One-dimensional Ito formula

Reduction to a bounded localized problem: fix $c>0$ and replace $X$ by ${\bar{X}}^{(c)}=1_{D_c}{X}^{{\rho }_c}$ as in [F7]. This is globally bounded by $c$, its drift variation and diffusion energy on $[0,T]$ are at most $c$, and on $D_c\cap \{{\rho }_c\ge T\}$ both sides of the desired formula agree with those for the original process. Thus the stochastic integrand has finite energy bounded by $c\sup \mathrm{\mid }{\mathrm{\partial }}_{x}f{\mathrm{\mid }}^2$, and every continuous function of $(s,{\bar{X}}_s^{(c)})$ is bounded on $[0,T]$. It remains to prove the identity for this bounded process; localization is removed at the end. To simplify notation, call the bounded process and its coefficients again $X,b,\sigma$.

Setup of the $C^3$ case: assume first that $f\in C^3$ on a neighbourhood of the cylinder $[0,T]\times [-c,c]$ with finite bounds $M_0,M_1,M_2,M_3$ for its partial derivatives of orders $0,1,2,3$ there. For the partition ${\pi }_n$ write $\mathrm{\Delta }{t}_j=t_{j+1}-t_j$, $\mathrm{\Delta }{X}_j=X_{t_{j+1}}-X_{t_j}=\mathrm{\Delta }{A}_j+\mathrm{\Delta }{M}_j$.

Weighted pullback setup: for continuous adapted $w$ with $\mathrm{\mid }w\mathrm{\mid }\le K$ put $H=\sigma$. In steps 1.4 and 2.1 we prove the weighted limit needed for the second-order term. Write $Q_n=Q_n^M$ from [F3] throughout the remainder of the proof; then $Q_n(T)$ is bounded in probability and converges to ${\int }_0^T{H}^{2}ds$.

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 fixed deterministic block points. For the cumulative sums $Q_n(v)$ on the original partition, [F3] gives uniform-in-$v$ convergence in probability to ${\int }_0^v{H}^{2}ds$. The sum over those original intervals lying wholly in $(u_a,u_{a+1}]$ is $Q_n(u_{a+1})-Q_n(u_a)$ up to the at most two boundary intervals. Their contribution is bounded by $2\mathrm{\parallel }{\lambda }_a{\mathrm{\parallel }}_{\mathrm{\infty }}{\max }_j\mathrm{\mid }{\mathrm{\Delta }}_{j}M{\mathrm{\mid }}^2$, which tends to zero almost surely by continuity. Summing over the finitely many blocks gives ${\sum }_{j}w(t_j){\mathrm{\Delta }}_j{M}^2\to {\sum }_a{\lambda }_a{\int }_{u_a}^{u_{a+1}}{H}^{2}ds={\int }_0^{T}w{H}^{2}ds$ in probability.

Remainder of the Taylor expansion: with $h_j=(t_{j+1}-t_j,X_{t_{j+1}}-X_{t_j})$ the Taylor expansion [F4] gives, for each $j$, $f(t_{j+1},X_{t_{j+1}})-f(t_j,X_{t_j})={\mathrm{\partial }}_{t}f{h}_1+{\mathrm{\partial }}_{x}f{h}_2+\frac{1}{2}({\mathrm{\partial }}_{tt}^{2}f{h}_1^2+2{\mathrm{\partial }}_t{\mathrm{\partial }}_{x}f{h}_1{h}_2+{\mathrm{\partial }}_{xx}^{2}f{h}_2^2)+R_j$ with $\mathrm{\mid }{R}_j\mathrm{\mid }\le M_3(\mathrm{\mid }{h}_1\mathrm{\mid }+\mathrm{\mid }{h}_2\mathrm{\mid }{)}^3\le 4M_3(\mathrm{\mid }\mathrm{\Delta }{t}_j{\mathrm{\mid }}^3+\mathrm{\mid }\mathrm{\Delta }{X}_j{\mathrm{\mid }}^3)$. Summing: ${\sum }_j\mathrm{\mid }\mathrm{\Delta }{t}_j{\mathrm{\mid }}^3\le \mathrm{mesh}({\pi }_n{)}^{2}T\to 0$ and ${\sum }_j\mathrm{\mid }\mathrm{\Delta }{X}_j{\mathrm{\mid }}^3\le {\max }_j\mathrm{\mid }\mathrm{\Delta }{X}_j\mathrm{\mid }{\sum }_j\mathrm{\mid }\mathrm{\Delta }{X}_j{\mathrm{\mid }}^2\le {\max }_j\mathrm{\mid }\mathrm{\Delta }{X}_j\mathrm{\mid }(2{\sum }_j\mathrm{\Delta }{A}_j^2+2Q_n(T))$, where ${\max }_j\mathrm{\mid }\mathrm{\Delta }{X}_j\mathrm{\mid }\to 0$ almost surely by continuity of the paths of $A$ and $M$ and $Q_n(T)-{\int }_0^T{\sigma }_s^{2}ds\to 0$ in probability by [F3], so the sum of remainders tends to $0$ 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 ${\sum }_j{\mathrm{\partial }}_{x}f(t_j,X_{t_j})\mathrm{\Delta }{A}_j\to {\int }_0^T{\mathrm{\partial }}_{x}f(s,X_s)b_{s}ds$ both almost surely by the Riemann estimate [F6] applied with the continuous bounded weights ${\mathrm{\partial }}_{t}f(\cdot ,X)$ and ${\mathrm{\partial }}_{x}f(\cdot ,X)$; and the stochastic part satisfies ${\sum }_j{\mathrm{\partial }}_{x}f(t_j,X_{t_j}){\mathrm{\Delta }}_{j}M={\int }_0^T{g}^{(n)}\sigma dB$ for the left-endpoint staircase $g^{(n)}$ of $s\mapsto {\mathrm{\partial }}_{x}f(s,X_s)$, which converges in $L^2(P)$ to ${\int }_0^T{\mathrm{\partial }}_{x}f(s,X_s){\sigma }_{s}dB$ by F5, now with $D=\mathrm{\Omega }$, and [F2].

