Itô's lemma

For an Itô process $d{X}_t=b_{t}dt+{\sigma }_{t}d{B}_t$ and a twice differentiable function $f$,The second-order term reflects the nonzero quadratic variation of Brownian motion.

The one-dimensional Itô formula for polynomials follows by induction on powers using the stochastic product rule. If it holds for $X^k$, its martingale part gives $[X^k,X]=k{X}^{k-1}\cdot [X]$. The product rule for $X^{k}X$ then yields the coefficient $k(k+1)/2$ of the bracket term for $X^{k+1}$.

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For a semimartingale with càdlàg paths $X$ and $f\in C^2$, the Itô formula uses the left-limit gradient in $\int f^{\prime }(X_{s-})d{X}_s$, the continuous quadratic variation in its second-order integral, and the jump correction ${\sum }_{s\le t}[f(X_s)-f(X_{s-})-f^{\prime }(X_{s-})\Delta X_s]$. In several dimensions use the corresponding gradient, Hessian and continuous bracket matrix. The correction is the part of each jump not captured by first-order linearization.

For continuous semimartingales,It is the stochastic counterpart of the ordinary product rule; the quadratic covariation supplies the additional second-order term.

For a continuous semimartingale $X$,where $L^0(X)$ is the local time of a semimartingale at zero.

Choose $f_n(0)=0$ and $f_n^{\prime }(x)=\varphi (nx)$, where $\varphi$ is continuously differentiable, nondecreasing, equal to $-1$ on $(-\infty ,0]$ and $1$ on $[1,\infty )$. Then $f_n^{\prime \prime }\ge 0$ and $0\le |x|-f_n(x)\le 2/n$. For $X=M+A$ with continuous adapted finite-variation $A$, the Itô isometry and Doob L2 maximal inequality give local uniform convergence in probability of the martingale integrals. Dominated convergence against the total-variation measure gives pathwise uniform convergence of the drift integrals. The nondecreasing corrections $\frac{1}{2}\int f_n^{\prime \prime }(X)d[M]$ therefore converge locally uniformly in probability to $|X|-\int {\mathrm{sgn}}_{-}(X)dX$. A subsequence on one probability-one event shows that the limit is continuous and nondecreasing. Localization removes the deterministic bounds used in the argument.

For integer paths with increments in $\{-1,0,1\}$, the positive part obeys a telescoping identity with slope $f(x)=1_{\{x>0\}}+\frac{1}{2}{1}_{\{x=0\}}$ and a correction $\frac{1}{2}{1}_{\{S_{t-1}=K\}}(\Delta S_t{)}^2$ at the kink. For a martingale, the slope term is a martingale transform.

Let a simple symmetric random walk start from a deterministic integer $S_0=i$, and put $L_n={\sum }_{k=0}^{n-1}{1}_{\{S_k=0\}}$. Then $M_n=|S_n|-L_n$ is a martingale with $M_0=|i|$. The visit count includes time $0$ if $i=0$.

For every integer $s$ and step $\xi \in \{-1,1\}$, the pathwise identityholds with $\mathrm{sgn}(0)=0$. Subtract the indicator function and sum over time to obtainThe sign function of the current state is a bounded predictable process for the next increment, so the sum is a martingale transform. Alternatively, averaging the two possible next absolute values shows directly that the conditional drift is $1_{\{S_k=0\}}$. Bounds $|S_n|\le |i|+n$ and $L_n\le n$ verify integrability at each finite time.

This is an absolute-value version of the discrete Tanaka formula. The correction records the occasions when the random walk is at the corner of the absolute-value function; away from zero that function is locally linear along a single step.