Vector Itô Processes
Everything so far ran on a single source of noise. Real systems rarely oblige. A robot arm
receives independent disturbances at every joint, a portfolio is shaken by many market
factors at once, and the reverse process of a diffusion model runs in the dimension of the
data. The previous pages already prepared the probabilistic ground for this. The
multi-dimensional Itô integral
integrates matrix-valued processes against a vector of independent Brownian motions, over the
filtration \(\{\mathcal{F}_t^{(n)}\}\) generated by all components with null sets adjoined,
and the componentwise increment independence that makes each coordinate a legitimate
integrator was verified alongside it. What remains is the calculus. This page assembles
vector Itô processes, proves the general Itô formula, repairs the product rule that a
previous page broke on purpose, and puts the domain refinement recorded with the
one-dimensional formula to its intended work.
Throughout, \(\mathbf{w}_t = (w_t^{(1)}, \ldots, w_t^{(n)})\) is \(n\)-dimensional standard
Brownian motion started at the origin, and \(\{\mathcal{F}_t^{(n)}\}_{t \geq 0}\) is its
joint filtration as above. We state our dimension convention once. The driving noise has
\(n\) components, the process has \(m\) components, and smooth maps will land in \(p\)
components, so that coefficient matrices are \(m \times n\) and Jacobians are
\(p \times m\).
One bookkeeping remark connects the one-dimensional machinery to the enlarged filtration,
and after the generalized statements of the previous page it is a matter of checking
hypotheses rather than reproving anything.
Progressive measurability
and its
two criteria
are stated for an arbitrary filtration and apply to \(\{\mathcal{F}_t^{(n)}\}\) as they
stand. The
pathwise time integral
asks only that the filtration contain every null set, which
\(\{\mathcal{F}_t^{(n)}\}\) does by construction. The
weighted increment lemma
asks that the integrating motion's increments, singly for parts (i) and (ii) and in
independent pairs for part (iii), be independent of the filtration. For single components
\(w^{(j)}\) against the joint past this is exactly what the previous page verified in
building the multi-dimensional integral, and the paired version is the same Gaussian block
computation with one more row, every cross-covariance vanishing for the same two reasons,
the same-component formula \(\min(t_i, v) - \min(t_i, u) = 0\) and the absence of
cross-component entries in the covariance \(\min(s, t)\, I_n\). Every hypothesis is met, so
all four results are
available over \(\{\mathcal{F}_t^{(n)}\}\) with any component, or pair of components, in the
role of the driving motion.
Definition: Vector Itô Process
An \(m\)-dimensional (or vector) Itô process is a stochastic process
\(\mathbf{X}_t = (X_t^{(1)}, \ldots, X_t^{(m)})\), \(t \geq 0\), of the form
\[
\mathbf{X}_t
= \mathbf{X}_0
+ \int_0^t \mathbf{u}(s, \omega)\, ds
+ \int_0^t \mathbf{v}(s, \omega)\, d\mathbf{w}_s ,
\]
read componentwise as
\[
X_t^{(i)}
= X_0^{(i)}
+ \int_0^t u_i(s, \omega)\, ds
+ \sum_{j=1}^{n} \int_0^t v_{ij}(s, \omega)\, dw_s^{(j)} ,
\quad
i = 1, \ldots, m ,
\]
where:
(M1)
\(\mathbf{X}_0\) is an \(\mathcal{F}_0^{(n)}\)-measurable random vector in
\(\mathbb{R}^m\);
(M2)
each \(u_i\) is progressively measurable with respect to
\(\{\mathcal{F}_t^{(n)}\}\) and satisfies
\(\mathbb{P}\bigl[\int_0^t |u_i|\, ds \lt \infty \quad \forall\, t \geq 0\bigr] = 1\),
the \(ds\)-term being the pathwise time integral applied in each coordinate;
(M3)
\(\mathbf{v} \in \mathcal{V}^{m \times n}(0, T)\) for every \(T \gt 0\), each entry's
integral taken in its everywhere-continuous realization.
