First Computations
The previous page proved the
Itô formula
and promised that it turns the evaluation of stochastic integrals into calculus. This
page collects the payoff. We evaluate integrals without returning to the definition,
prove the stochastic version of integration by parts, and settle the account the
construction story left open, the correction separating the Itô and Stratonovich
readings of a stochastic differential.
Throughout, \(w_t\) is the standard one-dimensional Brownian motion of the last four pages,
started at \(w_0 = 0\), with its Brownian filtration \(\{\mathcal{F}_t\}_{t \geq 0}\)
containing all null sets. The
Itô process
notation \(dX_t = u\, dt + v\, dw_t\) carries its meaning from the previous page.
One discipline accompanies every application. The Itô formula carries a membership hypothesis,
that the diffusion coefficient of the image, \(v\, \frac{\partial g}{\partial x}(s, X_s)\),
lies in
\(\mathcal{V}(0, T)\)
for every \(T\). The check is one line in each case below, and we write the line each time,
because a hypothesis that is never checked eventually gets forgotten.
The Integral That Started It
First, the formula must reproduce the one integral this track ever computed from the
definition. Take \(X_t = w_t\), an Itô process with \(u = 0\) and \(v = 1\), and
\(g(t, x) = \tfrac{1}{2} x^2\), which is \(C^2\) with
\(\frac{\partial g}{\partial t} = 0\), \(\frac{\partial g}{\partial x} = x\),
and \(\frac{\partial^2 g}{\partial x^2} = 1\).
Membership asks that \(v\, \frac{\partial g}{\partial x}(s, X_s) = w_s\) belong to
\(\mathcal{V}(0, T)\), and it does. It is adapted with continuous paths, hence
progressively measurable
and in particular jointly measurable, and it satisfies
\(\mathbb{E}\bigl[ \int_0^T w_s^2\, ds \bigr] = \int_0^T s\, ds \lt \infty\), where the
exchange of expectation and integral is the
Tonelli
exchange for a nonnegative integrand. The formula returns, with its initial term
\(g(0, w_0) = 0\) vanishing,
\[
\begin{align*}
\tfrac{1}{2} w_t^2
&= \int_0^t \tfrac{1}{2}\, ds + \int_0^t w_s\, dw_s \\\\
&= \tfrac{1}{2} t + \int_0^t w_s\, dw_s ,
\end{align*}
\]
which is
the integral of Brownian motion against itself
read from the other side. What the construction page reached through truncation, telescoping,
and the quadratic variation theorem is now the statement that the chain rule's correction
term equals \(\tfrac{1}{2} \cdot 1^2 \cdot 1 = \tfrac{1}{2}\).
A First New Integral
We turn now to an integral the definition never touched. What is \(\int_0^t s\, dw_s\)? Classical
calculus suggests that a term \(t\, w_t\) should appear, so we aim the formula at it. Take
\(X_t = w_t\) again and \(g(t, x) = t x\), which is \(C^2\) on
\([0, \infty) \times \mathbb{R}\) with
\(\frac{\partial g}{\partial t} = x\), \(\frac{\partial g}{\partial x} = t\), and all second
partial derivatives either \(1\) or \(0\).
Membership asks that \(v\, \frac{\partial g}{\partial x}(s, X_s) = s\) belong to
\(\mathcal{V}(0, T)\), and the deterministic integrand \(s\) is adapted, jointly measurable,
and square integrable on \([0, T]\), so it does. Since
\(\frac{\partial^2 g}{\partial x^2} = 0\), the correction term vanishes, the initial term
\(g(0, w_0) = 0\) vanishes as well, and the formula gives
\[
t\, w_t = \int_0^t w_s\, ds + \int_0^t s\, dw_s ,
\]
that is,
\[
\int_0^t s\, dw_s = t\, w_t - \int_0^t w_s\, ds .
\]
The \(ds\)-term on the right is the
pathwise time integral
of a continuous adapted process, well defined path by path. The identity has exactly the shape
classical integration by parts predicts, and no correction appears. That is not an accident.
