The Itô Formula

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The Itô Process The Chain Rule Acquires a Correction Proof of the Itô Formula

The Itô Process

Throughout this page, \(w_t\) is the standard one-dimensional Brownian motion of the last three pages, started at \(w_0 = 0\) and carried on \((\Omega, \mathcal{F}, \mathbb{P})\), and \(\{\mathcal{F}_t\}_{t \geq 0}\) is its Brownian filtration, with the convention that every set of \(\mathbb{P}\)-measure zero belongs to each \(\mathcal{F}_t\).

Two pages ago we computed the integral of Brownian motion against itself, \(\int_0^t w_s\, dw_s = \tfrac{1}{2} w_t^2 - \tfrac{1}{2} t\). Rearranged, the identity reads

\[ \tfrac{1}{2} w_t^2 = \tfrac{1}{2} t + \int_0^t w_s\, dw_s . \]

We now read this as a statement about the map \(g(x) = \tfrac{1}{2} x^2\). The process \(w_t\) is itself an Itô integral, namely \(\int_0^t 1\, dw_s\), and the left side is its image under \(g\). That image is not an Itô integral. Every integral \(\int_0^t f\, dw_s\) with \(f \in \mathcal{V}(0, t)\) has zero mean, while \(\mathbb{E}\bigl[\tfrac{1}{2} w_t^2\bigr] = \tfrac{1}{2} t \gt 0\) for \(t \gt 0\). So the class of Itô integrals is not closed under smooth maps. But the identity also shows exactly how the closure fails. The image escaped the class by acquiring an ordinary \(ds\)-integral, and nothing else. This suggests the repair. Enlarge the class to processes assembled from a \(dw\)-integral and a \(ds\)-integral together, and prove that the enlarged class is stable under smooth maps. Both steps are the business of this page. The stability statement is the Itô formula, the chain rule of stochastic calculus.

The Time Integral as a Process

Before we can define the enlarged class we must make sense of its drift term. We want

\[ A_t(\omega) = \int_0^t u(s, \omega)\, ds \]

as an ordinary Lebesgue integral in \(s\), computed path by path for each fixed \(\omega\). This is a new kind of object for our track. The Itô integral was built globally, as an \(L^2\) limit over \(\Omega\), and never asked whether any individual path could be integrated. Here the integral is pathwise, and three things need checking. The sliced function \(s \mapsto u(s, \omega)\) must be measurable, so that the integral of each path exists. The random variable \(A_t\) must be \(\mathcal{F}_t\)-measurable, so that the new process is adapted. And the paths \(t \mapsto A_t(\omega)\) should be continuous, so that the drift term matches the continuous version we adopted for the \(dw\)-term.

The subtlety sits in the second requirement. Membership in the class \(\mathcal{V}\) asks for measurability with respect to \(\mathcal{B}([0, \infty)) \otimes \mathcal{F}\) and for adaptedness one time at a time. Neither condition controls the product structure below a fixed level \(t\). The integral \(\int_0^t\) mixes all times \(s \leq t\) at once, so its measurability in \(\omega\) calls for a product \(\sigma\)-algebra whose \(\Omega\)-side is \(\mathcal{F}_t\) rather than all of \(\mathcal{F}\). For the Itô integral this issue never surfaced, because \(\mathcal{F}_t\)-measurability passed through the \(L^2\) limit of elementary integrals. The pathwise integral has no such limit to hide behind, and we record the condition tailored to what it needs.

Definition: Progressively Measurable Process

Let \(\{\mathcal{F}_t\}_{t \geq 0}\) be a filtration on \((\Omega, \mathcal{F}, \mathbb{P})\). A function \(u : [0, \infty) \times \Omega \to \mathbb{R}\) is called progressively measurable with respect to \(\{\mathcal{F}_t\}\) if, for every \(t \geq 0\), the restriction of \(u\) to \([0, t] \times \Omega\) is measurable with respect to \(\mathcal{B}([0, t]) \otimes \mathcal{F}_t\).

Progressive measurability strengthens both conditions of the class \(\mathcal{V}\). For joint measurability, note that for any Borel set \(E \subseteq \mathbb{R}\) the preimage \(u^{-1}(E)\) is the union over \(n \in \mathbb{N}\) of its intersections with \([0, n] \times \Omega\), each of which lies in \(\mathcal{B}([0, n]) \otimes \mathcal{F}_n \subseteq \mathcal{B}([0, \infty)) \otimes \mathcal{F}\). For adaptedness, fix \(t\) and consider the insertion map \(\iota_t : \omega \mapsto (t, \omega)\) from \((\Omega, \mathcal{F}_t)\) to \(([0, t] \times \Omega,\ \mathcal{B}([0, t]) \otimes \mathcal{F}_t)\). The preimage of a rectangle \(I \times B\) under \(\iota_t\) is \(B\) when \(t \in I\) and is empty otherwise, and the rectangles generate the product \(\sigma\)-algebra, so \(\iota_t\) is measurable, and \(u(t, \cdot) = u \circ \iota_t\) is \(\mathcal{F}_t\)-measurable as a composition of measurable maps. The same insertion argument in the other variable shows that for each fixed \(\omega\) the path \(s \mapsto u(s, \omega)\) is Borel measurable on every \([0, t]\). We will use both slicing arguments repeatedly below.

The condition would be useless if it were hard to verify. The two classes of integrands this track actually meets satisfy it automatically.

Theorem: Two Criteria for Progressive Measurability

Let \(\{\mathcal{F}_t\}_{t \geq 0}\) be a filtration on \((\Omega, \mathcal{F}, \mathbb{P})\) and let \(u : [0, \infty) \times \Omega \to \mathbb{R}\) be \(\mathcal{F}_t\)-adapted. Then \(u\) is progressively measurable in each of the following cases.

(i) Piecewise constancy.
\(u(s, \omega) = \sum_{j} e_j(\omega)\, \mathbf{1}_{[t_j, t_{j+1})}(s)\) for a finite partition \(0 \leq t_0 \lt t_1 \lt \cdots \lt t_N\), where each \(e_j\) is \(\mathcal{F}_{t_j}\)-measurable. In particular every elementary process is progressively measurable.

(ii) Path continuity.
\(s \mapsto u(s, \omega)\) is continuous for every \(\omega \in \Omega\).

Proof.

Case (i).
Fix \(t \geq 0\). On \([0, t] \times \Omega\) the restriction of \(u\) is the finite sum of the maps \((s, \omega) \mapsto e_j(\omega)\, \mathbf{1}_{[t_j, t_{j+1}) \cap [0, t]}(s)\), and only the terms with \(t_j \leq t\) contribute. For such a term the indicator factor is a Borel function of \(s\) alone and the level \(e_j\) is \(\mathcal{F}_{t_j}\)-measurable with \(\mathcal{F}_{t_j} \subseteq \mathcal{F}_t\). Each factor is therefore measurable with respect to \(\mathcal{B}([0, t]) \otimes \mathcal{F}_t\) after composing with the coordinate projections, and products and finite sums of measurable functions are measurable.