The definition is the one-dimensional one run \(m\) times in parallel, with all coordinates
listening to the same enlarged information flow. Every component \(X^{(i)}\) is a sum of one
initial value, one pathwise time integral, and \(n\) one-dimensional Itô integrals, so each
component is adapted with continuous paths, and progressively measurable by the continuity
criterion. A map into \(\mathbb{R}^m\) is progressively measurable exactly when its
coordinates are, measurability into a product being coordinatewise, so the vector process
\(\mathbf{X}\) inherits all three properties at once.
The differential shorthand extends verbatim. We write
\[
\begin{align*}
d\mathbf{X}_t &= \mathbf{u}\, dt + \mathbf{v}\, d\mathbf{w}_t , \\\\
dX_t^{(i)} &= u_i\, dt + \sum_{j=1}^{n} v_{ij}\, dw_t^{(j)} ,
\end{align*}
\]
with the same standing disclaimer as before. The differentials have no independent meaning,
and every line written in them is a name for the integral identity in the definition. Note
what the definition deliberately does not require. The components of \(\mathbf{X}\) are not
independent, and in the interesting cases they are strongly coupled, several coordinates
listening to the same noise component through the matrix \(\mathbf{v}\). The coupling is the
point, and the calculus must track it. When a smooth map mixes the coordinates of
\(\mathbf{X}\), the correction term of the chain rule will involve products
\(dX^{(i)}\, dX^{(k)}\) of increments of different components, and the entire new
content of the multi-dimensional theory sits in one question. What is
\(dw^{(i)}\, dw^{(j)}\) when \(i \neq j\)? The next section answers it, and the answer is
zero.
The General Itô Formula
The multiplication table needs one new entry. For a single motion the rules were
\(dt \cdot dt = dt \cdot dw = 0\) and \((dw)^2 = dt\). With several independent motions the
complete table reads
\[
\begin{align*}
dt \cdot dt &= dt \cdot dw_t^{(i)} = 0 , \\\\
dw_t^{(i)} \cdot dw_t^{(j)} &= \delta_{ij}\, dt ,
\end{align*}
\]
and the new claim is the off-diagonal zero. The heuristic is quick. A product
\(\Delta w^{(i)} \Delta w^{(j)}\) of increments of distinct components is a product of
independent centered variables, so unlike the square \((\Delta w^{(i)})^2\), whose mean
\(\Delta t\) is what survives, it has mean zero, and its fluctuations are too small to
survive the sum. The honest form of this claim is part (iii) of the
weighted increment lemma,
proved in advance for exactly this moment, and it is the only limit statement in the entire
multi-dimensional theory that the one-dimensional theory itself never needed.
Within this section and its proof we abbreviate, for a scalar component \(g_k\) of the map,
\(\partial_t g_k\), \(\partial_i g_k = \frac{\partial g_k}{\partial x_i}\), and
\(\partial_{il} g_k = \frac{\partial^2 g_k}{\partial x_i \partial x_l}\), together with
\(\partial_{tt} g_k\) and \(\partial_{t i} g_k\) for the second derivatives involving
time, each evaluated at \((s, \mathbf{X}_s)\) unless displayed otherwise.
Theorem: The General Itô Formula
Let \(\mathbf{X}_t\) be an \(m\)-dimensional Itô process with coefficients
\(\mathbf{u}, \mathbf{v}\), and let
\(g = (g_1, \ldots, g_p) : [0, \infty) \times \mathbb{R}^m \to \mathbb{R}^p\) be twice
continuously differentiable. Assume the membership condition that for every component
pair \(k, j\) and every \(T \gt 0\),
\[
\Bigl( (s, \omega) \mapsto
\sum_{i=1}^{m} \partial_i g_k(s, \mathbf{X}_s)\, v_{ij}(s, \omega) \Bigr)
\in \mathcal{V}(0, T) .