The function \(g(t, x) = t x\) is linear in \(x\), its second \(x\)-derivative vanishes, and
the Itô correction is proportional to precisely that derivative. Whenever the noise enters
linearly, Itô calculus and classical calculus agree. The next section turns this observation
into a theorem.
Integration by Parts
The second computation of the last section is begging to be generalized. Nothing about the
argument used the specific function \(t\), only that the noise entered linearly through
\(g(t, x) = f(t)\, x\). Running the same argument with a general deterministic \(f\) yields
the stochastic version of integration by parts.
Theorem: Integration by Parts
Let \(f \in C^2([0, \infty))\) be deterministic. Then, for every \(t \geq 0\), almost
surely,
\[
\int_0^t f(s)\, dw_s
= f(t)\, w_t - \int_0^t f'(s)\, w_s\, ds ,
\]
the \(ds\)-term being the pathwise time integral of the continuous adapted process
\(f'(s)\, w_s\).
Proof.
Take \(X_t = w_t\), an Itô process with \(u = 0\) and \(v = 1\), and
\(g(t, x) = f(t)\, x\). The partial derivatives are
\(\frac{\partial g}{\partial t} = f'(t)\, x\),
\(\frac{\partial g}{\partial x} = f(t)\),
\(\frac{\partial^2 g}{\partial t^2} = f''(t)\, x\),
\(\frac{\partial^2 g}{\partial t \partial x} = f'(t)\), and
\(\frac{\partial^2 g}{\partial x^2} = 0\), all continuous on
\([0, \infty) \times \mathbb{R}\) because \(f \in C^2\),
so \(g \in C^2([0, \infty) \times \mathbb{R})\).
Membership asks that \(v\, \frac{\partial g}{\partial x}(s, X_s) = f(s)\) belong to
\(\mathcal{V}(0, T)\) for every \(T\), and the deterministic continuous integrand \(f\) is
adapted, jointly measurable, and bounded on \([0, T]\), hence square integrable there. The
Itô formula
applies, the correction term vanishes with \(\frac{\partial^2 g}{\partial x^2}\), the
initial term is \(g(0, w_0) = f(0) \cdot 0 = 0\), and
\[
f(t)\, w_t
= \int_0^t f'(s)\, w_s\, ds + \int_0^t f(s)\, dw_s .
\]
Rearranging gives the claim.
Two extensions are worth recording, both beyond what the formula's hypotheses deliver
directly. First, the identity holds already for \(f \in C^1([0, \infty))\), by smoothing
\(f\) with mollifiers and passing to the limit. The
Itô isometry
controls the stochastic side, and uniform convergence on \([0, t]\) controls the classical
side. We record this without executing the limit passage. Second, in its strongest
form the identity reads
\[
\int_0^t f(s, \omega)\, dw_s
= f(t, \omega)\, w_t - \int_0^t w_s\, df_s
\]
for adapted processes \(f\) whose paths are continuous and of bounded variation, with the
integrability the left side requires. The last term is a pathwise Riemann-Stieltjes
integral against the path of \(f\). Determinism is not required. Our track has not built
Riemann-Stieltjes integration, so this version stays a recorded signpost, to be collected if
and when that machinery arrives. Its key hypothesis is bounded
variation, which makes a pathwise Riemann-Stieltjes integral against \(df_s\) exist, and which
Brownian motion itself fails, on
every interval.
Where the Classical Rule Breaks
The bounded variation hypothesis is doing real work, and the cleanest way to see it is to
break the rule on purpose. Brownian motion has unbounded variation on every interval, so
\(f = w\) sits outside every version of the identity above. Pretend otherwise. The symmetric
product rule applied to \(f = w\) would predict
\(w_t \cdot w_t = \int_0^t w_s\, dw_s + \int_0^t w_s\, dw_s\), that is,
\[
\int_0^t w_s\, dw_s = \tfrac{1}{2}\, w_t^2 ,
\]
while the true value carries the correction, \(\int_0^t w_s\, dw_s = \tfrac{1}{2} w_t^2 -
\tfrac{1}{2} t\). The classical prediction is wrong by exactly \(\tfrac{1}{2} t\), half the
quadratic variation of the path. This is the same half that the previous page located in the
Taylor expansion. There is a third place to locate it, and it is the most illuminating one,
because it explains what a calculus would have to give up to keep the classical rules. That is
the business of the final section.
