Multidimensional Ito formula for Brownian-driven processes

Reduction to a bounded localized problem: fix $c$ and replace $X$ by ${\bar{X}}^{(c)}=1_{D_c}{X}^{{\rho }_c}$ as in [F7]. This process is globally bounded by $c$, its total drift variations and diffusion energies on $[0,T]$ are at most $c$, and on $D_c\cap \{{\rho }_c\ge T\}$ both sides of its formula agree with those for the original process. Its stochastic integrands have finite energy bounded by $c\sup \mathrm{\mid }{\mathrm{\partial }}_{i}f{\mathrm{\mid }}^2$, and all continuous functions of $(s,{\bar{X}}_s^{(c)})$ are bounded. It suffices to prove the identity for this bounded process; rename it and its coefficients $X,b,\sigma$.

Setup of the $C^3$ case: assume $f\in C^3$ on a neighbourhood of the compact cylinder $[0,T]\times [-c,c{]}^d$ with finite bounds $M_0,M_1,M_2,M_3$ on partial derivatives of orders $0,1,2,3$; write $\mathrm{\Delta }{t}_j=t_{j+1}-t_j$ and $\mathrm{\Delta }{X}_j=X_{t_{j+1}}-X_{t_j}$.

Remainder control: Taylor's formula [F4] gives for each $j$ an expansion of $f(t_{j+1},X_{t_{j+1}})-f(t_j,X_{t_j})$ with third-order remainder $R_j$, $\mathrm{\mid }{R}_j\mathrm{\mid }\le C_d{M}_3(\mathrm{\mid }\mathrm{\Delta }{t}_j\mathrm{\mid }+\mathrm{\mid }\mathrm{\Delta }{X}_j\mathrm{\mid }{)}^3\le 4C_d{M}_3(\mathrm{\mid }\mathrm{\Delta }{t}_j{\mathrm{\mid }}^3+\mathrm{\mid }\mathrm{\Delta }{X}_j{\mathrm{\mid }}^3)$; summing, ${\sum }_j\mathrm{\mid }\mathrm{\Delta }{t}_j{\mathrm{\mid }}^3\le \mathrm{mesh}({\pi }_n{)}^{2}T\to 0$ and ${\sum }_j\mathrm{\mid }\mathrm{\Delta }{X}_j{\mathrm{\mid }}^3\le {\max }_j\mathrm{\mid }\mathrm{\Delta }{X}_j\mathrm{\mid }{\sum }_j\mathrm{\mid }\mathrm{\Delta }{X}_j{\mathrm{\mid }}^2={\max }_j\mathrm{\mid }\mathrm{\Delta }{X}_j\mathrm{\mid }{\sum }_i{Q}_n(X^i)(T)$, where $Q_n(X^i)(t):=S_n^{ii}(t)$, ${\max }_j\mathrm{\mid }\mathrm{\Delta }{X}_j\mathrm{\mid }\to 0$ almost surely by everywhere continuity, and each $Q_n(X^i)(T)\to [X^i{]}_T$ in probability by [F3]. The finite sum is bounded in probability; multiplying it by a quantity tending to zero almost surely gives convergence to zero in probability (split the probability at a fixed large bound for the sum). Thus ${\sum }_j\mathrm{\mid }{R}_j\mathrm{\mid }\to 0$, and hence ${\sum }_j{R}_j\to 0$ in probability.

Weighted pullback, elementary weights: let $w={\sum }_a{\lambda }_a{1}_{(u_a,u_{a+1}]}$ be elementary with bounded ${\mathcal{F}}_{u_a}$-measurable coefficients and deterministic block points. With the cumulative cross sums $S_n(v)$ on the original partition, [F3] gives uniform-in-$v$ convergence in probability to ${\int }_0^v(\sigma {\sigma }^{\mathsf{T}}{)}^{ij}ds$. The sum over original intervals lying wholly in one block is the difference of its endpoint cumulative sums up to at most two boundary intervals. Each boundary contribution is bounded by $\mathrm{\parallel }{\lambda }_a{\mathrm{\parallel }}_{\mathrm{\infty }}{\max }_l\mathrm{\mid }{\mathrm{\Delta }}_l{X}^i\mathrm{\mid }\mathrm{\mid }{\mathrm{\Delta }}_l{X}^j\mathrm{\mid }$ and tends to zero almost surely by continuity. The finite block sum therefore converges to ${\int }_0^T{w}_s(\sigma {\sigma }^{\mathsf{T}}{)}_s^{ij}ds$ in probability.

