t_i})\), and later terms vanish. At \(t = T\) every increment is complete. In
both cases \(I^{\phi}(t, \cdot)\) is precisely the elementary integral of
\(\phi\, \mathbf{1}_{[0, t)}\), the representative of \(\int_0^t \phi\, dw_s\) used throughout
the opening section. The formula is unchanged under refinement of the partition, since an
inserted point splits one term into two that telescope, and over a common partition it is
linear in \(\phi\).
For almost every \(\omega\) the path \(s \mapsto w_s(\omega)\) is continuous by (W3), and then
each map \(t \mapsto w_{t \wedge c}(\omega)\) is continuous, being a composition with the
continuous \(t \mapsto t \wedge c\). Hence \(t \mapsto I^{\phi}(t, \omega)\), a finite linear
combination of such maps, is continuous.
Step 2: the maximal estimate.
Fix bounded elementary \(\phi_n\) with \(\|f - \phi_n\|_{\mathcal{V}} \to 0\) and write
\(I_n(t, \omega) = I^{\phi_n}(t, \omega)\). For \(n, m\), pass to a common refinement. Then
\(\phi_n - \phi_m\) is bounded elementary and, by the linearity and refinement-invariance of
Step 1, \(I_n - I_m = I^{\phi_n - \phi_m}\) pointwise. For each fixed \(t\) this is a
representative of \(\int_0^t (\phi_n - \phi_m)\, dw_s\). By the
martingale theorem applied
to \(\phi_n - \phi_m \in \mathcal{V}(0, T)\), the process
\(\{I_n(t) - I_m(t)\}_{0 \leq t \leq T}\) is therefore a martingale with respect to
\(\{\mathcal{F}_t\}\). Conditions (M2) and (M3) concern only almost-sure classes, and (M1)
holds for every choice of representatives by part (iv) of the properties theorem. By Step 1,
the paths of this martingale are continuous almost surely.
Doob's inequality
with \(p = 2\), followed by the
Itô isometry
at the terminal time, gives for every \(\varepsilon \gt 0\)
\[
\begin{align*}
\mathbb{P}\Bigl[ \sup_{0 \leq t \leq T} |I_n(t) - I_m(t)| \geq \varepsilon \Bigr]
&\leq \frac{1}{\varepsilon^2}\, \mathbb{E}\bigl[ (I_n(T) - I_m(T))^2 \bigr] \\\\
&= \frac{1}{\varepsilon^2}\, \|\phi_n - \phi_m\|_{\mathcal{V}}^2 ,
\end{align*}
\]
second moments of almost-surely equal variables being equal. The supremum is read along
dyadic times, as in the previous section. On the almost-sure event of path continuity it is
the full supremum.
Step 3: a fast subsequence.
Since
\[
\|\phi_n - \phi_m\|_{\mathcal{V}} \leq \|\phi_n - f\|_{\mathcal{V}} + \|f - \phi_m\|_{\mathcal{V}} \to 0
\]
as \(n, m \to \infty\), we may choose indices \(n_1 \lt n_2 \lt \cdots\) such that
\(\|\phi_n - \phi_m\|_{\mathcal{V}}^2 \lt 2^{-3k}\) whenever \(n, m \geq n_k\). Taking
\(\varepsilon = 2^{-k}\) in Step 2 yields
\[
\begin{align*}
\mathbb{P}\Bigl[ \sup_{0 \leq t \leq T} |I_{n_{k+1}}(t) - I_{n_k}(t)| \geq 2^{-k} \Bigr]
&\leq 2^{2k} \cdot 2^{-3k} \\\\
&= 2^{-k} .
\end{align*}
\]
Step 4: Borel-Cantelli.
The bounds \(2^{-k}\) are summable, so by the
lemma, for almost every
\(\omega\) there exists \(k_1(\omega)\) such that, for all \(k \geq k_1(\omega)\),
\[
\sup_{0 \leq t \leq T} \bigl| I_{n_{k+1}}(t, \omega) - I_{n_k}(t, \omega) \bigr| \lt 2^{-k} .
\]
Step 5: uniform convergence.