Case (ii).
Fix \(t \gt 0\), the case \(t = 0\) being the adaptedness of \(u(0, \cdot)\) read through the insertion map. For \(n \in \mathbb{N}\) define on \([0, t] \times \Omega\) the right-endpoint discretization

\[ u_n(s, \omega) = u(0, \omega)\, \mathbf{1}_{\{0\}}(s) + \sum_{k=1}^{n} u\Bigl(\tfrac{k t}{n}, \omega\Bigr)\, \mathbf{1}_{\left(\frac{(k-1) t}{n},\, \frac{k t}{n}\right]}(s) . \]

Each summand is a product of a Borel function of \(s\) and an \(\mathcal{F}_{k t / n}\)-measurable function of \(\omega\), and \(\mathcal{F}_{k t / n} \subseteq \mathcal{F}_t\), so each \(u_n\) is \(\mathcal{B}([0, t]) \otimes \mathcal{F}_t\)-measurable by the argument of case (i). For fixed \((s, \omega)\) with \(s \gt 0\), the evaluation point used by \(u_n\) is the smallest grid point \(s_n \geq s\), and \(s \leq s_n \leq s + \tfrac{t}{n}\), so \(s_n \to s\) and \(u_n(s, \omega) = u(s_n, \omega) \to u(s, \omega)\) by continuity of the path. At \(s = 0\) the sequence is constant. A pointwise limit of measurable functions is measurable, so the restriction of \(u\) is \(\mathcal{B}([0, t]) \otimes \mathcal{F}_t\)-measurable.

With the measurability language in place, the pathwise time integral can be built in one theorem. The only property of the filtration the proof consumes, beyond the product bookkeeping already in place, is the convention quoted at the top of the page, that every set of \(\mathbb{P}\)-measure zero belongs to each \(\mathcal{F}_t\), so we state the theorem for any filtration with that property.

Theorem: The Pathwise Time Integral

Let \(\{\mathcal{F}_t\}_{t \geq 0}\) be a filtration on \((\Omega, \mathcal{F}, \mathbb{P})\) containing every set of \(\mathbb{P}\)-measure zero, let \(u : [0, \infty) \times \Omega \to \mathbb{R}\) be progressively measurable with respect to \(\{\mathcal{F}_t\}\), and suppose

\[ \mathbb{P}\Bigl[ \int_0^t |u(s, \omega)|\, ds \lt \infty \quad \forall\, t \geq 0 \Bigr] = 1 , \]

the event being measurable by Step 1 of the proof. Then there exists a stochastic process \(\{A_t\}_{t \geq 0}\) on \((\Omega, \mathcal{F}, \mathbb{P})\) such that:

(i) Pathwise identity.
For almost every \(\omega\), \(A_t(\omega) = \int_0^t u(s, \omega)\, ds\) simultaneously for all \(t \geq 0\).

(ii) Adaptedness.
\(A_t\) is \(\mathcal{F}_t\)-measurable for every \(t \geq 0\).

(iii) Continuity.
\(t \mapsto A_t(\omega)\) is continuous for every \(\omega \in \Omega\).

(iv) Progressive measurability.
\(A\) is itself progressively measurable.

Proof.

Step 1 (Measurability of the absolute integrals).
Fix \(n \in \mathbb{N}\). The restriction of \(|u|\) to \([0, n] \times \Omega\) is \(\mathcal{B}([0, n]) \otimes \mathcal{F}_n\)-measurable and nonnegative. By the slicing argument recorded after the definition, every path \(s \mapsto |u(s, \omega)|\) is Borel measurable, so \(\int_0^n |u(s, \omega)|\, ds\) is defined in \([0, \infty]\) for every \(\omega\). We claim the map

\[ \omega \mapsto \int_0^n |u(s, \omega)|\, ds \]

is \(\mathcal{F}_n\)-measurable. This is the measurability half of Tonelli's theorem applied on the finite product \(([0, n] \times \Omega,\ \mathcal{B}([0, n]) \otimes \mathcal{F}_n,\ \lambda \otimes \mathbb{P})\). The iterated integrals in that statement are meaningful only because the partial integral is measurable in the outer variable, and the proof of this fact runs on the same monotone-class machinery that our track has already taken on faith. We extend that act of trust to cover this one clause, and nothing else new.

Step 2 (The good set is harmless).
Since \(t \mapsto \int_0^t |u(s, \omega)|\, ds\) is nondecreasing, finiteness for all \(t \geq 0\) is equivalent to finiteness at every integer time, so the event in the hypothesis is

\[ \Omega^* = \bigcap_{n \in \mathbb{N}} \Bigl\{ \omega : \int_0^n |u(s, \omega)|\, ds \lt \infty \Bigr\} , \]

an intersection of \(\mathcal{F}_n\)-measurable sets by Step 1, hence measurable, and of probability one by hypothesis. Its complement is a set of measure zero, and the filtration contains every such set by hypothesis. Therefore \(\Omega^* \in \mathcal{F}_t\) for every \(t \geq 0\). Once again the null-set convention converts an almost-sure statement into an adapted one, and it will not be the last time.

Step 3 (Definition and adaptedness).
Write \(u = u^+ - u^-\) with \(u^\pm = \max(\pm u, 0)\). Both parts are progressively measurable, being compositions of \(u\) with continuous functions. Fix \(t \geq 0\). By Step 1 run at level \(t\) in place of \(n\), the maps \(A_t^\pm(\omega) = \int_0^t u^\pm(s, \omega)\, ds\) are \(\mathcal{F}_t\)-measurable with values in \([0, \infty]\). On \(\Omega^*\) both are finite, being dominated by \(\int_0^t |u|\, ds\). Define

\[ A_t(\omega) = \begin{cases} A_t^+(\omega) - A_t^-(\omega) , & \omega \in \Omega^* , \\\\ 0 , & \omega \notin \Omega^* . \end{cases} \]

On the \(\mathcal{F}_t\)-measurable set \(\Omega^*\) this is a difference of two finite \(\mathcal{F}_t\)-measurable functions, and off it a constant, so \(A_t\) is \(\mathcal{F}_t\)-measurable. For \(\omega \in \Omega^*\) it equals \(\int_0^t u(s, \omega)\, ds\) for every \(t\) at once, which proves (i) and (ii).

Step 4 (Continuity of every path).
Off \(\Omega^*\) the path is constant. Fix \(\omega \in \Omega^*\), a time \(t_0 \geq 0\), and an integer \(N \gt t_0 + 1\). For \(t \in [0, N]\),

\[ \bigl| A_t(\omega) - A_{t_0}(\omega) \bigr| \leq \int_0^N \mathbf{1}_{(\min(t, t_0),\, \max(t, t_0)]}(s)\, |u(s, \omega)|\, ds . \]

As \(t \to t_0\), the integrand tends to zero at every \(s \neq t_0\), a set of full Lebesgue measure, and is dominated by \(|u(\cdot, \omega)|\), which is integrable on \([0, N]\) because \(\omega \in \Omega^*\). The dominated convergence theorem applied along an arbitrary sequence \(t_k \to t_0\) sends the bound to zero, which is the sequential criterion for continuity at \(t_0\).

Step 5 (Progressive measurability).
\(A\) is adapted by Step 3 and every path is continuous by Step 4, so \(A\) is progressively measurable by criterion (ii) of the preceding theorem.

The drift term is now a legitimate process, and the enlarged class can be defined.

Definition: Itô Process

Let \(w_t\) be a one-dimensional Brownian motion on \((\Omega, \mathcal{F}, \mathbb{P})\) with Brownian filtration \(\{\mathcal{F}_t\}_{t \geq 0}\). A (one-dimensional) Itô process is a stochastic process \(\{X_t\}_{t \geq 0}\) of the form

\[ X_t = X_0 + \int_0^t u(s, \omega)\, ds + \int_0^t v(s, \omega)\, dw_s , \]

where:

(I1)
\(X_0\) is an \(\mathcal{F}_0\)-measurable random variable;

(I2)
\(u\) is progressively measurable with \(\mathbb{P}\bigl[\int_0^t |u|\, ds \lt \infty \quad \forall\, t \geq 0\bigr] = 1\), and the \(ds\)-term denotes the pathwise time integral of the theorem above;

(I3)
\(v \in \mathcal{V}(0, T)\) for every \(T \gt 0\), and the \(dw\)-term denotes the continuous version of the Itô integral process.