The Stratonovich Conversion
The half has now been found in two places, in the Taylor expansion behind the Itô formula and
in the discrepancy of the broken product rule. The third place is the oldest. The construction
page committed the integral to evaluating integrands at the left endpoint of each interval,
the causal choice, and named the midpoint alternative the
Stratonovich integral. On
the properties page, the track then computed its
value for the test integrand at statement level and displayed the drift correction
\(\tfrac{1}{2} \sigma' \sigma\) that translates between the two calculi, deferring the
derivation. Everything needed to pay these debts is now on the table.
Definition: The Stratonovich Integral
Let \(\{Z_s\}_{s \geq 0}\) be an adapted process with continuous paths, and fix
\(t \gt 0\). For a partition \(\pi : 0 = t_0 \lt \cdots \lt t_N = t\) write
\(m_j = \tfrac{1}{2}(t_j + t_{j+1})\) for the midpoints and form the midpoint sums
\[
S_\pi(Z) = \sum_j Z_{m_j} \bigl( w_{t_{j+1}} - w_{t_j} \bigr) .
\]
If there is a random variable \(I\) such that \(S_{\pi_n}(Z) \to I\) in probability along
every sequence of partitions of \([0, t]\) with mesh tending to zero, we call \(I\) the
Stratonovich integral of \(Z\) and write
\[
\int_0^t Z_s \circ dw_s = I .
\]
The definition is an obligation, not a gift. Nothing guarantees that the midpoint sums
converge, and existence must be proved integrand by integrand, or theorem by theorem. Both
proofs below run on the factorization mechanism of the previous page's lemma, which we
reassemble in one line for use here. If \(0 \leq a \lt b\), \(Y\) is integrable and
measurable with respect to \(\mathcal{F}_a\), and \(h\) is a polynomial, then \(Y\) and
\(h(w_b - w_a)\) are
independent,
because the increment is
independent of the past.
The product rule,
applicable because a polynomial of a Gaussian variable is integrable, then gives
\(\mathbb{E}[Y\, h(w_b - w_a)] = \mathbb{E}[Y]\, \mathbb{E}[h(w_b - w_a)]\).
The Test Integrand, Settled
Proposition: The Stratonovich Integral of Brownian Motion
The Stratonovich integral of \(Z = w\) exists on every \([0, t]\), and
\[
\int_0^t w_s \circ dw_s = \tfrac{1}{2}\, w_t^2 .
\]
Proof.
Fix a sequence of partitions with mesh tending to zero, and abbreviate
\(\Delta w_j = w_{t_{j+1}} - w_{t_j}\), together with the half-increments
\(\delta w_j = w_{m_j} - w_{t_j}\) and \(\delta w_j' = w_{t_{j+1}} - w_{m_j}\), so that
\(\Delta w_j = \delta w_j + \delta w_j'\). Splitting the midpoint value against the left
endpoint, we obtain
\[
\begin{align*}
S_\pi(w)
&= \sum_j w_{t_j}\, \Delta w_j
+ \sum_j \delta w_j \bigl( \delta w_j + \delta w_j' \bigr) \\\\
&= \sum_j w_{t_j}\, \Delta w_j
+ \sum_j (\delta w_j)^2
+ \sum_j \delta w_j\, \delta w_j' .