\]
Then \(\mathbf{Y}_t = g(t, \mathbf{X}_t)\) is again a \(p\)-dimensional Itô process, and
for each \(k\) and every \(t \geq 0\), almost surely,
\[
\begin{align*}
g_k(t, \mathbf{X}_t)
&= g_k(0, \mathbf{X}_0)
+ \int_0^t \Bigl(
\partial_t g_k
+ \sum_{i=1}^{m} u_i\, \partial_i g_k
+ \tfrac{1}{2} \sum_{i, l = 1}^{m}
\bigl( \mathbf{v} \mathbf{v}^\top \bigr)_{il}\, \partial_{il} g_k
\Bigr)(s, \mathbf{X}_s)\, ds \\\\
&\quad + \sum_{j=1}^{n} \int_0^t
\Bigl( \sum_{i=1}^{m} \partial_i g_k\, v_{ij} \Bigr)(s, \mathbf{X}_s)\, dw_s^{(j)} ,
\end{align*}
\]
where the partial derivatives of \(g_k\) are evaluated at \((s, \mathbf{X}_s)\) and
the coefficients \(u_i\), \(v_{ij}\), and the entries of
\(\mathbf{v}\mathbf{v}^\top\) at \((s, \omega)\).
In differential shorthand, expanded by the table,
\[
dY_t^{(k)}
= \partial_t g_k\, dt
+ \sum_{i=1}^{m} \partial_i g_k\, dX_t^{(i)}
+ \tfrac{1}{2} \sum_{i, l = 1}^{m} \partial_{il} g_k\, dX_t^{(i)}\, dX_t^{(l)} ,
\]
which is the one-dimensional formula with the second-order term promoted to a quadratic form.
The table computes the increment products,
\[
\begin{align*}
dX_t^{(i)}\, dX_t^{(l)}
&= \Bigl( u_i\, dt + \sum_j v_{ij}\, dw^{(j)} \Bigr)
\Bigl( u_l\, dt + \sum_{j'} v_{lj'}\, dw^{(j')} \Bigr) \\\\
&= \sum_{j} v_{ij}\, v_{lj}\, dt \\\\
&= \bigl( \mathbf{v} \mathbf{v}^\top \bigr)_{il}\, dt ,
\end{align*}
\]
every cross pairing \(j \neq j'\) dying on the off-diagonal rule. The matrix
\(\mathbf{v} \mathbf{v}^\top\) that lands in the drift is symmetric and positive
semidefinite, and it is the object a later page will call the diffusion matrix of the
process. The correction is no longer one term but a full quadratic form in the second
derivatives, weighted by how strongly each pair of coordinates listens to shared noise.
Proof.
Step 1 (Reduction).
The claim is componentwise in \(k\), so take \(p = 1\) and write \(g\) for the scalar
map. We prove the identity in the same core case as the one-dimensional theorem, \(g\)
with all partial derivatives up to second order bounded by \(M\), and every \(u_i\),
\(v_{ij}\) elementary on \([0, t]\) over one common partition with levels bounded by
\(C\). The two approximation passages that carry the core case to the statement are
the same two passages recorded in the one-dimensional proof, applied coordinate
by coordinate and entry by entry, and we do not multiply debts by restating them. In the
core case the membership condition holds automatically, each diffusion entry being a
bounded progressively measurable process.
Step 2 (Taylor and the master decomposition).
Fix a sequence of partitions of \([0, t]\) refining the common partition, with mesh
tending to zero. Exactly as before, outside a single null set the increments are exact,
\[
\Delta X_j^{(i)}
= u_i(t_j)\, \Delta t_j + \sum_{r=1}^{n} v_{ir}(t_j)\, \Delta w_j^{(r)} ,
\]
by the pathwise integral of a constant level and
additivity
in each coordinate.