Two remarks make the definition fully precise. First, the \(dw\)-term in (I3) is a single process on \([0, \infty)\). For each \(T\) the continuous version theorem provides a continuous process on \([0, T]\) agreeing with \(\int_0^t v\, dw_s\) at each fixed time, and two such processes for \(T \lt T'\) agree at each fixed \(t \leq T\) almost surely, hence are indistinguishable on \([0, T]\) by continuity. The versions over all \(T \in \mathbb{N}\) therefore patch into one continuous adapted process on \([0, \infty)\) outside a single null set. Second, under the Brownian filtration the condition (I1) is less generous than it looks. The starting value \(w_0\) is zero, so \(\mathcal{F}_0\) consists of null sets and their complements, and \(X_0\) is almost surely a constant. We state (I1) in measurable form anyway, since the definition will later run unchanged over richer filtrations, where a genuinely random initial condition is the point.

Assembling the three terms, every Itô process is adapted and has continuous paths outside a null set. Redefining the \(dw\)-term as zero on its exceptional null set makes every path continuous while the filtration convention preserves adaptedness, and after this harmless surgery the process is progressively measurable. From here on, the \(dw\)-term of every Itô process is taken in this everywhere-continuous realization. The constant-in-time term \(X_0\) contributes trivially, the drift term qualifies by the pathwise time integral theorem, and the \(dw\)-term by criterion (ii) applied to its continuous adapted realization.

Differential Notation

Equations of the displayed shape are abbreviated in differential form,

\[ dX_t = u\, dt + v\, dw_t . \]

The differentials carry no independent meaning. The line is defined to be a name for the integral identity in the definition, nothing more, and every computation with differentials on this page resolves into a statement about integrals. In this notation the opening identity of the page reads

\[ d\bigl(\tfrac{1}{2} w_t^2\bigr) = \tfrac{1}{2}\, dt + w_t\, dw_t , \]

exhibiting \(\tfrac{1}{2} w_t^2\) as an Itô process with drift \(u = \tfrac{1}{2}\) and diffusion coefficient \(v = w_s\).

One more remark records what we deliberately give up. A broader definition of the Itô process asks of \(v\) only the pathwise condition \(\mathbb{P}\bigl[\int_0^t v^2\, ds \lt \infty \quad \forall\, t\bigr] = 1\), in exact analogy with (I2). Constructing \(\int_0^t v\, dw_s\) for such \(v\) requires localization by stopping times, a tool this track has not yet introduced, so we record the wider class as a signpost and work inside \(\mathcal{V}\). No example on this page is lost by the restriction.

The question the opening paragraph raised can now be posed precisely. Is the class of Itô processes closed under smooth maps? For Itô integrals the answer was no, and the failure produced the definition. For Itô processes the answer is yes, and the exact bookkeeping of the closure is the content of the next section. The image \(g(t, X_t)\) is again an Itô process, and its drift picks up a term that classical calculus does not predict.

The Chain Rule Acquires a Correction

Classical calculus computes the image of a smooth motion under a smooth map by the chain rule. If \(x(t)\) is continuously differentiable and \(g(t, x)\) is smooth, then

\[ \frac{d}{dt}\, g(t, x(t)) = \frac{\partial g}{\partial t}(t, x(t)) + \frac{\partial g}{\partial x}(t, x(t))\, x'(t) , \]

and the same conclusion survives, in integrated form, as long as \(x(t)\) is continuous with bounded variation. Brownian paths refuse both hypotheses. They have infinite total variation on every interval, so no version of the classical argument applies to \(g(t, w_t)\), and the opening identity of the page already told us the conclusion itself is false. The term \(\tfrac{1}{2} t\) in \(\tfrac{1}{2} w_t^2 = \tfrac{1}{2} t + \int_0^t w_s\, dw_s\) is a correction that classical calculus does not predict.

A second-order Taylor expansion locates the source of the correction before any proof. For a small time step, writing \(\Delta X = X_{t + \Delta t} - X_t\),

\[ \begin{align*} \Delta g &= \frac{\partial g}{\partial t}\, \Delta t + \frac{\partial g}{\partial x}\, \Delta X \\\\ &\quad + \tfrac{1}{2} \frac{\partial^2 g}{\partial t^2} (\Delta t)^2 + \frac{\partial^2 g}{\partial t\, \partial x}\, \Delta t\, \Delta X + \tfrac{1}{2} \frac{\partial^2 g}{\partial x^2} (\Delta X)^2 + \cdots \end{align*} \]

For a differentiable motion, \(\Delta X\) is of order \(\Delta t\), every second-order term is of order \((\Delta t)^2\), and summing over a partition of \([0, t]\) kills them all in the limit. For an Itô process the increment is dominated by its noise part \(v\, \Delta w\), and a Brownian increment has size \(\sqrt{\Delta t}\), since \(\mathbb{E}[(\Delta w)^2] = \Delta t\). The square \((\Delta X)^2\) is therefore of order \(\Delta t\), first order rather than second, and its sum over a partition survives the limit. The quadratic variation of Brownian motion says exactly what it survives to. The sums \(\sum (\Delta w)^2\) converge to \(t\) in \(L^2\), so the term \(\tfrac{1}{2} g_{xx} (\Delta X)^2\) contributes \(\tfrac{1}{2} v^2 g_{xx}\, \Delta t\) worth of drift, while the cross term of size \(\Delta t \cdot \sqrt{\Delta t}\) and the pure \((\Delta t)^2\) term still vanish. One second-order term is promoted to first order, and it is the only one.

This bookkeeping is compressed into a multiplication table for differentials,

\[ \begin{align*} dt \cdot dt &= dt \cdot dw_t = dw_t \cdot dt = 0 , \\\\ dw_t \cdot dw_t &= dt , \end{align*} \]

whose one nontrivial entry is the quadratic variation theorem wearing differential clothing. The table is a mnemonic, not a definition. Its precise content is the theorem below, stated in integral form, and every use of the table on this page resolves into that statement.

Theorem: The One-Dimensional Itô Formula

Let \(X_t = X_0 + \int_0^t u\, ds + \int_0^t v\, dw_s\) be an Itô process, and let \(g \in C^2([0, \infty) \times \mathbb{R})\), meaning that \(g\) and all its partial derivatives up to second order exist and are continuous on \([0, \infty) \times \mathbb{R}\). Assume the membership condition that for every \(T \gt 0\),

\[ \Bigl( (s, \omega) \mapsto v(s, \omega)\, \frac{\partial g}{\partial x}(s, X_s(\omega)) \Bigr) \in \mathcal{V}(0, T) . \]

Then \(Y_t = g(t, X_t)\) is again an Itô process, and for every \(t \geq 0\), almost surely,

\[ g(t, X_t) = g(0, X_0) + \int_0^t \Bigl( \frac{\partial g}{\partial s} + u\, \frac{\partial g}{\partial x} + \tfrac{1}{2}\, v^2\, \frac{\partial^2 g}{\partial x^2} \Bigr)(s, X_s)\, ds + \int_0^t v\, \frac{\partial g}{\partial x}(s, X_s)\, dw_s , \]

where \(\frac{\partial g}{\partial s}\) denotes the derivative of \(g\) in its first argument, each partial derivative is evaluated at \((s, X_s)\), and \(u, v\) at \((s, \omega)\).

