Multi-Dimensional Itô Formula

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. In that case \(g\) has all partial derivatives up to second order bounded by \(M\), and every \(u_i\), \(v_{ij}\) is 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. We do not multiply debts by restating them. In the core case the membership condition holds automatically on \([0, t]\), 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 two groups of terms and nothing else. The first-order terms are \(\partial_t g\, \Delta t_j + \sum_i \partial_i g\, \Delta X_j^{(i)}\). The second-order terms carry weighted-average coefficients \(\theta_j\) over every pair drawn from \((\Delta t_j, \Delta X_j^{(1)}, \ldots, \Delta X_j^{(m)})\).

The boundary patch at \(t_0 = 0\) and the separation \(\theta_j = \partial g(t_j, \mathbf{X}_j) + \varepsilon_j\) are word for word the one-dimensional Step 2. So is the pathwise control \(\eta(\pi, \omega) \to 0\) of the deviations, by uniform continuity of the second derivatives on the compact \([0, t] \times \{ |\mathbf{x}| \leq R_\omega \}\), where \(R_\omega = \max_{0 \leq s \leq t} |\mathbf{X}_s(\omega)|\). The only changes are that the segment argument runs in \(\mathbb{R}^{1 + m}\) and that 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 fixed random variable \(g(t, \mathbf{X}_t) - g(0, \mathbf{X}_0)\) equals the first-order sums, plus the second-order sums with left-endpoint coefficients, plus 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, and the left side is fixed along the sequence. The identity at fixed \(t\), its extension to all \(t\) by continuity of both sides, and the fact that the image is an Itô process then follow exactly as in the one-dimensional Step 6. The well-posedness of the integrands transfers by the same composition arguments, with \(\mathbf{X}\) progressively measurable.

Proof.

Apply the general Itô formula to the two-dimensional process \(\mathbf{X} = (X, Y)\) and the time-independent map \(g(x_1, x_2) = x_1 x_2\). The map 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 displayed formula of the theorem.

Proof.

The accumulated square.
The event \(G\) is measurable, since \(h\) is product measurable by (V1) and the inner integral is therefore a random variable by Tonelli's theorem. We apply the pathwise time integral theorem to \(c = h^2\, \mathbf{1}_{[0, t]}\). The integrand \(c\) is progressively measurable, and \(\int_0^a |c|\, ds \leq \int_0^t h^2\, ds\) for every \(a\), the right side being finite almost surely by (V3), while \(\{\mathcal{F}_s^{(n)}\}\) contains every null set. The theorem then gives a process \(A\) with every path continuous and each \(A_a\) measurable with respect to \(\mathcal{F}_a^{(n)}\), and a null set \(N\) off which \(A_a = \int_0^a h^2\, ds\) simultaneously for all \(a \in [0, t]\). Off \(N\) each path of \(A\) is nondecreasing on \([0, t]\) and starts at zero, so the set of \(a \in [0, t]\) with \(A_a = 0\) is a closed interval \([0, \sigma(\omega)]\).

Switching off the approximants.
By the construction behind the definition of the Itô integral, carried out over \(\{\mathcal{F}_s^{(n)}\}\) as the multi-dimensional integral prescribes, there are bounded elementary processes \(\phi_\ell\) with \(\mathbb{E}\bigl[ \int_0^t (h - \phi_\ell)^2\, ds \bigr] \to 0\). Adding partition points does not change an elementary process, so we may assume that the partition \(\pi_\ell\) of \(\phi_\ell\) has mesh \(|\pi_\ell| \to 0\). Write \(\phi_\ell = \sum_j e_j\, \mathbf{1}_{[t_j, t_{j+1})}\) and set

\[ \chi_\ell = \sum_j e_j\, \mathbf{1}_{\{ A_{t_j} \gt 0 \}}\, \mathbf{1}_{[t_j, t_{j+1})} . \]