\end{align*}
\]
The left-endpoint sums converge in \(L^2(\mathbb{P})\) to the Itô integral. They are the
elementary integrals of \(\Phi_\pi = \sum_j w_{t_j} \mathbf{1}_{[t_j, t_{j+1})}\). The
identification survives the unbounded levels \(w_{t_j}\) by the truncation bridge run
for exactly this process inside the proof of
the first computation.
Exchanging expectation and integral by
Tonelli
for a nonnegative integrand and using the increment law
\(\mathbb{E}[(w_{t_j(s)} - w_s)^2] = s - t_j(s)\), where \(t_j(s)\) is the left endpoint of the
partition interval containing \(s\), we find
\[
\begin{align*}
\mathbb{E}\bigl[ \int_0^t (\Phi_\pi - w)^2\, ds \bigr]
&= \int_0^t (s - t_j(s))\, ds \\\\
&\leq t\, |\pi| ,
\end{align*}
\]
so
L² continuity
gives \(\sum_j w_{t_j} \Delta w_j \to \int_0^t w_s\, dw_s\).
The squared half-increments converge to \(\tfrac{1}{2} t\) in \(L^2(\mathbb{P})\). Consider
the refined partition \(\pi'\) obtained by inserting every midpoint, and apply part (ii)
of the
weighted increment lemma
over \(\pi'\) with weights equal to one on the first-half intervals and zero on the
second-half intervals. The weights are measurable at the left endpoints and bounded by
one. This gives \(\sum_j \bigl( (\delta w_j)^2 - \tfrac{1}{2} \Delta t_j \bigr) \to 0\) in
\(L^2(\mathbb{P})\), and \(\sum_j \tfrac{1}{2} \Delta t_j = \tfrac{1}{2} t\).
The mixed sum dies. Expand its square. The off-diagonal terms with \(i \lt j\) vanish
by factorization, since \((\delta w_i\, \delta w_i')(\delta w_j)\) is
\(\mathcal{F}_{m_j}\)-measurable and integrable while
\(\mathbb{E}[\delta w_j'] = 0\), and the diagonal factorizes,
\[
\begin{align*}
\mathbb{E}\Bigl[ \Bigl( \sum_j \delta w_j\, \delta w_j' \Bigr)^{2} \Bigr]
&= \sum_j \mathbb{E}\bigl[ (\delta w_j)^2 \bigr]\,
\mathbb{E}\bigl[ (\delta w_j')^2 \bigr] \\\\
&= \sum_j \tfrac{(\Delta t_j)^2}{4} \\\\
&\leq \tfrac{1}{4}\, t\, |\pi| \\\\
&\longrightarrow 0 .
\end{align*}
\]
Adding the three limits gives \(S_\pi(w) \to \int_0^t w_s\, dw_s + \tfrac{1}{2} t\) in
probability along every sequence with vanishing mesh, and
the first computation
identifies the limit as
\(\tfrac{1}{2} w_t^2 - \tfrac{1}{2} t + \tfrac{1}{2} t = \tfrac{1}{2} w_t^2\).
The claim that the properties page recorded at statement level is now a theorem.
The Conversion Theorem
Theorem: The Itô-Stratonovich Conversion
Let \(X_t = X_0 + \int_0^t u\, ds + \int_0^t v\, dw_s\) be an Itô process whose
coefficients \(u\) and \(v\) are, on \([0, t]\), elementary over a common partition
\(R\) with all levels bounded by \(C\). Let \(\sigma : [0, \infty) \times \mathbb{R}
\to \mathbb{R}\) have continuous partial derivatives
\(\frac{\partial \sigma}{\partial t}\) and \(\frac{\partial \sigma}{\partial x}\), with
\(\sigma\), \(\frac{\partial \sigma}{\partial t}\), and
\(\frac{\partial \sigma}{\partial x}\) bounded by \(M\). Then, along every sequence of
partitions of \([0, t]\) refining \(R\) with mesh tending to zero, the midpoint sums of
\(Z_s = \sigma(s, X_s)\) converge in probability, and
\[
\int_0^t \sigma(s, X_s) \circ dw_s
= \int_0^t \sigma(s, X_s)\, dw_s
+ \tfrac{1}{2} \int_0^t \frac{\partial \sigma}{\partial x}(s, X_s)\, v(s, \omega)\, ds ,
\]
the left side denoting the common limit along such sequences. Dropping the restriction to
refining sequences, which makes the limit the Stratonovich integral in the unrestricted
sense of the definition, is part of the approximation passage recorded in the first remark
after the proof, and we take it on faith.