In differential shorthand the conclusion reads

\[ dY_t = \frac{\partial g}{\partial t}\, dt + \frac{\partial g}{\partial x}\, dX_t + \tfrac{1}{2}\, \frac{\partial^2 g}{\partial x^2}\, (dX_t)^2 , \]

where \((dX_t)^2 = (u\, dt + v\, dw_t)^2 = v^2\, dt\) after expanding by the table. The formula is the classical chain rule plus exactly one correction, the term \(\tfrac{1}{2} v^2 g_{xx}\, dt\), and the correction sits in the drift. As a first sanity check, take \(g(t, x) = \tfrac{1}{2} x^2\) and \(X_t = w_t\), so \(u = 0\) and \(v = 1\). The table gives

\[ \begin{align*} d\bigl(\tfrac{1}{2} w_t^2\bigr) &= w_t\, dw_t + \tfrac{1}{2}\, (dw_t)^2 \\\\ &= w_t\, dw_t + \tfrac{1}{2}\, dt , \end{align*} \]

which is the opening identity of the page. The formula we are about to prove reproduces the one integral this track has computed by hand.

The Statement Is Well Posed

Before proving the identity we check that every object in it qualifies for the roles the definition of an Itô process assigns. Write \(\Phi(s, \omega) = \phi(s, X_s(\omega))\) generically for any of the coefficient processes appearing above, where \(\phi\) is one of the continuous functions \(\frac{\partial g}{\partial s}\), \(\frac{\partial g}{\partial x}\), or \(\frac{\partial^2 g}{\partial x^2}\).

First, measurability. The map \((s, \omega) \mapsto (s, X_s(\omega))\), restricted to \([0, t] \times \Omega\), is measurable from \(\mathcal{B}([0, t]) \otimes \mathcal{F}_t\) to \(\mathcal{B}([0, t]) \otimes \mathcal{B}(\mathbb{R})\), the first coordinate being a projection and the second the progressively measurable process \(X\). Composing with the continuous, hence Borel, function \(\phi\) shows each \(\Phi\) is progressively measurable, and products and sums of progressively measurable processes remain so. Every integrand in the theorem is therefore progressively measurable.

Second, the drift term satisfies condition (I2) automatically. Fix \(\omega\) in the full-measure set on which \(X\) has continuous paths and the pathwise conditions of (I2) and (I3) hold. On \([0, t]\) the path \(s \mapsto X_s(\omega)\) is continuous, so it stays in a compact interval, and each \(\phi(s, X_s(\omega))\) is bounded on \([0, t]\) by continuity of \(\phi\), say by \(C_t(\omega)\). Then

\[ \begin{align*} \int_0^t \Bigl| \frac{\partial g}{\partial s} + u\, \frac{\partial g}{\partial x} + \tfrac{1}{2} v^2 \frac{\partial^2 g}{\partial x^2} \Bigr|(s, X_s)\, ds &\leq C_t(\omega) \Bigl( t + \int_0^t |u|\, ds + \tfrac{1}{2} \int_0^t v^2\, ds \Bigr) \\\\ &\lt \infty , \end{align*} \]

where \(\int_0^t |u|\, ds\) is finite by (I2) and \(\int_0^t v^2\, ds\) is finite almost surely because its expectation is finite by (V3). So the new drift is a legitimate input to the pathwise time integral theorem.

Third, the diffusion term. The same compactness argument shows \(\int_0^t (v\, \frac{\partial g}{\partial x}(s, X_s))^2\, ds \lt \infty\) almost surely, but membership in \(\mathcal{V}\) demands finiteness of the expectation of this integral, and continuity of paths bounds nothing uniformly over \(\Omega\). This is why the theorem carries the membership condition as a hypothesis. It holds whenever \(\frac{\partial g}{\partial x}\) is bounded, since then the integrand is dominated by a constant multiple of \(v \in \mathcal{V}\), and it is checked by direct inspection in every concrete case this track meets. For the sanity example the check is already on record, since \(v\, \frac{\partial g}{\partial x}(s, X_s) = w_s\) is the integrand of the opening identity. Dropping the condition entirely is possible, at the price of the same localization machinery that the wider integrand class of Section 1 signposted, and it waits with that class.

One more degree of freedom costs nothing. The theorem is stated for \(g\) twice continuously differentiable on all of \([0, \infty) \times \mathbb{R}\), but the proof only ever evaluates \(g\) and its derivatives along the path \((s, X_s)\). If every path of \(X\) stays in an open set \(U \subseteq \mathbb{R}\), the same proof runs verbatim for \(g \in C^2([0, \infty) \times U)\). We record this without separate proof and will collect it when a process confined to a region first appears.

The heuristic said one Taylor term is promoted and all others die. The proof's job is to make each of those deaths, and the one survival, an honest limit theorem. That is the next section.

Proof of the Itô Formula

The proof follows the heuristic with the hand-waving removed. Fix \(t \gt 0\). We telescope \(g(t, X_t) - g(0, X_0)\) over a partition of \([0, t]\), expand each increment by Taylor's theorem with an exact remainder, and sort the resulting sums by order. Each sum then meets one of three fates as the mesh shrinks. The first-order sums converge to the two integrals of the formula. Every sum carrying a factor \((\Delta t)^2\) dies by a pathwise estimate, and every sum carrying \(\Delta t\, \Delta w\) dies by an \(L^2\) estimate. The \((\Delta X)^2\) sum splits into dying pieces and one survivor, and the survivor converges to the correction term by the quadratic variation mechanism, upgraded from the Brownian statement to weighted sums. All modes of convergence are stronger than convergence in probability, which is the common currency in which the pieces are finally added.

Two of the deaths deserve to be stated on their own. The multiplication table's vanishing entries, \(dt \cdot dw_t = 0\) and the centered half of \(dw_t \cdot dw_t = dt\), have an honest form as sums of increments against adapted bounded weights, dying in \(L^2\) as the mesh shrinks. The mechanism recurs on this page and again beyond it, so we isolate it as a lemma with explicit constants.

Lemma: Weighted Increment Limits

Let \(\{\mathcal{F}_t\}_{t \geq 0}\) be a filtration on \((\Omega, \mathcal{F}, \mathbb{P})\) and let \(w\) be a standard Brownian motion, adapted to \(\{\mathcal{F}_t\}\), whose increments \(w_b - w_a\) with \(0 \leq a \lt b\) are independent of \(\mathcal{F}_a\). Let \(t \gt 0\), let \(0 = t_0 \lt t_1 \lt \cdots \lt t_N = t\) be a partition with mesh \(|\pi| = \max_j \Delta t_j\), where \(\Delta t_j = t_{j+1} - t_j\), and write \(\Delta w_j = w_{t_{j+1}} - w_{t_j}\). For each \(j\) let \(b_j\) be an \(\mathcal{F}_{t_j}\)-measurable random variable with \(|b_j| \leq K\), for a constant \(K\) independent of \(j\). Then:

(i)
\[ \mathbb{E}\Bigl[ \Bigl( \sum_j b_j\, \Delta t_j\, \Delta w_j \Bigr)^{2} \Bigr] \leq K^2\, t\, |\pi|^2 . \]

(ii)
\[ \mathbb{E}\Bigl[ \Bigl( \sum_j b_j \bigl( (\Delta w_j)^2 - \Delta t_j \bigr) \Bigr)^{2} \Bigr] \leq 2 K^2\, t\, |\pi| . \]

If moreover \(\tilde{w}\) is a second standard Brownian motion, independent of \(w\), such that the paired increments \((w_b - w_a,\ \tilde{w}_b - \tilde{w}_a)\) are independent of \(\mathcal{F}_a\), then, writing \(\Delta \tilde{w}_j = \tilde{w}_{t_{j+1}} - \tilde{w}_{t_j}\):

(iii)
\[ \mathbb{E}\Bigl[ \Bigl( \sum_j b_j\, \Delta w_j\, \Delta \tilde{w}_j \Bigr)^{2} \Bigr] \leq K^2\, t\, |\pi| . \]

In particular, along any sequence of partitions of \([0, t]\) with mesh tending to zero, all three sums tend to zero in \(L^2(\mathbb{P})\).