Taylor's theorem with integral remainder
at order one, now in the \(1 + m\) variables \((s, x_1, \ldots, x_m)\), expands each
increment of \(g(t_j, \mathbf{X}_j)\) into the first-order terms
\(\partial_t g\, \Delta t_j + \sum_i \partial_i g\, \Delta X_j^{(i)}\), the second-order
terms with weighted-average coefficients \(\theta_j\) over every pair drawn from
\((\Delta t_j, \Delta X_j^{(1)}, \ldots, \Delta X_j^{(m)})\), and nothing else. The
boundary patch at \(t_0 = 0\), the separation
\(\theta_j = \partial g(t_j, \mathbf{X}_j) + \varepsilon_j\), and the pathwise control
\(\eta(\pi, \omega) \to 0\) of the deviations by uniform continuity of the second
derivatives on the compact
\([0, t] \times \{ |\mathbf{x}| \leq R_\omega \}\) are word for word the one-dimensional
Step 2, the only change being that the segment argument runs in \(\mathbb{R}^{1 + m}\)
and the modulus is taken at the displacement
\(|\pi| + \max_{j} |\Delta \mathbf{X}_j|\), which tends to zero by uniform continuity of
each coordinate path. Summing over \(j\) yields the master decomposition, the constant
\(g(t, \mathbf{X}_t) - g(0, \mathbf{X}_0)\) equal to the first-order sums, the
second-order sums with left-endpoint coefficients, and an error \(E(\pi)\) bounded by
\(\eta(\pi)\) times \(\sum_j ( (\Delta t_j)^2 + |\Delta \mathbf{X}_j|^2 )\) up to a fixed
factor.
Step 3 (The fates of the sums).
Insert the exact increments and expand. Every resulting sum belongs to one of five
families, and every family's fate is already on the shelf.
First-order time and drift sums.
\(\sum_j \partial_t g\, \Delta t_j\) and \(\sum_j \partial_i g\, u_i\, \Delta t_j\)
converge pathwise to \(\int_0^t \partial_t g\, ds\) and
\(\int_0^t \partial_i g\, u_i\, ds\), by the left-endpoint Riemann estimate of the
one-dimensional Step 3, run interval by interval of the common partition.
First-order noise sums.
For each pair \((i, r)\), \(\sum_j \partial_i g\, v_{ir}\, \Delta w_j^{(r)}\) converges
in \(L^2(\mathbb{P})\) to \(\int_0^t \partial_i g\, v_{ir}\, dw_s^{(r)}\), by the
elementary-approximant argument of the one-dimensional Step 3, bounded convergence
against the constant \(2 M C\) and
L² continuity,
with \(w^{(r)}\) as the integrator. There are finitely many pairs.
Sums carrying \((\Delta t_j)^2\).
The pure time square, the time-drift crosses, and the drift-drift products
\(u_i u_l (\Delta t_j)^2\) all die pathwise under deterministic bounds by constants
times \(t\, |\pi|\), as in the one-dimensional Step 4.
Sums carrying \(\Delta t_j\, \Delta w_j^{(r)}\).
The time-noise and drift-noise crosses are part (i) of the weighted increment lemma with
\(w^{(r)}\) as the driving motion and weights built from \(\partial g\)-factors,
\(u\)-levels, and \(v\)-levels, bounded by fixed multiples of \(M C\) and \(M C^2\).
They die in \(L^2(\mathbb{P})\), finitely many at a time.
Sums carrying \(\Delta w_j^{(r)}\, \Delta w_j^{(r')}\).