Proof.
Fix a sequence of partitions refining \(R\) with mesh tending to zero, keep the notation
of the previous proof, and write \(X_j = X_{t_j}\), \(Z_j = \sigma(t_j, X_j)\),
\(\sigma_x = \frac{\partial \sigma}{\partial x}\), and
\(\sigma_t = \frac{\partial \sigma}{\partial t}\). Because the partition refines \(R\)
and midpoints stay inside their intervals, \(u\) and \(v\) are constant on
\([t_j, t_{j+1})\) with \(\mathcal{F}_{t_j}\)-measurable values \(u_j, v_j\)
bounded by \(C\).
The drift piece of the half-increment is the pathwise integral of a constant level, and the
noise piece is the elementary integral of a constant level, identified with the increment
of the integral process by
additivity over intervals.
Outside a single null set, for every partition of the sequence at once,
\[
X_{m_j} - X_j = u_j\, \tfrac{\Delta t_j}{2} + v_j\, \delta w_j .
\]
Step 1 (Splitting at the left endpoint).
Decompose the midpoint value as
\(Z_{m_j} = Z_j + \sigma_x(t_j, X_j)\,(X_{m_j} - X_j) + \rho_j\), where the remainder
collects the time increment and the mean value error,
\[
\rho_j
= \bigl[ \sigma(m_j, X_{m_j}) - \sigma(t_j, X_{m_j}) \bigr]
+ \bigl[ \sigma(t_j, X_{m_j}) - \sigma(t_j, X_j)
- \sigma_x(t_j, X_j)\, (X_{m_j} - X_j) \bigr] .
\]
Multiplying by \(\Delta w_j\) and summing gives
\[
S_\pi(Z)
= \sum_j Z_j\, \Delta w_j
+ \sum_j \sigma_x(t_j, X_j)\, u_j\, \tfrac{\Delta t_j}{2}\, \Delta w_j
+ \sum_j \sigma_x(t_j, X_j)\, v_j\, \delta w_j\, \Delta w_j
+ \sum_j \rho_j\, \Delta w_j .
\]
Step 2 (The left sums and the drift-noise sum).
The left sums \(\sum_j Z_j \Delta w_j\) converge in \(L^2(\mathbb{P})\) to
\(\int_0^t \sigma(s, X_s)\, dw_s\), by the argument of the previous page's proof run with
\(\sigma\) in place of \(g_x v\). The approximants
\(\Phi_\pi = \sum_j Z_j \mathbf{1}_{[t_j, t_{j+1})}\) are elementary with
\(\mathcal{F}_{t_j}\)-measurable levels bounded by \(M\). They converge pointwise
\(\lambda \otimes \mathbb{P}\)-almost everywhere to \(\sigma(s, X_s)\) by continuity of the
paths, and the constant \(2 M\) dominates the differences, so the
dominated convergence theorem
on the finite product measure and
L² continuity
conclude. In passing, \(\sigma(s, X_s)\) is progressively measurable, being a continuous
function of \((s, X_s)\) as in the well-posedness check of the previous page, and it is
bounded by \(M\). It therefore lies in \(\mathcal{V}(0, t)\), so the Itô integral on the
right of the theorem is legitimate. The drift-noise sum is part (i) of the
weighted increment lemma
with weights \(\tfrac{1}{2} \sigma_x(t_j, X_j)\, u_j\) bounded by \(\tfrac{1}{2} M C\),
and dies in \(L^2(\mathbb{P})\).
Step 3 (The main sum).