On this page the lemma is consumed with \(\{\mathcal{F}_t\}\) the Brownian filtration of \(w\) itself, where the increment hypothesis is exactly the independence of increments from the past, and part (iii) sits idle, there being no second motion in sight. It is proved now because several independent motions arrive two pages ahead, the proof costs a few lines here, and it would cost a page of duplicated machinery there.

Proof.

Step 1 (The increment law).
Throughout, the increment \(w_b - w_a\) with \(0 \leq a \lt b\) is a centered Gaussian variable with variance \(b - a\). Gaussianity and the mean are part of the defining properties, an increment being a linear combination of the Gaussian process with \(\mathbb{E}[w_b - w_a] = \mathbb{E}[w_b] - \mathbb{E}[w_a] = 0\), and the variance is read off the defining covariance, \[ \begin{align*} \operatorname{Var}(w_b - w_a) &= \operatorname{Cov}(w_b, w_b) - 2 \operatorname{Cov}(w_a, w_b) + \operatorname{Cov}(w_a, w_a) \\\\ &= b - 2a + a = b - a . \end{align*} \] The same applies verbatim to any standard Brownian motion, in particular to \(\tilde{w}\) when part (iii) is in play.

Step 2 (The factorization principle).
The estimates repeatedly compute expectations of products in which one factor is measurable with respect to the past and the other is a function of a future increment. We isolate the mechanism once. Let \(0 \leq a \lt b\), let \(Y\) be an integrable \(\mathcal{F}_a\)-measurable random variable, and let \(h : \mathbb{R} \to \mathbb{R}\) be Borel with \(h(w_b - w_a)\) integrable. By hypothesis the increment is independent of \(\mathcal{F}_a\), meaning that the \(\sigma\)-algebras \(\sigma(w_b - w_a)\) and \(\mathcal{F}_a\) are independent. Since \(Y\) is measurable for the second and \(h(w_b - w_a)\) for the first, the pair is independent, and the product rule for expectations, which also delivers the integrability of the product, gives

\[ \mathbb{E}\bigl[ Y\, h(w_b - w_a) \bigr] = \mathbb{E}[Y]\ \mathbb{E}\bigl[ h(w_b - w_a) \bigr] . \]

In every application below \(h\) is a polynomial and \(Y\) is dominated by a constant multiple of a polynomial in earlier increments, so all integrability demands are met by the finiteness of Gaussian moments. In particular, taking \(h(z) = z\) or \(h(z) = z^2 - (b - a)\), whose expectations under the increment distribution are zero by the mean and variance of Step 1, the product expectation vanishes. For part (iii) the same argument runs with \(h\) a polynomial in the paired increment \((w_b - w_a,\ \tilde{w}_b - \tilde{w}_a)\), the pair being independent of \(\mathcal{F}_a\) by hypothesis.

Step 3 (Second and fourth moments).
For \(Z\) centered Gaussian with variance \(\sigma^2 \gt 0\) and density \(\varphi\), the identity \(z\, \varphi(z) = -\sigma^2\, \varphi'(z)\) and integration by parts give

\[ \begin{align*} \mathbb{E}[Z^2] &= \int_{-\infty}^{\infty} z \cdot z\, \varphi(z)\, dz = -\sigma^2 \Bigl[ z\, \varphi(z) \Bigr]_{-\infty}^{\infty} + \sigma^2 \int_{-\infty}^{\infty} \varphi(z)\, dz = \sigma^2 , \\\\ \mathbb{E}[Z^4] &= \int_{-\infty}^{\infty} z^3 \cdot z\, \varphi(z)\, dz = -\sigma^2 \Bigl[ z^3 \varphi(z) \Bigr]_{-\infty}^{\infty} + 3 \sigma^2 \int_{-\infty}^{\infty} z^2\, \varphi(z)\, dz = 3 \sigma^4 , \end{align*} \]

the boundary terms vanishing because the Gaussian density decays faster than any polynomial grows, and the total mass being one by the Gaussian integral after the change of variables that normalizes the density. Consequently

\[ \begin{align*} \mathbb{E}\Bigl[ \bigl( (\Delta w_j)^2 - \Delta t_j \bigr)^{2} \Bigr] &= 3 (\Delta t_j)^2 - 2 (\Delta t_j)^2 + (\Delta t_j)^2 \\\\ &= 2 (\Delta t_j)^2 . \end{align*} \]

Step 4 (The two estimates).
For (i), expand the square and apply the factorization principle to every term. The off-diagonal terms with \(i \lt j\) vanish, because \(Y = b_i b_j \Delta w_i\) is \(\mathcal{F}_{t_j}\)-measurable and bounded in absolute value by \(K^2 |\Delta w_i|\), hence integrable, while \(h(z) = z\) has \(\mathbb{E}[h(\Delta w_j)] = 0\). The diagonal factorizes with \(h(z) = z^2\), so

\[ \begin{align*} \mathbb{E}\Bigl[ \Bigl( \sum_j b_j\, \Delta t_j\, \Delta w_j \Bigr)^{2} \Bigr] &= \sum_j (\Delta t_j)^2\, \mathbb{E}\bigl[ b_j^2 \bigr]\, \mathbb{E}\bigl[ (\Delta w_j)^2 \bigr] \\\\ &\leq K^2 \sum_j (\Delta t_j)^3 \\\\ &\leq K^2\, t\, |\pi|^2 , \end{align*} \]

using \(\mathbb{E}[(\Delta w_j)^2] = \Delta t_j\) from Step 1. For (ii), the off-diagonal terms with \(i \lt j\) vanish by the factorization principle, with \(Y = b_i b_j ((\Delta w_i)^2 - \Delta t_i)\), which is \(\mathcal{F}_{t_j}\)-measurable and integrable, being dominated by \(K^2 ((\Delta w_i)^2 + \Delta t_i)\), and \(h(z) = z^2 - \Delta t_j\), whose expectation is zero. The diagonal factorizes with \(Y = b_j^2\) and \(h(z) = (z^2 - \Delta t_j)^2\), so by Step 3,

\[ \begin{align*} \mathbb{E}\Bigl[ \Bigl( \sum_j b_j \bigl( (\Delta w_j)^2 - \Delta t_j \bigr) \Bigr)^{2} \Bigr] &= \sum_j \mathbb{E}\bigl[ b_j^2 \bigr] \cdot 2 (\Delta t_j)^2 \\\\ &\leq 2 K^2\, t\, |\pi| . \end{align*} \]