These arrive from \(\tfrac{1}{2} \sum_{i,l} \partial_{il} g\, \Delta X^{(i)} \Delta X^{(l)}\)
with weights \(b_j = \tfrac{1}{2}\, \partial_{il} g(t_j, \mathbf{X}_j)\,
v_{ir}(t_j)\, v_{lr'}(t_j)\), which are \(\mathcal{F}_{t_j}^{(n)}\)-measurable and
bounded by \(\tfrac{1}{2} M C^2\). For \(r = r'\), part (ii) of the lemma centers the sum
and the left-endpoint Riemann sums of the piecewise continuous paths deliver, in
probability, exactly as in the one-dimensional Step 5,
\[
\sum_j b_j \bigl( \Delta w_j^{(r)} \bigr)^2
\longrightarrow
\tfrac{1}{2} \int_0^t \partial_{il} g\, v_{ir}\, v_{lr}\, ds
\]
Summing over \(i, l, r\) assembles precisely the quadratic-form drift
\(\tfrac{1}{2} \sum_{i,l} (\mathbf{v}\mathbf{v}^\top)_{il}\, \partial_{il} g\). For
\(r \neq r'\), the components \(w^{(r)}\) and \(w^{(r')}\) are independent standard
motions whose paired increments are independent of the joint past, so part (iii) of the
lemma applies verbatim and the sum dies in \(L^2(\mathbb{P})\). This is the off-diagonal
zero of the table, and it is the only place the multi-dimensional proof needs anything
new.
Step 4 (Error and assembly).
The error \(E(\pi)\) dies in \(L^1(\mathbb{P})\) by the one-dimensional Step 4 estimate,
with \(\sum_j |\Delta \mathbf{X}_j|^2\) bounded by a constant times
\(t |\pi| + \sum_{r} Q^{(r)}_\pi\) and each component's quadratic sum
\(Q^{(r)}_\pi \to t\) in \(L^2(\mathbb{P})\) by
quadratic variation.
Every family converges in probability, sums respect the mode, the left side is fixed
along the sequence, and the identity at fixed \(t\), its extension to all \(t\) by
continuity of both sides, and the Itô-process-hood of the image follow exactly as in the
one-dimensional Step 6, the well-posedness of the integrands transferring by the same
composition arguments with \(\mathbf{X}\) progressively measurable.
The proof is deliberately a dictionary rather than a duplication. Five families of sums, four
of them settled by the one-dimensional page and its lemma, one settled by the cross estimate
proved there in advance. The formula's new face is the matrix \(\mathbf{v}\mathbf{v}^\top\)
in the drift, and the first thing it pays for is an identity classical calculus loses.
The Product Rule, Repaired
A previous page broke the classical product rule on purpose, applying it to
\(w_t \cdot w_t\) and finding the prediction wrong by exactly half the quadratic variation.
The general formula now supplies the repaired rule, and the repair is one extra term. In the
statement, \(X\) and \(Y\) are one-dimensional Itô processes over the joint filtration,
each driven by the full vector of motions, with coefficients written
\(dX_t = u^X dt + \sum_j v^X_j\, dw^{(j)}_t\) and likewise for \(Y\), so that the pair
\((X, Y)\) is a two-dimensional Itô process.
Theorem: The Stochastic Product Rule
Let \(X\) and \(Y\) be one-dimensional Itô processes as above, and assume the membership
condition that for every \(j\) and \(T \gt 0\) the process
\(Y_s\, v^X_j + X_s\, v^Y_j\) belongs to \(\mathcal{V}(0, T)\). Then \(X_t Y_t\) is again
an Itô process, and for every \(t \geq 0\), almost surely,
\[
X_t Y_t = X_0 Y_0
+ \int_0^t \Bigl( X_s\, u^Y + Y_s\, u^X + \sum_{j=1}^{n} v^X_j\, v^Y_j \Bigr) ds
+ \sum_{j=1}^{n} \int_0^t \bigl( X_s\, v^Y_j + Y_s\, v^X_j \bigr)\, dw_s^{(j)} ,
\]
in differential shorthand
\[
\begin{align*}
d(X_t Y_t) &= X_t\, dY_t + Y_t\, dX_t + dX_t \cdot dY_t , \\\\
dX_t \cdot dY_t &= \sum_{j=1}^{n} v^X_j\, v^Y_j\, dt .