Each level \(e_j\, \mathbf{1}_{\{ A_{t_j} \gt 0 \}}\) is bounded and \(\mathcal{F}_{t_j}^{(n)}\)-measurable, so \(\chi_\ell\) is again a bounded elementary process. Let \(t_j(s)\) denote the left endpoint of the \(\pi_\ell\)-interval containing \(s\). Where \(A_{t_j(s)} \gt 0\) the processes \(\chi_\ell\) and \(\phi_\ell\) agree, and where \(A_{t_j(s)} = 0\) the process \(\chi_\ell\) vanishes. Off \(N\) the path of \(A\) is nonnegative, so these two cases are exhaustive, and at every \(s \in [0, t)\) and every \(\omega\) off \(N\),

\[ (h - \chi_\ell)^2 \leq (h - \phi_\ell)^2 + h^2\, \mathbf{1}_{\{ A_{t_j(s)} = 0 \}} . \]

The switched-off part is small.
Fix \(\omega\) off \(N\). If \(A_{t_j(s)}(\omega) = 0\), then \(t_j(s) \leq \sigma(\omega)\), and so \(s \lt \sigma(\omega) + |\pi_\ell|\). Since \(\int_0^{\sigma} h^2\, ds = A_\sigma = 0\), writing \(\sigma_\ell = \min(\sigma + |\pi_\ell|, t)\) we get

\[ \begin{align*} \int_0^t h^2\, \mathbf{1}_{\{ A_{t_j(s)} = 0 \}}\, ds &\leq \int_\sigma^{\sigma_\ell} h^2\, ds \\\\ &= A_{\sigma_\ell} - A_\sigma , \end{align*} \]

which tends to zero by continuity of the path of \(A\). The left side is dominated by \(\int_0^t h^2\, ds\), which is integrable by (V3), so its expectation tends to zero by the dominated convergence theorem. By the inequality above, which holds off the null set \(N\), \(\mathbb{E}\bigl[ \int_0^t (h - \chi_\ell)^2\, ds \bigr] \to 0\), and the definition of the integral gives the convergence in \(L^2(\mathbb{P})\) of \(\int_0^t \chi_\ell\, dw_s^{(r)}\) to \(\int_0^t h\, dw_s^{(r)}\).

Conclusion.
On \(G \setminus N\) every \(A_{t_j}\) vanishes, because \(0 \leq A_{t_j} \leq A_t = 0\). Every level of \(\chi_\ell\) is therefore zero there, and so is its elementary integral \(\sum_j e_j\, \mathbf{1}_{\{ A_{t_j} \gt 0 \}}\, \Delta w_j^{(r)}\). Multiplication by \(\mathbf{1}_G\) does not increase the norm of \(L^2(\mathbb{P})\), so it preserves the convergence, and

\[ \begin{align*} &\mathbb{E}\Bigl[ \Bigl( \int_0^t h\, dw_s^{(r)} \Bigr)^2 \mathbf{1}_G \Bigr] \\\\ &\quad = \lim_{\ell \to \infty} \mathbb{E}\Bigl[ \Bigl( \int_0^t \chi_\ell\, dw_s^{(r)} \Bigr)^2 \mathbf{1}_G \Bigr] \\\\ &\quad = 0 , \end{align*} \]

which says that \(\int_0^t h\, dw_s^{(r)}\) vanishes almost surely on \(G\).

Proof.

Step 1 (Exhaustion and twins).
When \(U = \mathbb{R}^m\) the general formula applies as it stands, so let the complement of \(U\) be nonempty. The claim is componentwise, so take \(g\) scalar. For each integer \(q \geq 1\) let

\[ U_q = \bigl\{ \mathbf{x} : |\mathbf{x}| \lt q,\ \operatorname{dist}(\mathbf{x}, \mathbb{R}^m \setminus U) \gt 1/q \bigr\} . \]

Each \(U_q\) is open, the norm and the distance to a set being continuous. Its closure lies in the set where \(|\mathbf{x}| \leq q\) and the distance is at least \(1/q\), which is compact and contained in \(U_{q+1}\). Every compact \(K \subseteq U\) lies in some \(U_q\), since \(K\) is bounded and the continuous distance to the closed complement attains a positive minimum on \(K\).