Step 5 (The cross estimate).
For (iii), expand the square. The off-diagonal terms with \(i \lt j\) vanish by the paired factorization in two moves. First, with \(Y' = b_i b_j\, \Delta w_i\, \Delta \tilde{w}_i\), which is \(\mathcal{F}_{t_j}\)-measurable and integrable, and \(h\) the paired polynomial \(h(z, \tilde{z}) = z \tilde{z}\), the term factors into \(\mathbb{E}[Y']\, \mathbb{E}[\Delta w_j\, \Delta \tilde{w}_j]\). Second, the two motions are independent of each other, so \(\mathbb{E}[\Delta w_j\, \Delta \tilde{w}_j] = \mathbb{E}[\Delta w_j]\, \mathbb{E}[\Delta \tilde{w}_j] = 0\) by the product rule and Step 1. The diagonal factorizes the same way twice,

\[ \begin{align*} \mathbb{E}\bigl[ b_j^2\, (\Delta w_j)^2 (\Delta \tilde{w}_j)^2 \bigr] &= \mathbb{E}\bigl[ b_j^2 \bigr]\, \mathbb{E}\bigl[ (\Delta w_j)^2 \bigr]\, \mathbb{E}\bigl[ (\Delta \tilde{w}_j)^2 \bigr] \\\\ &= \mathbb{E}\bigl[ b_j^2 \bigr]\, (\Delta t_j)^2 \\\\ &\leq K^2 (\Delta t_j)^2 , \end{align*} \]

summing to at most \(K^2\, t\, |\pi|\).

We prove the identity in full for a core case, bounded derivatives and bounded elementary integrands, and we are explicit below about the two approximation passages that carry the core case to the general statement. Everything else is proved.

Proof of the Itô formula.

Within this proof we abbreviate the partial derivatives of \(g\) as \(g_t, g_x, g_{tt}, g_{tx}, g_{xx}\), each evaluated at the point indicated.

Step 1 (Reduction to a core case).
We prove the identity under two additional assumptions.

(R1)
\(g\) and all its partial derivatives up to second order are bounded on \([0, \infty) \times \mathbb{R}\), by a common constant \(M\).

(R2)
On \([0, t]\), the coefficients \(u\) and \(v\) are elementary processes over a common partition \(0 = r_0 \lt r_1 \lt \cdots \lt r_m = t\), with all levels bounded by a common constant \(C\).

Two approximation passages carry the core case to the theorem as stated, and we record precisely what each one asserts. First, a function \(g \in C^2([0, \infty) \times \mathbb{R})\) admits approximants \(g_n\) satisfying (R1) such that \(g_n\) and its partial derivatives up to second order converge to those of \(g\) uniformly on compact sets. The construction is by smooth truncation and mollification and we take it on faith, this being a statement of classical analysis with no probabilistic content. Second, general coefficients satisfying (I2), (I3), and the membership condition admit approximating sequences as in (R2) for which every term of the identity converges to the corresponding term for \((u, v)\). For the \(dw\)-terms this is the three-step approximation scheme that built the Itô integral, and the uniform pathwise control needed for the \(ds\)-terms is exactly what the maximal bound for the Itô integral was forged for. We nevertheless take the passage as a whole on faith here, since executing it doubles the length of the proof, and we flag it as the one debt of this page. In the core case the membership condition of the theorem holds automatically, because \(|v\, g_x(s, X_s)| \leq C M\) makes the diffusion integrand a bounded progressively measurable process, hence a member of every \(\mathcal{V}(0, T)\).

Step 2 (Exact Taylor decomposition).
Let \(\pi : 0 = t_0 \lt t_1 \lt \cdots \lt t_N = t\) be a partition refining \(\{r_0, \ldots, r_m\}\), with mesh \(|\pi| = \max_j \Delta t_j\), where \(\Delta t_j = t_{j+1} - t_j\). Write \(X_j = X_{t_j}\), \(\Delta X_j = X_{t_{j+1}} - X_j\), and \(\Delta w_j = w_{t_{j+1}} - w_{t_j}\). Because \(\pi\) refines the elementary partition, \(u\) and \(v\) are constant on each \([t_j, t_{j+1})\), with values \(u_j = u(t_j, \cdot)\) and \(v_j = v(t_j, \cdot)\) that are \(\mathcal{F}_{t_j}\)-measurable and bounded by \(C\). The drift piece of the increment is the pathwise integral of a constant, and the noise piece is the elementary integral of a constant level, which additivity over intervals identifies with the increment of the integral process. Outside a single null set, simultaneously for every partition drawn from the countable family fixed below,

\[ \Delta X_j = u_j\, \Delta t_j + v_j\, \Delta w_j . \]

Now telescope and expand. By Taylor's theorem with integral remainder, applied at order \(k = 1\) in the two variables \((s, x)\) at the base point \((t_j, X_j)\) with increment \((\Delta t_j, \Delta X_j)\),

\[ \begin{align*} g(t_{j+1}, X_{j+1}) - g(t_j, X_j) &= g_t(t_j, X_j)\, \Delta t_j + g_x(t_j, X_j)\, \Delta X_j \\\\ &\quad + \tfrac{1}{2}\, \theta_j^{tt} (\Delta t_j)^2 + \theta_j^{tx}\, \Delta t_j\, \Delta X_j + \tfrac{1}{2}\, \theta_j^{xx} (\Delta X_j)^2 , \end{align*} \]

where, for each second-order index \(I\), the coefficient is the weighted average of the corresponding derivative along the segment,

\[ \theta_j^{I} = 2 \int_0^1 (1 - \tau)\, \partial_I g\bigl( (t_j, X_j) + \tau\, (\Delta t_j, \Delta X_j) \bigr)\, d\tau . \]

The cited theorem is stated on open sets, and the only boundary contact happens at \(t_0 = 0\). There the one-variable function \(\tau \mapsto g((t_j, X_j) + \tau (\Delta t_j, \Delta X_j))\) is still twice continuously differentiable on \([0, 1]\), with a one-sided derivative in time at \(\tau = 0\) supplied by the \(C^2\) hypothesis on \([0, \infty) \times \mathbb{R}\), and the one-dimensional integral-remainder expansion at the heart of the cited proof applies unchanged.

Each \(\theta_j^I\) is bounded by \(M\), since \(2 \int_0^1 (1 - \tau)\, d\tau = 1\), and each is a genuine random variable, being the pointwise limit of Riemann sums in \(\tau\) of integrands that are continuous in \(\tau\) and measurable in \(\omega\). More is true. Fix \(\omega\) outside the null set assembled above, so that the increments are exact and the paths of \(X\) and of the pathwise integrals are continuous, and let \(K_\omega = [0, t] \times [-R_\omega, R_\omega]\) with \(R_\omega = \max_{0 \leq s \leq t} |X_s(\omega)|\). Every segment \((t_j, X_j) + \tau (\Delta t_j, \Delta X_j)\) lies in the convex set \(K_\omega\), so writing

\[ \begin{align*} \varepsilon_j^{I} &= \theta_j^{I} - \partial_I g(t_j, X_j) , \\\\ \eta(\pi, \omega) &= \max_{j, I}\, \bigl| \varepsilon_j^{I} \bigr| , \end{align*} \]

the deviation \(\eta(\pi, \omega)\) is at most the modulus of continuity of the second derivatives of \(g\) on \(K_\omega\), evaluated at the largest displacement \(|\pi| + \max_j |\Delta X_j(\omega)|\). The path \(s \mapsto X_s(\omega)\) is uniformly continuous on \([0, t]\), so \(\max_j |\Delta X_j| \to 0\) as \(|\pi| \to 0\), the second derivatives of \(g\) are uniformly continuous on the compact \(K_\omega\), and therefore \(\eta(\pi, \omega) \to 0\) pathwise as \(|\pi| \to 0\), while the crude bound \(\eta(\pi, \omega) \leq 2 M\) holds for every partition.