We first fix the event on which we work. By (W3) the paths of \(w\) fail to be continuous
only inside some null set \(N \in \mathcal{F}\), and off \(N\) every path
\(I_{n_k}(\cdot, \omega)\) is continuous by Step 1. Let \(E\) be the event of Step 4, its suprema
read along dyadic times as in Step 2, with \(N\) removed. Every event entering \(E\) is built
from countably many of the \(\mathcal{F}\)-measurable values \(I_{n_k}(t)\), so \(E \in \mathcal{F}\),
and \(\mathbb{P}(E) = 1\). Off \(N\) the dyadic supremum of a continuous path is its supremum over
\([0, T]\), so on \(E\) the conclusion of Step 4 holds as displayed. For \(\omega \in E\),
\(l \gt k \geq k_1(\omega)\) and every \(t \in [0, T]\), the triangle inequality along the
chain of consecutive differences gives
\[
\begin{align*}
\bigl| I_{n_l}(t, \omega) - I_{n_k}(t, \omega) \bigr|
&\leq \sum_{j = k}^{l - 1} 2^{-j} \\\\
&\lt 2^{-k + 1} ,
\end{align*}
\]
so the sequence \(\{I_{n_k}(t, \omega)\}_k\) is Cauchy in \(\mathbb{R}\), uniformly in \(t\).
By completeness of \(\mathbb{R}\) the limit \(J_t(\omega) = \lim_{k} I_{n_k}(t, \omega)\)
exists for every \(t\). Letting \(l \to \infty\) in the display then gives
\[
\sup_{0 \leq t \leq T} |J_t(\omega) - I_{n_k}(t, \omega)| \leq 2^{-k+1}
\]
for \(k \geq k_1(\omega)\), so the convergence is uniform on \([0, T]\). Off \(E\), set
\(J_t(\omega) = 0\) for all \(t\). Each \(J_t\) is then \(\mathcal{F}\)-measurable, being the
limit constructed in part (iv) of the properties theorem, applied to the values \(I_{n_k}(t)\),
times the indicator of \(E\).
Step 6: continuity of the limit.
Fix \(\omega \in E\), a point \(t \in [0, T]\), and \(\varepsilon \gt 0\). Choose
\(k \geq k_1(\omega)\) with \(2^{-k+1} \lt \varepsilon / 3\), and then \(\delta \gt 0\) such
that \(|I_{n_k}(s, \omega) - I_{n_k}(t, \omega)| \lt \varepsilon / 3\) whenever
\(|s - t| \lt \delta\), by continuity of \(I_{n_k}(\cdot, \omega)\). For such \(s\),
\[
\begin{align*}
|J_s - J_t|
&\leq |J_s - I_{n_k}(s)| + |I_{n_k}(s) - I_{n_k}(t)| + |I_{n_k}(t) - J_t| \\\\
&\lt \varepsilon ,
\end{align*}
\]
suppressing \(\omega\). Hence \(t \mapsto J_t(\omega)\) is continuous. This is the
classical fact that a uniform limit of continuous functions is continuous, proved here
inline in its entirety.
Step 7: \(J\) is a version.
At \(t = 0\) the integral is \(0\) under the convention adopted with the martingale theorem,
and \(J_0 = 0\) by construction. Fix now \(t \in (0, T]\). As in the proof of the properties
theorem, \(\phi_{n_k} \mathbf{1}_{[0, t)}\) is bounded elementary and admissible for \(f\) on
\([0, t]\), so \(I_{n_k}(t) \to \int_0^t f\, dw_s\) in \(L^2(\mathbb{P})\). We pass to a further
subsequence converging almost surely.
Since \(I_{n_k}(t) \to J_t\) almost surely by Step 5, the two almost-sure limits coincide
almost surely. Hence \(J_t\) is a representative of the class \(\int_0^t f\, dw_s\), which is
the displayed claim.
The theorem earns an immediate reward. The continuous version is a martingale whose paths satisfy
the hypothesis of Doob's inequality, and feeding the Itô isometry through it yields a bound
on the entire trajectory of the integral in terms of the plain size of the integrand. This
estimate closes the circle of this page's first four sections.
Theorem: Maximal Bound for the Itô Integral
Let \(f \in \mathcal{V}(0, T)\) and let \(\{J_t\}_{0 \leq t \leq T}\) be a continuous version
of the integral process. Then for every \(\lambda \gt 0\),
\[
\mathbb{P}\Bigl[ \sup_{0 \leq t \leq T} |J_t| \geq \lambda \Bigr]
\leq \frac{1}{\lambda^2}\, \mathbb{E}\Bigl[ \int_0^T f(s, \omega)^2\, ds \Bigr] ,
\]
the supremum being taken over the dyadic times \(k T 2^{-N}\), with \(N \in \mathbb{N}\) and
\(k = 0, 1, \ldots, 2^N\), and for almost every \(\omega\) coinciding with the supremum over
all of \([0, T]\).