By the existence of smooth bump functions on the manifold \(\mathbb{R}^m\), applied to the closed set \(\overline{U_q}\) and the open set \(U_{q+1}\), there is a smooth \(\psi_q : \mathbb{R}^m \to [0, 1]\) equal to \(1\) on \(\overline{U_q}\) whose support lies in \(U_{q+1}\). That support is closed and bounded, hence compact.

Define the twin \(\tilde{g}_q = \psi_q\, g\) on \([0, \infty) \times U\) and \(\tilde{g}_q = 0\) elsewhere. The open sets \([0, \infty) \times U\) and \([0, \infty) \times (\mathbb{R}^m \setminus \operatorname{supp} \psi_q)\) cover \([0, \infty) \times \mathbb{R}^m\), and \(\tilde{g}_q\) is of class \(C^2\) on each, so \(\tilde{g}_q \in C^2([0, \infty) \times \mathbb{R}^m)\). For every \(T \gt 0\) its derivatives up to second order are bounded on \([0, T] \times \mathbb{R}^m\), being continuous on the compact set \([0, T] \times \operatorname{supp} \psi_q\) and zero off it. On the open set \([0, \infty) \times U_q\) the factor \(\psi_q\) is identically \(1\), so \(\tilde{g}_q\) and \(g\) agree there together with all their partial derivatives up to second order.

Step 2 (The integrands of the formula for \(g\)).
Each \(v_{ij}\) satisfies (V1) and (V2), and such a process admits a progressively measurable version, the measure-theoretic fact taken on faith with the one-dimensional formula. We replace each \(v_{ij}\) by such a version. Two versions differ only on a set of \(\lambda \otimes \mathbb{P}\)-measure zero, so neither the Itô integrals nor the time integrals of the formula change. Write \(f_r = \sum_i \partial_i g\, v_{ir}\) for the diffusion integrands of \(g\) and \(D\) for its drift integrand, all evaluated along \((s, \mathbf{X}_s)\).

The derivatives of \(g\), read as zero off \(U\), are Borel functions on \([0, \infty) \times \mathbb{R}^m\), being continuous on the open set \([0, \infty) \times U\) and zero on its complement, and so is \(g\) itself. Composed with the progressively measurable \(\mathbf{X}\) and multiplied by the coefficients, they make every integrand of the formula progressively measurable. The initial value \(g(0, \mathbf{X}_0)\) is \(\mathcal{F}_0^{(n)}\)-measurable by (M1). The diffusion integrands lie in \(\mathcal{V}(0, T)\) for every \(T \gt 0\) by the membership condition.

The drift integrand satisfies the pathwise integrability condition of the definition. Take an outcome for which the path stays in \(U\) and, for every integer \(T \geq 1\), the integrals \(\int_0^T |u_i|\, ds\) and \(\int_0^T v_{ij}^2\, ds\) are finite. By the hypothesis on the paths and by (M2) and (M3), almost every outcome is of this kind. On the compact set \([0, T] \times \{ \mathbf{X}_s : 0 \leq s \leq T \}\), contained in \([0, \infty) \times U\), the derivatives of \(g\) up to second order are bounded, so \(\int_0^T |D|\, ds\) is finite for every \(T\).

Step 3 (The formula for the twin).
Write \(\tilde{f}_{q, r} = \sum_i \partial_i \tilde{g}_q\, v_{ir}\) and \(\tilde{D}_q\) for the diffusion and drift integrands of \(\tilde{g}_q\). Fix \(T \gt 0\). The first derivatives of \(\tilde{g}_q\) are bounded on \([0, T] \times \mathbb{R}^m\), say by \(M_{q, T}\), so \(|\tilde{f}_{q, r}| \leq M_{q, T} \sum_i |v_{ir}|\). Each \(\tilde{f}_{q, r}\) is progressively measurable, being built from the continuous derivatives of \(\tilde{g}_q\), the progressively measurable \(\mathbf{X}\), and the coefficients. Since every \(v_{ir}\) lies in \(\mathcal{V}(0, T)\), the bound places \(\tilde{f}_{q, r}\) in \(\mathcal{V}(0, T)\) as well, and \(T\) was arbitrary. The membership condition for \(\tilde{g}_q\) holds, and the general Itô formula gives, for each \(t \geq 0\), almost surely,