First-order terms: ${\sum }_j{\mathrm{\partial }}_{t}f(t_j,X_{t_j})\mathrm{\Delta }{t}_j\to {\int }_0^T{\mathrm{\partial }}_{t}f(s,X_s)ds$ and ${\sum }_j{\sum }_i{\mathrm{\partial }}_{i}f(t_j,X_{t_j}){\mathrm{\Delta }}_j{A}^i\to {\int }_0^T{\sum }_i{\mathrm{\partial }}_{i}f(s,X_s)b_s^{i}ds$ almost surely by the Riemann estimate [F6]; and the martingale part equals ${\sum }_{i,k}{\int }_0^T{g}_i^{(n)}{\sigma }^{ik}d{B}^k$ for the left-endpoint staircases $g_i^{(n)}$ of $s\mapsto {\mathrm{\partial }}_{i}f(s,X_s)$, which converges in $L^2(P)$ to ${\sum }_{i,k}{\int }_0^T{\mathrm{\partial }}_{i}f(s,X_s){\sigma }_s^{ik}d{B}_s^k$ by [F6] with $D=\mathrm{\Omega }$ and the isometry.

Weighted pullback, continuous weights: for continuous adapted $w$ with $\mathrm{\mid }w\mathrm{\mid }\le K$ and its left-endpoint staircase $w^{(m)}$ on the grid of mesh $2^{-m}T$, uniform continuity gives ${\sup }_s\mathrm{\mid }{w}_s^{(m)}-w_s\mathrm{\mid }\to 0$ almost surely. Fix $m$ first and apply step 1.4 as $n\to \mathrm{\infty }$. The sum error is bounded by ${\sup }_s\mathrm{\mid }{w}_s^{(m)}-w_s\mathrm{\mid }{Q}_n(X^i{)}^{1\mathrm{/}2}{Q}_n(X^j{)}^{1\mathrm{/}2}$, where $Q_n(X^i)$ denotes its terminal value. Put $R_m={\sup }_s\mathrm{\mid }{w}_s^{(m)}-w_s\mathrm{\mid }$ and $Z_n=Q_n(X^i{)}^{1\mathrm{/}2}{Q}_n(X^j{)}^{1\mathrm{/}2}$. For $L>0$, $P(R_m{Z}_n>\epsilon )\le P(R_m>\epsilon \mathrm{/}L)+P(Z_n>L)$. Tightness of the quadratic factors makes the second term uniformly small for large $L$ (or for all sufficiently large $n$), and then large $m$ makes the first small. The limiting integral error is at most $R_m{\int }_0^T{\sum }_k\mathrm{\mid }{\sigma }^{ik}{\sigma }^{jk}\mathrm{\mid }ds\le c{R}_m$ by the localized total-energy bound and $2\mathrm{\mid }ab\mathrm{\mid }\le a^2+b^2$; it tends to zero almost surely and in $L^1$, since $R_m\le 2K$. Taking $n\to \mathrm{\infty }$ and then $m\to \mathrm{\infty }$ proves [F5].