Write \(b_j = \sigma_x(t_j, X_j)\, v_j\), which is \(\mathcal{F}_{t_j}\)-measurable and
bounded by \(M C\), and split \(\Delta w_j = \delta w_j + \delta w_j'\),
\[
\sum_j b_j\, \delta w_j\, \Delta w_j
= \sum_j b_j\, (\delta w_j)^2
+ \sum_j b_j\, \delta w_j\, \delta w_j' .
\]
The first sum converges in probability to
\(\tfrac{1}{2} \int_0^t \sigma_x(s, X_s)\, v(s, \omega)\, ds\). Centering it with part
(ii) of the lemma over the midpoint-refined partition, with weights \(b_j\) on the
first-half intervals and zero on the second-half intervals, leaves
\(\sum_j b_j\, \tfrac{\Delta t_j}{2}\). This is a left-endpoint Riemann sum of the
piecewise continuous path \(s \mapsto \tfrac{1}{2} \sigma_x(s, X_s)\, v(s)\), and it
converges pathwise interval by interval of \(R\), exactly as on the previous page.
The second sum dies in \(L^2(\mathbb{P})\) by the computation of the proposition,
decorated with the weights. Off-diagonal terms vanish by factorization, since
\((b_i \delta w_i \delta w_i')(b_j \delta w_j)\) is \(\mathcal{F}_{m_j}\)-measurable and
integrable while \(\mathbb{E}[\delta w_j'] = 0\), and the diagonal factorizes into
\(\mathbb{E}[b_j^2 (\delta w_j)^2]\, \mathbb{E}[(\delta w_j')^2]
\leq M^2 C^2 (\Delta t_j)^2 / 4\). The diagonal terms sum to at most
\(M^2 C^2\, t\, |\pi| / 4 \to 0\).
Step 4 (The remainder dies).
Write \(\rho_j^{\mathrm{time}}\) and \(\rho_j^{\mathrm{space}}\) for the two brackets
of \(\rho_j\). The bracket \(\rho_j^{\mathrm{time}}\) equals the integral of \(\sigma_t\) over
\([t_j, m_j]\) at frozen second argument, a representation legitimate for a \(C^1\) time
dependence with the one-sided derivative at the time origin. The bound on \(\sigma_t\)
therefore gives \(|\rho_j^{\mathrm{time}}| \leq M\, \tfrac{\Delta t_j}{2}\), and
\[
\begin{align*}
\mathbb{E}\Bigl[ \sum_j \tfrac{M}{2}\, \Delta t_j\, |\Delta w_j| \Bigr]
&\leq \tfrac{M}{2} \sum_j \Delta t_j\, (\Delta t_j)^{1/2} \\\\
&\leq \tfrac{M}{2}\, t\, |\pi|^{1/2} \\\\
&\longrightarrow 0 ,
\end{align*}
\]
using \(\mathbb{E}|\Delta w_j| \leq \mathbb{E}[(\Delta w_j)^2]^{1/2} = (\Delta t_j)^{1/2}\)
by the
Cauchy-Schwarz inequality.
For the space part, we write the increment of the \(C^1\) function
\(\tau \mapsto \sigma(t_j, X_j + \tau (X_{m_j} - X_j))\) as the integral of its
derivative over \([0, 1]\), so the bracket equals
\(\int_0^1 \bigl[ \sigma_x(t_j, X_j + \tau (X_{m_j} - X_j)) - \sigma_x(t_j, X_j) \bigr] d\tau \cdot (X_{m_j} - X_j)\).
Let \(\eta(\pi, \omega)\) be the maximum over \(j\) of the supremum over rational
\(\tau \in [0, 1]\) of
\(|\sigma_x(t_j, X_j + \tau (X_{m_j} - X_j)) - \sigma_x(t_j, X_j)|\). By continuity in
\(\tau\), this supremum is the one over all of \([0, 1]\), so the bracket is at most
\(\eta\, |X_{m_j} - X_j|\) in absolute value. As a countable supremum of random
variables, \(\eta\) is itself a random variable.