\end{align*}
\]
Proof.
Apply the general Itô formula to the two-dimensional process \(\mathbf{X} = (X, Y)\) and
the time-independent map \(g(x_1, x_2) = x_1 x_2\), which is \(C^2\) with
\(\partial_t g = 0\), \(\partial_1 g = x_2\), \(\partial_2 g = x_1\),
\(\partial_{12} g = \partial_{21} g = 1\), and \(\partial_{11} g = \partial_{22} g = 0\).
The membership condition of the general formula for the single image component reads
\(\partial_1 g\, v_{1j} + \partial_2 g\, v_{2j} = Y_s\, v^X_j + X_s\, v^Y_j
\in \mathcal{V}(0, T)\), which is the hypothesis. The drift of the image is
\(X u^Y + Y u^X\) from the first-order terms and, from the quadratic form,
\(\tfrac{1}{2} \bigl[ (\mathbf{v}\mathbf{v}^\top)_{12} + (\mathbf{v}\mathbf{v}^\top)_{21} \bigr]
= \sum_j v^X_j v^Y_j\), while the diffusion row is \(X v^Y_j + Y v^X_j\). This is the
display.
The correction \(dX \cdot dY\) is the polarization of the square correction, and this is
worth seeing once. The algebraic identity
\(X Y = \tfrac{1}{2}\bigl[ (X + Y)^2 - X^2 - Y^2 \bigr]\) expresses every product through
squares, and applying the chain rule for squares to the three Itô processes
\(X + Y\), \(X\), \(Y\) produces drift corrections
\(\sum_j (v^X_j + v^Y_j)^2\), \(\sum_j (v^X_j)^2\), and
\(\sum_j (v^Y_j)^2\), whose signed half-combination
\(\tfrac{1}{2}\bigl[ \sum_j (v^X_j + v^Y_j)^2 - \sum_j (v^X_j)^2 - \sum_j (v^Y_j)^2 \bigr]\)
is exactly \(\sum_j v^X_j v^Y_j\). The product correction carries no new information beyond the square
correction that a single computation with \(\int w\, dw\) first exposed. Specializing the
theorem to \(X = Y = w^{(1)}\), where \(v^X_1 = v^Y_1 = 1\), returns
\(d(w^2) = 2 w\, dw + dt\), the broken prediction plus its repair.
One notational corollary completes the classical toolkit. For an Itô process \(Y\) and a
process \(H\) that is progressively measurable with
\(\int_0^t |H_s\, u^Y|\, ds \lt \infty\) almost surely for all \(t\) and
\(H v^Y_j \in \mathcal{V}(0, T)\) for all \(j, T\), we write
\[
\int_0^t H_s\, dY_s
= \int_0^t H_s\, u^Y\, ds + \sum_{j=1}^{n} \int_0^t H_s\, v^Y_j\, dw_s^{(j)} ,
\]
the integral of \(H\) against the Itô process \(Y\). In this notation, and under the
membership hypotheses making every integral below legitimate, the product rule rearranges
into the general integration by parts formula,
\[
\int_0^t X_s\, dY_s
= X_t Y_t - X_0 Y_0
- \int_0^t Y_s\, dX_s
- \int_0^t dX_s \cdot dY_s ,
\]
where the last term abbreviates \(\int_0^t \sum_j v^X_j v^Y_j\, ds\). Against the version a
previous page proved for deterministic integrands, the price of letting both factors be
random is on display. Symmetric exchange of the two sides costs exactly one quadratic
covariation term, the term that vanishes when either factor has no noise and that classical
analysis, working with bounded variation, never had to see.