Proof
The process \(J\) is a martingale with respect to \(\{\mathcal{F}_t\}\). For (M1), each
\(J_t\) is \(\mathcal{F}\)-measurable, as is every value of a stochastic process on
\((\Omega, \mathcal{F}, \mathbb{P})\), and is a representative of \(\int_0^t f\, dw_s\),
hence, for \(t \gt 0\), \(\mathcal{F}_t\)-measurable by part (iv) of the
properties theorem,
while \(J_0\), almost surely \(0\), is \(\mathcal{F}_0\)-measurable because every null set
belongs to \(\mathcal{F}_0\). Here again the every-representative form of (iv), and behind it
the null-set convention, pays for itself. Adaptedness costs nothing to transfer. For (M2) and
(M3), each \(J_t\) is almost surely equal to the corresponding value \(M_t\) of the process
in the martingale theorem,
integrability transfers, and conditional expectations of almost-surely equal variables agree,
so \(\mathbb{E}[J_s \mid \mathcal{F}_t] = J_t\) almost surely for \(t \leq s\).
The paths of \(J\) are continuous almost surely, so
Doob's inequality
applies with \(p = 2\). Combined with the
Itô isometry
at time \(T\), it gives
\[
\begin{align*}
\mathbb{P}\Bigl[ \sup_{0 \leq t \leq T} |J_t| \geq \lambda \Bigr]
&\leq \frac{1}{\lambda^2}\, \mathbb{E}\bigl[ J_T^2 \bigr] \\\\
&= \frac{1}{\lambda^2}\, \mathbb{E}\Bigl[ \int_0^T f(s, \omega)^2\, ds \Bigr] .
\end{align*}
\]
From this point on, \(\int_0^t f\, dw_s\), viewed as a process in \(t\), will always denote a
continuous version. Since any two continuous versions agree for all \(t\) at once almost surely,
the choice is immaterial and the article the is deserved. Every pathwise statement of
stochastic calculus, beginning with
the Itô formula, is a statement about
this object.
What a Sampler Actually Samples
Every numerical scheme that simulates a stochastic differential equation, from Euler-Maruyama
in computational finance to the samplers that integrate the reverse-time dynamics of
diffusion models, outputs a
trajectory, one sample path per random seed, computed on a time grid. The object such a
scheme approximates is not the family of \(L^2\) classes constructed on the previous page,
which has no trajectories at all, but a solution whose noise term is an Itô integral
taken in its continuous version. This section proved that such a version exists for every
integrand in \(\mathcal{V}(0, T)\). The theorem is, in that sense, one part of the license
under which path simulation operates. Together with an existence theorem for the equation
itself, it guarantees that there is a path-valued object to converge to, and the maximal
bound above is the shape of estimate, uniform over the whole time horizon, by which such
convergence is measured.
What remains is a survey of how far the construction reaches, and the question of what becomes of
the calculus if the evaluation point abandons causality.
Extensions and the Stratonovich Question
The theory now standing was built under four commitments. It assumed one driving Brownian motion, its own
filtration, square-integrable integrands, and the left endpoint. The first three are technical
restrictions, and each can be relaxed. We survey the relaxations with precision about what has to
be re-verified, because the verification burden is where the mathematics lives. The fourth is not
a restriction but a decision, made in the endpoint experiment of the previous page and payable now. The page
closes by settling the account with the midpoint convention it set aside there.
Larger Filtrations and Several Driving Motions
Read back through the construction and mark every place the Brownian filtration entered. It
played two mathematical roles beyond containing the history of \(w\). The integrand was required
to be \(\mathcal{F}_t\)-adapted, and each increment \(w_v - w_u\) was
independent of \(\mathcal{F}_u\).
That independence is the input used by the elementary isometry, by the zero-mean and martingale
computations of this page, and by nothing else. The null-set convention, which part (iv) needed,
is bookkeeping that any larger filtration can repeat. It follows that the entire theory survives,
verbatim, if \(\{\mathcal{F}_t\}\) is replaced by any larger filtration \(\{\mathcal{H}_t\}\),
with \(\mathcal{F}_t \subseteq \mathcal{H}_t\) and each \(\mathcal{H}_t\) containing the
\(\mathbb{P}\)-null sets as before, such that each increment \(w_v - w_u\) remains independent of
\(\mathcal{H}_u\). Integrands may then depend on more information than the history of \(w\), so
long as the extra information never foresees the noise.
