Ito's formula is Taylor's theorem evaluated on paths whose squared increments do not disappear. The first-order terms become stochastic and finite-variation integrals; quadratic terms converge to covariation; jumps retain the exact nonlinear Taylor remainder.
Continuous one-dimensional formula
Theorem (Ito formula). Let \(X=X_0+M+A\) be a continuous semimartingale and \(f\in C^2(\mathbb R)\). Then \[ f(X_t) = f(X_0) +\int_0^tf'(X_s)\,\mathrm{d}X_s +\frac12\int_0^tf''(X_s)\,\mathrm{d}[X]_s. \] Since \([X]=[M]\), only the local-martingale part contributes to the second-order term.
Proof. Stop so that \(X\), \(f'\), and \(f''\) remain bounded on the relevant range. On a partition, \[ f(X_{t_{i+1}})-f(X_{t_i}) = f'(X_{t_i})\Delta X_i +\frac12f''(X_{t_i})(\Delta X_i)^2+r_i, \] where \[ |r_i| \le\frac12\omega_{f''} \!\left(\max_j|\Delta X_j|\right)(\Delta X_i)^2. \] Continuity makes the maximum increment tend to zero in probability, while the sums of squared increments are tight, so \(\sum_i r_i\to0\) in probability. The first sums converge by the definition of semimartingale integration, and the quadratic sums converge to \(\int f''(X_s)\,\mathrm{d}[X]_s\). Remove localization.
Corollary (Ito process). If \[ X_t=X_0+\int_0^tb_s\,\mathrm{d} s +\int_0^t\sigma_s\,\mathrm{d}B_s, \] then \[ \begin{aligned} f(X_t)=f(X_0) &+\int_0^tf'(X_s)\sigma_s\,\mathrm{d}B_s\\ &+\int_0^t \left\{f'(X_s)b_s+\frac12f''(X_s)\sigma_s^2\right\}\mathrm{d} s. \end{aligned} \]
Time-dependent and multidimensional forms
Theorem. Let \(X=(X^1,\ldots,X^d)\) be a continuous semimartingale and \(f\in C^{1,2}([0,T]\times\mathbb R^d)\). Then \[ \begin{aligned} f(t,X_t)=f(0,X_0) &+\int_0^t\partial_sf(s,X_s)\,\mathrm{d} s\\ &+\sum_i\int_0^t\partial_if(s,X_s)\,\mathrm{d}X_s^i\\ &+\frac12\sum_{i,j}\int_0^t \partial_{ij}f(s,X_s)\,\mathrm{d}[X^i,X^j]_s. \end{aligned} \]
Proof. Apply multivariate Taylor expansion to \((t,X_t)\). Time has finite variation, so \([t]=[t,X^i]=0\). Spatial quadratic products converge to the covariations. The same modulus-of-continuity remainder bound as in one dimension makes all terms of order higher than two vanish.
Corollary (product rule). For continuous semimartingales, \[ \mathrm{d}(X_tY_t) = X_t\,\mathrm{d}Y_t+Y_t\,\mathrm{d}X_t +\mathrm{d}[X,Y]_t. \]
Proof. Apply the two-dimensional formula to \(f(x,y)=xy\), whose mixed Hessian entries are one.
Formula with jumps
Theorem. Let \(X\) be a cadlag semimartingale and \(f\in C^2(\mathbb R)\). Then \[ \begin{aligned} f(X_t)=f(X_0) &+\int_0^tf'(X_{s-})\,\mathrm{d}X_s +\frac12\int_0^tf''(X_{s-})\,\mathrm{d}[X]^c_s\\ &+\sum_{0<s\le t} \left\{ f(X_s)-f(X_{s-})-f'(X_{s-})\Delta X_s \right\}. \end{aligned} \]
Proof outline. Separate jumps larger than \(\varepsilon\), of which there are finitely many on compact intervals after localization. Apply the continuous Ito argument between those jumps and insert the exact change \(f(X_s)-f(X_{s-})\) at each jump. The first-order part of each jump is already included in the stochastic integral, leaving the displayed correction. Control the small jumps by the local quadratic bound \[ |f(x+h)-f(x)-f'(x)h|\le C h^2, \] then let \(\varepsilon\downarrow0\).
The differential mnemonic
Remark. For an Ito process \(\mathrm{d}X_t=b_t\,\mathrm{d}t+\sigma_t\,\mathrm{d}B_t\), the rules \[ (\mathrm{d}B_t)^2=\mathrm{d}t,\qquad \mathrm{d}t\,\mathrm{d}B_t=0,\qquad (\mathrm{d}t)^2=0 \] summarize quadratic-variation limits. They correctly reproduce \[ \mathrm{d}f(X_t) = f'(X_t)\,\mathrm{d}X_t +\frac12f''(X_t)\sigma_t^2\,\mathrm{d}t, \] but the theorem and its integrals, not formal infinitesimal algebra, provide the definition.