Second-order terms: by step 2.1 applied to $w={\mathrm{\partial }}_i{\mathrm{\partial }}_{j}f(\cdot ,X)$, ${\sum }_l\frac{1}{2}{\mathrm{\partial }}_i{\mathrm{\partial }}_{j}f(t_l,X_{t_l}){\mathrm{\Delta }}_l{X}^i{\mathrm{\Delta }}_l{X}^j\to \frac{1}{2}{\int }_0^T{\mathrm{\partial }}_i{\mathrm{\partial }}_{j}f(s,X_s)(\sigma {\sigma }^{\mathsf{T}}{)}_s^{ij}ds$ in probability for each pair $i,j$, hence for the finite sum. These are the complete spatial Hessian terms; no separate drift expansion is added. The time--space terms are bounded in absolute value by $M_{2}T{\sum }_i{\max }_l\mathrm{\mid }{\mathrm{\Delta }}_l{X}^i\mathrm{\mid }$ and the pure time term by $\frac{1}{2}{M}_{2}T\mathrm{mesh}({\pi }_n)$, so both tend to zero by continuity.

Assemble the $C^3$ case: summing the exact expansions of step 1.3 over $j$, the left side telescopes to $f(T,X_T)-f(0,X_0)$ and the right side is controlled by steps 1.3, 1.5 and 3.1; passing to the limit along ${\pi }_n$ gives the identity at $T$ in probability, hence almost surely, and then at each fixed $t\in [0,T]$ by restricting and augmenting the partition sequence. Intersect the full events for rational times and the endpoint $T$; continuity of both sides extends the equality to all times on that single full event.

Reduction to $C^{1,2}$ by cutoff and mollification: for $f\in C^{1,2}$ take the collar extension, smooth cutoff and mollifications from [F8]. On the inner cylinder $[0,T]\times [-c,c{]}^d$ the smoothed functions and their derivatives ${\mathrm{\partial }}_t,{\mathrm{\partial }}_i,{\mathrm{\partial }}_i{\mathrm{\partial }}_j$ converge uniformly to those of $f$. Apply step 4.1 and let the mollification scale tend to zero: the drift integral converges by dominated convergence with bound $K^{\mathrm{\prime }}(1+{\sum }_i\mathrm{\mid }{b}^i\mathrm{\mid }+{\sum }_{i,k}({\sigma }^{ik}{)}^2)$; for each $i,k$ the stochastic difference has squared $L^2$ norm at most the uniform squared derivative error times $E{\int }_0^T({\sigma }_s^{ik}{)}^{2}ds$, which tends to zero; and the left side converges uniformly on the inner cylinder.

Removal of the localization and conclusion: the identity for ${\bar{X}}^{(c)}$ agrees with the desired identity on $D_c\cap \{{\rho }_c\ge T\}$, and these events increase to a probability-one event by [F7]: every integer $c$ larger than the path supremum, total drift variation and total energy has $D_c$ true and ${\rho }_c=T$. Take countably many integer $c$ and horizons $T$ to obtain one full event. Hence the identity holds almost surely at every deterministic time and, by continuity of both sides, up to indistinguishability; the integrands are predictable and locally square-integrable by [F1] and [F2].

Boundary and consistency cases: for $d=1$ and $m=1$ the formula is the one-dimensional formula of One-dimensional Ito formula; for $f(t,x)=x^i$ it reduces to the defining display of $X^i$; for $f(t,x)=\mathrm{\mid }x{\mathrm{\mid }}^2$ it gives $\mathrm{\mid }{X}_t{\mathrm{\mid }}^2=\mathrm{\mid }{X}_0{\mathrm{\mid }}^2+2{\sum }_i{\int }_0^t{X}_s^{i}d{X}_s^i+{\sum }_{i,k}{\int }_0^t({\sigma }_s^{ik}{)}^{2}ds$; if $\sigma \equiv 0$ the covariation matrix vanishes and the formula is the chain rule along an absolutely continuous path; at $t=0$ both sides equal $f(0,X_0)$; if $d=0$ is excluded there is nothing degenerate to treat, and a singular dispersion matrix is allowed because only the products $(\sigma {\sigma }^{\mathsf{T}}{)}^{ij}$ enter the quadratic term. Independence and unit covariance of the coordinates are exactly the standard vector-Brownian convention already encoded in [F3]. AC enters only through [F10], and all localization and mollification parameters are canonical.