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 deterministic grid of mesh $2^{-m}T$, uniform continuity of $s\mapsto w_s$ gives ${\sup }_s\mathrm{\mid }{w}_s^{(m)}-w_s\mathrm{\mid }\to 0$ almost surely. Fix $m$ first. Step 1.4 gives the asserted convergence with $w^{(m)}$ as $n\to \mathrm{\infty }$. Put $R_m={\sup }_s\mathrm{\mid }{w}_s^{(m)}-w_s\mathrm{\mid }$. The error in the sums is at most $R_m{Q}_n(T)$. For every $L>0$ its probability of exceeding $\epsilon$ is at most $P(R_m>\epsilon \mathrm{/}L)+P(Q_n(T)>L)$. First take $L$ large using tightness from [F3], then $m$ large; this bound needs no independence of the two factors. the error in the limiting integrals tends to zero as $m\to \mathrm{\infty }$ by dominated convergence with integrable bound $2K{\int }_0^T{H}^{2}ds$. Taking these two limits successively proves F5.

Second-order terms: by step 2.1 applied to $w={\mathrm{\partial }}_{xx}^{2}f(\cdot ,X)$, ${\sum }_j\frac{1}{2}{\mathrm{\partial }}_{xx}^{2}f(t_j,X_{t_j}){\mathrm{\Delta }}_j{M}^2\to \frac{1}{2}{\int }_0^T{\mathrm{\partial }}_{xx}^{2}f(s,X_s){\sigma }_s^{2}ds$ in probability. The mixed drift--martingale term satisfies $\mid {\sum }_j{\mathrm{\partial }}_{xx}^{2}f{\mathrm{\Delta }}_{j}A{\mathrm{\Delta }}_{j}M\mid \le M_2({\sum }_j{\mathrm{\Delta }}_j{A}^2{)}^{1\mathrm{/}2}({\sum }_j{\mathrm{\Delta }}_j{M}^2{)}^{1\mathrm{/}2}\le M_2({\max }_j\mathrm{\mid }{\mathrm{\Delta }}_{j}A\mathrm{\mid }{\int }_0^T\mathrm{\mid }b\mathrm{\mid }{)}^{1\mathrm{/}2}{Q}_n(T{)}^{1\mathrm{/}2}\to 0$ in probability, because $A$ is continuous and hence ${\max }_j\mathrm{\mid }{\mathrm{\Delta }}_{j}A\mathrm{\mid }\to 0$ by [F6] and $Q_n(T)$ is bounded in probability; and the two purely drift/time terms satisfy $\mid {\sum }_j\frac{1}{2}{\mathrm{\partial }}_{xx}^{2}f{\mathrm{\Delta }}_j{A}^2\mid \le \frac{M_2}{2}{\max }_j\mathrm{\mid }{\mathrm{\Delta }}_{j}A\mathrm{\mid }{\int }_0^T\mathrm{\mid }b\mathrm{\mid }\to 0$ and $\mid {\sum }_j{\mathrm{\partial }}_t{\mathrm{\partial }}_{x}f\mathrm{\Delta }{t}_j\mathrm{\Delta }{X}_j\mid +\mid {\sum }_j\frac{1}{2}{\mathrm{\partial }}_{tt}^{2}f\mathrm{\Delta }{t}_j^2\mid \le M_{2}T{\max }_j\mathrm{\mid }\mathrm{\Delta }{X}_j\mathrm{\mid }+\frac{M_2}{2}T{\delta }_n\to 0$.