We stress that conditional centering \(\mathbb{E}[w_v - w_u \mid \mathcal{H}_u] = 0\) alone would
not carry the diagonal of the elementary isometry. The factorization
\[
\mathbb{E}\bigl[ e_j^2 (\Delta w_j)^2 \bigr] = \mathbb{E}\bigl[ e_j^2 \bigr]\, \Delta t_j
\]
uses the conditional second moment
\(\mathbb{E}[(\Delta w_j)^2 \mid \mathcal{H}_{t_j}] = \Delta t_j\), a statement about the squared
increment that independence supplies and centering does not.
The example that makes the enlargement indispensable is several Brownian motions at
once. For an \(n\)-dimensional standard
Brownian motion
\(\mathbf{w}_t = (w_t^{(1)}, \ldots, w_t^{(n)})\), the relevant information at time
\(t\) is \(\mathcal{F}_t^{(n)}\), generated by all components up to time
\(t\), that is, the
Brownian filtration
with the generating events ranging over every coordinate, null sets adjoined as
before. An integrand may now read the entire vector history, and the single fact to
check is that each component's increment \(w_v^{(j)} - w_u^{(j)}\) is independent of
the joint past \(\mathcal{F}_u^{(n)}\).
The verification is the Gaussian block computation of the increment lemma over again. The
covariance of the increment with an earlier value of the same component vanishes by the
formula \(\min(t_i, v) - \min(t_i, u) = 0\). For an earlier value of a different
component, the covariance vanishes because the covariance matrix \(\min(s, t)\, I_n\) in (W1) has
no cross-component entries. The passage from generating events to the full \(\sigma\)-algebra is
the same monotone-class step taken on faith there. With that single input re-verified, each
component is a legitimate integrator for \(\mathcal{F}_t^{(n)}\)-adapted integrands, and
matrix-valued integrands assemble the components into one object.
Definition: The Multi-Dimensional Itô Integral
Let \(0 \leq S \lt T\), let \(\mathbf{w}_t = (w_t^{(1)}, \ldots, w_t^{(n)})\) be
\(n\)-dimensional standard Brownian motion, and let \(\{\mathcal{F}_t^{(n)}\}\) be the
filtration generated by all of its components up to time \(t\), null sets adjoined. Let
\(\mathcal{V}^{m \times n}(S, T)\) denote the class of
\(m \times n\) matrix-valued processes
\(\mathbf{v}(t, \omega) = [v_{ij}(t, \omega)]\) whose every entry satisfies conditions
(V1)-(V3) of the
class \(\mathcal{V}(S, T)\)
with \(\mathcal{F}_t^{(n)}\) in place of \(\mathcal{F}_t\). For
\(\mathbf{v} \in \mathcal{V}^{m \times n}(S, T)\), the
multi-dimensional Itô integral
\[
\int_S^T \mathbf{v}(t, \omega)\, d\mathbf{w}_t
\]
is the \(\mathbb{R}^m\)-valued random vector whose \(i\)-th component is
\[
\sum_{j=1}^{n} \int_S^T v_{ij}(t, \omega)\, dw_t^{(j)} ,
\]
each summand being the one-dimensional Itô integral driven by \(w^{(j)}\), constructed with
respect to the filtration \(\{\mathcal{F}_t^{(n)}\}\).
The notation is chosen so that the mnemonic is the statement. The product
\(\mathbf{v}\, d\mathbf{w}\) is a matrix multiplying a column of differentials, row by row.
Integrands like \(v(t, \omega) = w_t^{(2)}\) driving \(dw_t^{(1)}\), one noise source modulating
exposure to another, are now admissible. They were unreachable in the single-component theory,
because \(w^{(2)}\) is not measurable for the filtration of \(w^{(1)}\) alone. Every result of
this page applies to each component sum. That covers zero mean, the martingale property with
respect to \(\{\mathcal{F}_t^{(n)}\}\), and a continuous version, and the isometry of the
previous page applies summand by summand. The
multi-dimensional Itô formula,
developed on a coming page, is built on precisely this object.
Weaker Integrability
Condition (V3) can also be relaxed. It suffices to demand
\[
\mathbb{P}\Bigl[ \int_S^T f(t, \omega)^2\, dt \lt \infty \Bigr] = 1 ,
\]
with no expectation at all. For such integrands one can still produce approximating step
processes, but the approximation and the resulting integral converge only in the sense of
convergence in probability,
and the price is steep.
Without (V3) the isometry has no finite right-hand side, the zero-mean property may fail, and the
integral need not be a martingale. It retains only a localized remnant of the property, the notion
of a local martingale, which we leave undefined. We record this extension at statement
level and take none of it on. The class \(\mathcal{V}\) and its martingale calculus suffice for
everything on our horizon, and when the weaker class is eventually wanted, the honest construction
will be built, not borrowed.