Summing the expansion over \(j\) and separating each \(\theta_j^I\) into \(\partial_I g(t_j, X_j) + \varepsilon_j^I\) yields the master decomposition

\[ \begin{align*} g(t, X_t) - g(0, X_0) &= \sum_j g_t\, \Delta t_j + \sum_j g_x\, \Delta X_j \\\\ &\quad + \tfrac{1}{2} \sum_j g_{tt}\, (\Delta t_j)^2 + \sum_j g_{tx}\, \Delta t_j\, \Delta X_j + \tfrac{1}{2} \sum_j g_{xx}\, (\Delta X_j)^2 \\\\ &\quad + E(\pi) , \end{align*} \]

where every derivative is evaluated at \((t_j, X_j)\) and the error term collects the \(\varepsilon\)-contributions,

\[ \begin{align*} E(\pi) &= \sum_j \Bigl( \tfrac{1}{2}\, \varepsilon_j^{tt} (\Delta t_j)^2 + \varepsilon_j^{tx}\, \Delta t_j\, \Delta X_j + \tfrac{1}{2}\, \varepsilon_j^{xx} (\Delta X_j)^2 \Bigr) , \\\\ |E(\pi)| &\leq \eta(\pi) \sum_j \bigl( (\Delta t_j)^2 + |\Delta t_j \Delta X_j| + (\Delta X_j)^2 \bigr) . \end{align*} \]

From here on, fix a sequence of partitions \(\pi_n\), each refining \(\{r_0, \ldots, r_m\}\), with \(|\pi_n| \to 0\). The left side of the master decomposition does not depend on \(n\). We show that each sum on the right converges in probability, identify the limits, and conclude that their sum, which is constant along the sequence, equals the limit almost surely. The error term \(E(\pi_n)\) is estimated in Step 4 together with its second-order companions.

Step 3 (The first-order sums).
The time sum converges pathwise. For \(\omega\) in the full-measure set of Step 2, the map \(s \mapsto g_t(s, X_s(\omega))\) is continuous on \([0, t]\), hence uniformly continuous, and a direct estimate against its Riemann, hence Lebesgue, integral gives

\[ \begin{align*} \Bigl| \sum_j g_t(t_j, X_j)\, \Delta t_j - \int_0^t g_t(s, X_s)\, ds \Bigr| &\leq \sum_j \int_{t_j}^{t_{j+1}} \bigl| g_t(t_j, X_j) - g_t(s, X_s) \bigr|\, ds \\\\ &\leq t\, \omega_\pi , \end{align*} \]

where \(\omega_\pi\) is the modulus of continuity of \(s \mapsto g_t(s, X_s(\omega))\) evaluated at \(|\pi|\), and \(\omega_\pi \to 0\) as \(|\pi| \to 0\). So \(\sum_j g_t\, \Delta t_j \to \int_0^t g_t(s, X_s)\, ds\) almost surely.

The space sum splits along \(\Delta X_j = u_j \Delta t_j + v_j \Delta w_j\) into

\[ \sum_j g_x(t_j, X_j)\, \Delta X_j = \sum_j g_x(t_j, X_j)\, u_j\, \Delta t_j + \sum_j g_x(t_j, X_j)\, v_j\, \Delta w_j . \]

The drift half converges pathwise to \(\int_0^t g_x(s, X_s)\, u(s)\, ds\) by the estimate just used, applied on each interval \([r_i, r_{i+1})\) of the common partition separately. On such an interval \(u\) is a constant random level, the integrand \(s \mapsto g_x(s, X_s)\, u(s)\) is continuous there, and \(\pi_n\) refines the \(r_i\), so the left-endpoint sums over each piece converge to the integral over that piece, and there are finitely many pieces.

The noise half converges in \(L^2(\mathbb{P})\) to the stochastic integral of the theorem. Define elementary processes over \(\pi_n\) by

\[ \Phi_n(s, \omega) = \sum_j g_x(t_j, X_j)\, v_j\, \mathbf{1}_{[t_j, t_{j+1})}(s) , \]

whose levels are \(\mathcal{F}_{t_j}\)-measurable and bounded by \(C M\), so \(\Phi_n \in \mathcal{V}(0, t)\), and whose elementary Itô integral is precisely the noise half. Set \(\Phi(s, \omega) = g_x(s, X_s)\, v(s, \omega)\), the integrand of the theorem. At every point \((s, \omega)\) with \(s \in [0, t)\) and \(\omega\) in the full-measure set, refinement makes \(v(t_j(s)) = v(s)\) exactly, where \(t_j(s)\) is the left endpoint of the \(\pi_n\)-interval containing \(s\), while \(t_j(s) \to s\) and continuity of \(r \mapsto g_x(r, X_r)\) give \(g_x(t_j(s), X_{t_j(s)}) \to g_x(s, X_s)\). So \(\Phi_n \to \Phi\) pointwise \(\lambda \otimes \mathbb{P}\)-almost everywhere on \([0, t] \times \Omega\), with \(|\Phi_n - \Phi| \leq 2 C M\) throughout. The constant is integrable for the finite measure \(\lambda \otimes \mathbb{P}\), so the dominated convergence theorem gives \(\mathbb{E}\bigl[ \int_0^t (\Phi_n - \Phi)^2\, ds \bigr] \to 0\), and L² continuity of the Itô integral concludes

\[ \begin{align*} \sum_j g_x(t_j, X_j)\, v_j\, \Delta w_j &= \int_0^t \Phi_n\, dw_s \\\\ &\ \xrightarrow{\ L^2\ }\ \int_0^t g_x(s, X_s)\, v(s, \omega)\, dw_s . \end{align*} \]

Step 4 (The second-order sums die, except one).
The sums carrying a factor \((\Delta t_j)^2\) die pathwise, by deterministic bounds. With every derivative bounded by \(M\) and every level by \(C\),

\[ \begin{align*} \Bigl| \tfrac{1}{2} \sum_j g_{tt}\, (\Delta t_j)^2 \Bigr| &\leq \tfrac{1}{2} M\, t\, |\pi| , \\\\ \Bigl| \sum_j g_{tx}\, u_j\, (\Delta t_j)^2 \Bigr| &\leq M C\, t\, |\pi| , \\\\ \Bigl| \tfrac{1}{2} \sum_j g_{xx}\, u_j^2\, (\Delta t_j)^2 \Bigr| &\leq \tfrac{1}{2} M C^2\, t\, |\pi| , \end{align*} \]

each right side tending to zero with the mesh. Here the middle sum is the drift half of the cross sum \(\sum_j g_{tx} \Delta t_j \Delta X_j\) and the third is the pure drift piece of \(\tfrac{1}{2} \sum_j g_{xx} (\Delta X_j)^2\), after inserting \(\Delta X_j = u_j \Delta t_j + v_j \Delta w_j\) and expanding.

The mixed pieces share one shape, a sum \(\sum_j b_j\, \Delta t_j\, \Delta w_j\) with \(b_j\) an \(\mathcal{F}_{t_j}\)-measurable variable bounded by a constant, which is exactly the shape part (i) of the weighted increment lemma kills. Applied with \(b_j = g_{tx}(t_j, X_j)\, v_j\) and \(K = M C\), it kills the noise half of the cross sum, and applied with \(b_j = g_{xx}(t_j, X_j)\, u_j v_j\) and \(K = M C^2\), it kills the mixed piece of the squared-increment sum, both in \(L^2(\mathbb{P})\).