A Process Confined to a Region
The one-dimensional page recorded a promise. The Itô formula only ever evaluates \(g\) along
the path, so if the process never leaves an open set, smoothness should only be required
there, and the refinement would be collected when a process confined to a region first
appeared. The process has appeared, and the multi-dimensional setting forces us to collect
the debt honestly rather than by the one sentence the promise suggested. The proof expands
\(g\) along straight segments between consecutive path points, and an open set in
\(\mathbb{R}^m\), unlike a connected open set in \(\mathbb{R}\), need not contain the
segment between two of its points. The repair is a compactness argument.
Theorem: The Itô Formula on an Open Domain
Let \(\mathbf{X}_t\) be an \(m\)-dimensional Itô process, let
\(U \subseteq \mathbb{R}^m\) be open, and suppose that almost every path of
\(\mathbf{X}\) stays in \(U\) for all time. Then the general Itô formula, with its
membership condition unchanged, holds for every
\(g \in C^2([0, \infty) \times U)\), scalar or vector valued.
Proof.
Two modifications to the proof of the general formula suffice, and everything else runs
unchanged, the reduction to bounded derivatives now performed with approximants
controlled on compact subsets of \([0, \infty) \times U\) by the same trusted smoothing
passage.
First, the segments. Fix \(\omega\) in the full-measure set where the path stays in
\(U\), is continuous, and the increments are exact. The path image
\(K_\omega = \{ \mathbf{X}_s(\omega) : 0 \leq s \leq t \}\) is compact, being the
continuous image of \([0, t]\), and it is contained in the open set \(U\), so its
distance to the complement is a positive number \(\delta_\omega\), the continuous
distance function to the closed complement attaining a minimum on the compact
\(K_\omega\) and vanishing nowhere on it. When \(U = \mathbb{R}^m\) there is nothing
to prove, the original theorem applying as it stands. By uniform continuity
of the path, \(\max_j |\Delta \mathbf{X}_j(\omega)| \lt \delta_\omega / 2\) once the
mesh is small enough, and from that stage on every Taylor segment lies within distance
\(\delta_\omega / 2\) of \(K_\omega\), hence inside \(U\). The Taylor expansion, the
weighted-average coefficients, and the modulus control then run exactly as before, the
modulus taken on the compact set
\(\{ (s, \mathbf{x}) : s \in [0, t],\ \operatorname{dist}(\mathbf{x}, K_\omega) \leq
\delta_\omega / 2 \}\), which is contained in \([0, t] \times U\). The well-posedness checks run on the same
compact set, which sits inside the domain of \(g\).
Second, the assembly. Here the argument needs one moment of care, and it is worth
spelling out slowly. For each outcome \(\omega\), the Taylor decomposition of the fixed
quantity \(L = g(t, \mathbf{X}_t) - g(0, \mathbf{X}_0)\) into the stage-\(n\) sums
\(S_n\) is valid as soon as the partition is fine enough for that particular path, in
the sense of the first modification. How fine is fine enough depends on the path, so at
any fixed stage \(n\) there may be outcomes for which the decomposition is not yet
valid. Nothing is claimed for them. Let \(A_n\) be the event that stage \(n\) is fine
enough, that is, \(\max_j |\Delta \mathbf{X}_j| \lt \delta_\omega / 2\) for the
stage-\(n\) partition, so that \(L = S_n\) on \(A_n\). For almost every outcome the
maximal increment tends to zero, so the indicator of \(A_n\) tends to one almost
surely, and \(\mathbb{P}[A_n] \to 1\) by the
dominated convergence theorem.
The limits of the individual sums are unchanged, so \(S_n \to R\) in probability with
\(R\) the right side of the formula, and for every \(\epsilon \gt 0\),
\[
\mathbb{P}\bigl[ |L - R| \geq \epsilon \bigr]
\leq \mathbb{P}\bigl[ A_n^{c} \bigr]
+ \mathbb{P}\bigl[ |S_n - R| \geq \epsilon \bigr] ,
\]
because on \(A_n\) the quantity \(|L - R|\) equals \(|S_n - R|\). Both terms tend to
zero, the left side does not depend on \(n\), and \(\epsilon\) is arbitrary, so
\(L = R\) almost surely. The continuity upgrade is untouched.