\[ \begin{align*} \tilde{g}_q(t, \mathbf{X}_t) &= \tilde{g}_q(0, \mathbf{X}_0) + \int_0^t \tilde{D}_q\, ds \\\\ &\quad + \sum_{r=1}^{n} \int_0^t \tilde{f}_{q, r}\, dw_s^{(r)} . \end{align*} \]

Step 4 (Where the twin is invisible).
Fix \(t \gt 0\) and \(q\), and let \(\Omega_q(t)\) be the set of outcomes for which \(\mathbf{X}_s \in U_q\) for every \(s \in [0, t]\). This set is an event. By continuity of the paths it is the set where the maximum of \(|\mathbf{X}_s|\) over \(s \in [0, t]\) is less than \(q\) and the minimum of \(\operatorname{dist}(\mathbf{X}_s, \mathbb{R}^m \setminus U)\) exceeds \(1/q\). Both extrema are attained, and they equal the supremum and the infimum over the rational \(s \in [0, t]\) together with \(s = t\), which are random variables.

Now take an outcome in \(\Omega_q(t)\). Every point \((s, \mathbf{X}_s)\) with \(s \in [0, t]\) lies in \([0, \infty) \times U_q\), where \(\tilde{g}_q\) and \(g\) agree with their derivatives. So \(\tilde{g}_q(t, \mathbf{X}_t) = g(t, \mathbf{X}_t)\), \(\tilde{g}_q(0, \mathbf{X}_0) = g(0, \mathbf{X}_0)\), the drift integrands \(\tilde{D}_q\) and \(D\) agree on \([0, t]\), and so do their time integrals, that of \(D\) being finite for almost every outcome by Step 2.

For the stochastic integrals, fix a component \(r\) and put \(h = \tilde{f}_{q, r} - f_r\). It is progressively measurable by Step 2 and Step 3, and it satisfies (V1)-(V3) on \([0, t]\), because \(\tilde{f}_{q, r}\) does and \(f_r\) does by the membership condition. At every outcome in \(\Omega_q(t)\) the integrand \(h\) vanishes on \([0, t]\), so \(\Omega_q(t)\) lies inside the event where \(\int_0^t h^2\, ds = 0\). The local property of the integral and linearity give, almost surely on \(\Omega_q(t)\),

\[ \int_0^t \tilde{f}_{q, r}\, dw_s^{(r)} = \int_0^t f_r\, dw_s^{(r)} . \]

Substituting these identities into the formula of Step 3, we find that the formula for \(g\) at time \(t\) holds almost surely on \(\Omega_q(t)\).

Step 5 (Exhausting the paths).
Fix an outcome whose path stays in \(U\). Such outcomes form a set of full measure. The image \(\{ \mathbf{X}_s : 0 \leq s \leq t \}\) is compact, being the continuous image of \([0, t]\), and it lies in \(U\), so by Step 1 it lies in some \(U_q\). The outcome therefore belongs to \(\Omega_q(t)\). The countably many events \(\Omega_q(t)\) thus cover every outcome outside a null set, and the formula for \(g\) holds almost surely at the fixed time \(t\). At \(t = 0\) both sides equal \(g(0, \mathbf{X}_0)\).

Step 6 (All times at once).
Both sides of the formula are continuous in \(t\) outside one null set. The left side is continuous because \(g\) is continuous on \([0, \infty) \times U\) and the path stays in \(U\), and the right side is an Itô process in its everywhere-continuous realization, legitimately so by Step 2. Two processes with continuous paths that agree almost surely at each fixed time agree at all rational times off one null set, hence at all times. So \(\mathbf{Y}_t = g(t, \mathbf{X}_t)\) is the Itô process given by the formula.