The Stratonovich Question
One account remains open. On the previous page, comparing right and left endpoints showed that the
would-be integral of \(w\) against itself depends on the evaluation point, the discrepancy between
the two being exactly the quadratic variation. That page committed to the left endpoint and named
the midpoint alternative, the Stratonovich integral, written
\(\int f \circ dw_t\), with a promise that it would reappear. The midpoint integral reappears now,
because for the test integrand everything is computable.
The first computation
gave
\[
\int_0^T w_s\, dw_s = \tfrac{1}{2} w_T^2 - \tfrac{1}{2} T .
\]
The right-endpoint sums exceed the left-endpoint sums by the quadratic sums, which converge to
\(T\), and the average of the two therefore converges to the classical value
\(\tfrac{1}{2} w_T^2\). The midpoint sums can be shown to share this limit, an estimate we record
here at statement level and prove on a later page as the
Stratonovich integral of Brownian motion, so
\[
\int_0^T w_s \circ dw_s = \tfrac{1}{2} w_T^2 .
\]
The Stratonovich integral of \(w\) thus obeys the classical chain rule on the nose. The
Itô-Stratonovich discrepancy for this integrand is exactly half the quadratic variation,
\(\tfrac{1}{2} T\), rigorously accounted for by the
path properties theorem.
The agreement with the classical chain rule is no isolated coincidence, and the deeper reason
lies in the modeling question that opened the previous page. Which integral makes the scalar
white-noise equation posed there an honest idealization? Approximate the rough path by processes
\(\gamma_n\) with continuously differentiable paths that converge to \(w\) uniformly on bounded
intervals for almost every \(\omega\), and for each \(\omega\) solve the ordinary differential
equation obtained by using the derivative of \(\gamma_n\) as the noise. The Wong-Zakai
approximation theorem assumes regularity conditions on the drift \(b\) and the noise coefficient
\(\sigma\) of the white-noise equation. Under these conditions the theorem says that for almost
every \(\omega\) the solutions converge uniformly on bounded intervals and that the limit solves
the equation in the Stratonovich sense. We state it without proof, since it rests on a
finer analysis of pathwise approximation than this page undertakes.
Smooth physics, pushed to the white-noise limit, lands on the midpoint calculus. Classical
mechanisms have no arrow-of-time asymmetry inside an infinitesimal interval, and the midpoint is
where that symmetry lives.
Yet nothing is lost by refusing the midpoint, because the two calculi are intertranslatable.
We state the translation at the level of statements only, since the notion of solving the
integral equations below is itself the business of a coming page. For sufficiently regular
\(b\) and \(\sigma\), with \(\sigma\) in particular continuously differentiable in \(x\),
the Stratonovich equation
\[
X_t = X_0 + \int_0^t b(s, X_s)\, ds + \int_0^t \sigma(s, X_s) \circ dw_s
\]
describes the same process as the Itô equation with a corrected drift,
\[
\begin{align*}
X_t = X_0
&+ \int_0^t \Bigl( b(s, X_s) + \tfrac{1}{2}\, \sigma'(s, X_s)\, \sigma(s, X_s) \Bigr) ds \\\\
&+ \int_0^t \sigma(s, X_s)\, dw_s ,
\end{align*}
\]
where \(\sigma'\) denotes the derivative of \(\sigma(t, x)\) in \(x\). The correction
\(\tfrac{1}{2} \sigma' \sigma\) is the equation-level face of the half-quadratic-variation
gap computed above. The derivation of this correction is
taken up once the Itô formula is
available. When \(\sigma\) does not depend on \(x\) (additive noise), the correction vanishes
and the two interpretations coincide.
The choice, then, is a genuine trade. The Stratonovich calculus keeps the classical chain
rule and transforms cleanly under changes of variables, which makes it the natural language
where geometry dominates, since much of stochastic calculus on manifolds is built on it.
For the scalar equation, Wong-Zakai moreover makes the Stratonovich solution the limit of
smooth approximating models.
The Itô calculus pays a correction term in its chain rule and receives, in exchange,
everything proved on this page and the previous one. The integral never reads the future, has
mean zero, is an isometry, is a martingale, obeys Doob's inequality and the maximal bound, and
admits a continuous version. For estimation, prediction, and control (settings where the
filtration is the point), the martingale structure is not a convenience but the subject matter,
and it is unavailable on the Stratonovich side. Since the corrected drift converts any model that
arrives in Stratonovich form with sufficiently regular \(b\) and \(\sigma\), we commit to
Itô and lose nothing.