The Taylor error dies in \(L^1(\mathbb{P})\). From \(2 |\Delta t_j\, \Delta X_j| \leq (\Delta t_j)^2 + (\Delta X_j)^2\) and \((\Delta X_j)^2 \leq 2 u_j^2 (\Delta t_j)^2 + 2 v_j^2 (\Delta w_j)^2\),

\[ \begin{align*} |E(\pi)| &\leq \tfrac{3}{2}\, \eta(\pi) \Bigl( \sum_j (\Delta t_j)^2 + \sum_j (\Delta X_j)^2 \Bigr) \\\\ &\leq \tfrac{3}{2}\, \eta(\pi) \Bigl( (1 + 2 C^2)\, t\, |\pi| + 2 C^2\, Q_\pi \Bigr) , \end{align*} \]

where \(Q_\pi = \sum_j (\Delta w_j)^2\). Take expectations along the sequence \(\pi_n\). The first contribution is at most a constant times \(|\pi_n|\, \mathbb{E}[\eta(\pi_n)]\) and tends to zero, since \(\eta \leq 2 M\). For the second, the Cauchy-Schwarz inequality gives \(\mathbb{E}[\eta(\pi_n)\, Q_{\pi_n}] \leq \mathbb{E}[\eta(\pi_n)^2]^{1/2}\, \mathbb{E}[Q_{\pi_n}^2]^{1/2}\). The first factor tends to zero by the dominated convergence theorem, because \(\eta(\pi_n) \to 0\) almost surely with the constant bound \(2 M\). The second factor is bounded along the sequence, because quadratic variation gives \(Q_{\pi_n} \to t\) in \(L^2(\mathbb{P})\), so the \(L^2\) norms \(\mathbb{E}[Q_{\pi_n}^2]^{1/2}\) converge to \(t\) and in particular stay bounded. Therefore \(\mathbb{E}\bigl[ |E(\pi_n)| \bigr] \to 0\).

Step 5 (The weighted quadratic variation limit).
One sum survives. Set

\[ \begin{align*} a_j &= g_{xx}(t_j, X_j)\, v_j^2 , \\\\ a(s, \omega) &= g_{xx}(s, X_s)\, v(s, \omega)^2 , \\\\ |a_j| &\leq K = M C^2 , \end{align*} \]

so that the survivor is \(\tfrac{1}{2} \sum_j a_j (\Delta w_j)^2\). We claim \(\sum_j a_j (\Delta w_j)^2 \to \int_0^t a(s, \omega)\, ds\) in probability. This is the multiplication rule \(dw_t \cdot dw_t = dt\) in its honest form, quadratic variation with adapted weights. Split

\[ \sum_j a_j (\Delta w_j)^2 = \sum_j a_j \Delta t_j + \sum_j a_j \bigl( (\Delta w_j)^2 - \Delta t_j \bigr) . \]

The first sum converges pathwise to \(\int_0^t a(s, \omega)\, ds\). On each interval \([r_i, r_{i+1})\) of the common partition, \(v\) is a constant level and \(s \mapsto g_{xx}(s, X_s)\) is continuous, so the left-endpoint sums over each piece converge to the integral over that piece by the modulus estimate of Step 3, and the pieces are finitely many.

The second sum is exactly the shape part (ii) of the weighted increment lemma kills, its weights \(a_j\) being \(\mathcal{F}_{t_j}\)-measurable and bounded by \(K\), so it dies in \(L^2(\mathbb{P})\) with the explicit bound \(2 K^2\, t\, |\pi|\).

The survivor therefore converges in probability,

\[ \tfrac{1}{2} \sum_j a_j (\Delta w_j)^2 \longrightarrow \tfrac{1}{2} \int_0^t g_{xx}(s, X_s)\, v(s, \omega)^2\, ds . \]

Step 6 (Assembly).
Along the sequence \(\pi_n\), the master decomposition, with the two increment splittings of Steps 3 and 4 inserted, expresses the fixed random variable \(g(t, X_t) - g(0, X_0)\) as a sum of ten terms, and Steps 3 through 5 assign each one a limit. Five converge almost surely, two to the integrals \(\int_0^t g_t\, ds\) and \(\int_0^t g_x\, u\, ds\) and three to zero, being the sums carrying \((\Delta t)^2\). Two die in \(L^2\), being the sums carrying \(\Delta t\, \Delta w\). One converges in \(L^2\) to the stochastic integral, one converges in probability to \(\tfrac{1}{2} \int_0^t g_{xx}\, v^2\, ds\), and the Taylor error dies in \(L^1\). Almost sure convergence implies convergence in probability, and so do the two integral modes by the elementary bound obtained from integrating the pointwise inequality \(\epsilon^2\, \mathbf{1}_{\{|Z| \geq \epsilon\}} \leq Z^2\), namely \(\mathbb{P}[|Z| \geq \epsilon] \leq \epsilon^{-2}\, \mathbb{E}[Z^2]\), together with its first-power analogue. Sums respect the mode, since for any two sequences \(\mathbb{P}[|A_n + B_n - A - B| \geq \epsilon] \leq \mathbb{P}[|A_n - A| \geq \epsilon / 2] + \mathbb{P}[|B_n - B| \geq \epsilon / 2]\). The right side of the master decomposition therefore converges in probability to

\[ \int_0^t \Bigl( g_t + u\, g_x + \tfrac{1}{2}\, v^2\, g_{xx} \Bigr)(s, X_s)\, ds + \int_0^t v\, g_x(s, X_s)\, dw_s , \]

while the left side is the same random variable for every \(n\). A constant sequence converging in probability identifies its limit, since \(\mathbb{P}[|c - L| \geq \epsilon] = \lim_n \mathbb{P}[|c_n - L| \geq \epsilon] = 0\) for every \(\epsilon \gt 0\) when \(c_n = c\). This proves the identity at the fixed time \(t \gt 0\), almost surely, and at \(t = 0\) both sides are \(g(0, X_0)\).

Finally, the identity upgrades from each fixed time to all times at once. Both sides are continuous in \(t\) outside one null set, the left as a continuous function of the continuous \((t, X_t)\), the right as an Itô process in its everywhere-continuous realization, legitimately so by the well-posedness checks of the previous section. Two continuous processes agreeing at each fixed time agree at all rational times off one null set, hence everywhere by continuity, which is the uniqueness of versions with continuous paths. So \(Y_t = g(t, X_t)\) is the Itô process with initial value \(g(0, X_0)\), drift \(g_t + u\, g_x + \tfrac{1}{2} v^2 g_{xx}\), and diffusion coefficient \(v\, g_x\), and the proof of the core case is complete.

The theorem earns its keep the moment it stops being proved. The integral of Brownian motion against itself consumed two pages of bare-hands analysis, and it is the only Itô integral this track has ever evaluated from the definition. On this page it became a one-line remark, the case \(g(x) = \tfrac{1}{2} x^2\) read off the multiplication table. That is the trade the formula offers in general. Evaluation from the definition is replaced by calculus, the correction term doing silently what the quadratic variation argument did by hand. Practicing that calculus has its own page. There the formula evaluates integrals that no bare-hands argument would reach, produces the stochastic version of integration by parts, and pays out the correction \(\tfrac{1}{2} \sigma' \sigma\) that the previous page attached to the Itô and Stratonovich readings of a stochastic differential, settling the last account the construction story left open.