The theorem earns its keep immediately, on the most natural function that is smooth
everywhere except one point.
The Bessel Process
Let \(\mathbf{w}_t\) be \(n\)-dimensional standard Brownian motion with \(n \geq 2\),
started, for this example only and in deviation from the standing convention of the page, at
a point \(\mathbf{x}_0 \neq 0\), as the defining properties of Brownian motion permit. View
it as the Itô process with \(\mathbf{u} = 0\) and \(\mathbf{v} = I_n\), and consider its
distance to the origin,
\[
R_t = |\mathbf{w}_t| = \Bigl( \bigl( w_t^{(1)} \bigr)^2 + \cdots +
\bigl( w_t^{(n)} \bigr)^2 \Bigr)^{1/2} .
\]
The function \(g(\mathbf{x}) = |\mathbf{x}|\) is not differentiable at the origin, and no
version of the formula on all of \(\mathbb{R}^n\) can touch it. On
\(U = \mathbb{R}^n \setminus \{0\}\) it is smooth, with
\[
\begin{align*}
\partial_i g &= \frac{x_i}{|\mathbf{x}|} , \\\\
\partial_{il} g &= \frac{\delta_{il}}{|\mathbf{x}|} - \frac{x_i x_l}{|\mathbf{x}|^3} ,
\end{align*}
\]
and the domain theorem asks one thing, that almost every path of \(\mathbf{w}\) avoid the
origin for all time. Starting away from the origin handles time zero, and for the rest this
is true for \(n \geq 2\), and we take it on faith here as
a recorded debt. It is a genuine theorem about the polarity of points for planar and higher
Brownian motion, its proof runs through machinery this track has not built, and it is the
single external input of this example. Granting it, the membership condition is immediate,
each diffusion entry \(\partial_i g(\mathbf{w}_s) = w_s^{(i)} / R_s\) being bounded by one
and progressively measurable, and the formula delivers
\[
dR_t
= \sum_{i=1}^{n} \frac{w_t^{(i)}}{R_t}\, dw_t^{(i)}
+ \frac{n - 1}{2 R_t}\, dt ,
\]
the drift arising from the trace,
\[
\begin{align*}
\tfrac{1}{2} \sum_{i} \partial_{ii} g
&= \tfrac{1}{2} \bigl( \tfrac{n}{|\mathbf{x}|} - \tfrac{|\mathbf{x}|^2}{|\mathbf{x}|^3} \bigr) \\\\
&= \tfrac{n - 1}{2 |\mathbf{x}|} ,
\end{align*}
\]
since \(\mathbf{v}\mathbf{v}^\top = I_n\). The process \(R\) is the
\(n\)-dimensional Bessel process, the radial part of Brownian motion, and
its equation repays a moment of reading. The noise coefficients \(w^{(i)} / R\) form a unit
vector, so radially the process fluctuates with the strength of a single one-dimensional
Brownian motion. Everything dimensional sits in the drift \((n - 1) / 2 R\), a deterministic
outward push that grows as the process approaches the origin and vanishes in dimension one.
Curvature of the sphere, felt through the second derivatives of the distance function,
converts ambient noise into radial drift. That a geometric quantity enters through the
correction term of a chain rule is not a coincidence of this example, and it is one of the
doors through which stochastic calculus walks onto manifolds.
The page closes where the chapter of the formula closes. One noise source became many, the
correction became a quadratic form in \(\mathbf{v}\mathbf{v}^\top\), the product rule came
back with its price tag attached, and a function smooth away from one point met a process
that politely never visits it. The calculus is now strong enough to be pointed at equations
whose coefficients depend on the unknown process itself, and that is where the track goes
next.