Fix \(\omega\) outside the null set chosen at the start of the proof. Then
\(\eta(\pi, \omega)\) is at most the modulus of continuity of \(\sigma_x\) on the compact
set \([0, t] \times [-r_\omega, r_\omega]\), evaluated at \(\max_j |X_{m_j} - X_j|\),
where \(r_\omega = \max_{s \leq t} |X_s|\). This maximal displacement tends to zero by
uniform continuity of the path on \([0, t]\). Hence \(\eta \to 0\) pathwise. Moreover
\(\eta \leq 2 M\). With \(|X_{m_j} - X_j| \leq C \Delta t_j + C |\delta w_j|\) and
\(2 |\delta w_j| |\Delta w_j| \leq 3 (\delta w_j)^2 + (\delta w_j')^2\), we obtain
\[
\sum_j \bigl| \rho_j^{\mathrm{space}} \bigr|\, |\Delta w_j|
\leq \eta\, C \Bigl( \sum_j \Delta t_j |\Delta w_j|
+ \tfrac{3}{2}\, Q_{\pi'} \Bigr) ,
\]
where \(Q_{\pi'}\) is the quadratic sum over the midpoint-refined partition. The first
term inside dies in \(L^1(\mathbb{P})\) by the time-part estimate above, uniformly
against \(\eta \leq 2 M\). The second is handled exactly as on the previous page. By
Cauchy-Schwarz it needs only \(\mathbb{E}[\eta^2] \to 0\), from the
dominated convergence theorem,
and the \(L^2\)-boundedness of \(Q_{\pi'}\), from
quadratic variation.
So \(\sum_j \rho_j \Delta w_j \to 0\) in \(L^1(\mathbb{P})\).
Step 5 (Assembly).
Every piece of the splitting converges in probability, so the midpoint sums converge in
probability to
\(\int_0^t \sigma(s, X_s)\, dw_s + \tfrac{1}{2} \int_0^t \sigma_x(s, X_s)\, v\, ds\),
and the limit does not depend on the chosen sequence, which is existence and the claimed
identity at once.
Two remarks finish the account. First, the boundedness and elementary-coefficient
hypotheses are the core-case regime of the Itô formula's proof. Carrying the
conversion to general coefficients, and dropping the restriction to partitions that refine
\(R\), takes a further approximation passage, which, like the approximation passages of
that proof, we take on faith.
Second, granted that passage, take the equation-level case displayed on the properties page,
where the noise coefficient of \(X\) is itself \(v = \sigma(s, X_s)\). The correction becomes
\(\tfrac{1}{2} \int_0^t \sigma_x\, \sigma\, ds\), which is exactly the drift correction
\(\tfrac{1}{2} \sigma' \sigma\) of the promised display. Reading a stochastic differential in
the Stratonovich sense and reading it in the Itô sense with that drift added are the same
reading, and the derivation deferred there is the theorem above.
Two Calculi, One Division of Labor
Within the scope of the conversion theorem, nothing is at stake mathematically in the
choice of calculus, and everything is at stake ergonomically. The Itô integral
evaluates at the left endpoint, never peeks ahead, and inherits the zero-mean and
martingale structure that makes expectations computable, which is why much of
probability, statistics, and finance lives on the Itô side. The Stratonovich
integral obeys the classical chain rule, which is why modeling limits of smooth physics
and calculus on manifolds, where a chain rule that respects coordinate changes is not
negotiable, mostly live on the Stratonovich side.
In the continuous-time formulation of a
diffusion model, the forward noising
process is an Itô equation whose noise coefficient depends on time alone, so the
correction vanishes and the two readings coincide. The distinction becomes real when the
noise coefficient depends on the state. Gradient methods for learned stochastic differential
equations have been derived in the Stratonovich form, whose symmetry under time reversal
gives clean backward dynamics, and a model posed in the Itô form reaches them
through the multi-dimensional form of this conversion.