Assemble the $C^3$ case at $T$: summing the exact expansion [step 1.5] over $j$, the left side telescopes to $f(T,X_T)-f(0,X_0)$, while the right side is the sum of the terms controlled in steps 1.5, 1.6 and 3.1; passing to the limit along ${\pi }_n$ gives $f(T,X_T)-f(0,X_0)={\int }_0^T({\mathrm{\partial }}_{t}f+b{\mathrm{\partial }}_{x}f+\frac{1}{2}{\sigma }^2{\mathrm{\partial }}_{xx}^{2}f)(s,X_s)ds+{\int }_0^T{\sigma }_s{\mathrm{\partial }}_{x}f(s,X_s)d{B}_s$ in probability, and uniqueness of limits in probability makes the difference of the two fixed random variables zero almost surely. For an arbitrary $t\in [0,T]$ apply the same argument on $[0,t]$ with the restricted, $t$-augmented partition sequence; both sides are continuous in $t$, so the identity holds for all $t$ up to indistinguishability.

Reduction to $C^{1,2}$ by cutoff and mollification: let $f\in C^{1,2}$ and take the reflected extension, smooth cutoff and mollifications $g_{\epsilon }$ of [F8]. On the nonnegative inner cylinder $[0,T]\times [-c,c]$, where the extension equals $f$, the functions $g_{\epsilon }$ and their derivatives ${\mathrm{\partial }}_t{g}_{\epsilon },{\mathrm{\partial }}_x{g}_{\epsilon },{\mathrm{\partial }}_{xx}^2{g}_{\epsilon }$ converge uniformly to the corresponding derivatives of $f$. Applying step 4.1 to the smooth $g_{\epsilon }$ and passing to the limit, the drift term converges by dominated convergence with bound $K^{\mathrm{\prime }}(1+\mathrm{\mid }{b}_s\mathrm{\mid }+{\sigma }_s^2)$, whose integral is at most $K^{\mathrm{\prime }}(T+2c)$; the stochastic term converges in $L^2(P)$ by [F2], since its squared norm is bounded by the uniform squared derivative error times $E{\int }_0^T{\sigma }_s^{2}ds\le c$; and the left side converges uniformly on the inner cylinder. Hence the identity holds for $f$ at $T$, and then for every $t$ by the same continuity argument.

Removal of the localization and conclusion: the identity for ${\bar{X}}^{(c)}$ agrees with the desired identity on $D_c\cap \{{\rho }_c\ge T\}$. For each path with the normalized local bounds, every integer $c$ larger than ${\sup }_{t\le T}\mathrm{\mid }{X}_t\mathrm{\mid }$, ${\int }_0^T\mathrm{\mid }b\mathrm{\mid }$ and ${\int }_0^T{\sigma }^2$ has $D_c$ true and ${\rho }_c=T$. Thus these events increase to a probability-one event by [F7], so the identity holds almost surely at every deterministic time, and by continuity of both sides up to indistinguishability. The stochastic integrand $\sigma {\mathrm{\partial }}_{x}f(\cdot ,X)$ is predictable and locally square-integrable by [F1] and [F2]. The Lebesgue integrand is progressively measurable and integrable almost surely on every finite horizon: after localization its continuous derivative factors are bounded, while $\mathrm{\mid }b\mathrm{\mid }$ and ${\sigma }^2$ are integrable. Thus the displayed statement follows.

Boundary and consistency cases: for $t=0$ both sides equal $f(0,X_0)$; for $f$ independent of $x$ the formula reduces to $f(t)=f(0)+{\int }_0^t{\mathrm{\partial }}_{t}f(s)ds$, the fundamental theorem for the deterministic continuous function ${\mathrm{\partial }}_{t}f$; for $f(t,x)=x$ it reduces to the definition of $X$; for $f(t,x)=x^2$ and $X=B$ (so $X_0=0$, $b=0,\sigma =1$) it gives $B_t^2=2{\int }_0^t{B}_{s}d{B}_s+t$; if $\sigma \equiv 0$ then $X$ is pathwise absolutely continuous and the formula is the chain rule with the second-order term absent, consistent with the vanishing covariation of finite-variation parts; if $X\equiv X_0$ is deterministic the formula is the fundamental theorem along the deterministic time variable; and if the diffusion coefficient is unbounded the localization of step 1.1 is what makes every integral finite, with no additional hypothesis. AC enters only through [F10], and all localization and mollification parameters are canonical.