Itô’s formula
0.1 Case of single space dimension
Let ${X}_{t}$ be an Itô process satisfying the stochastic differential equation
$$d{X}_{t}={\mu}_{t}dt+{\sigma}_{t}d{W}_{t},$$ 
with ${\mu}_{t}$ and ${\sigma}_{t}$ being adapted processes, adapted to the same filtration as the Brownian motion^{} ${W}_{t}$. Let $f$ be a function with continuous^{} partial derivatives^{} $\frac{\partial f}{\partial t}$, $\frac{\partial f}{\partial x}$ and $\frac{{\partial}^{2}f}{\partial {x}^{2}}$.
Then ${Y}_{t}=f({X}_{t})$ is also an Itô process, and its stochastic differential equation is
$d{Y}_{t}$  $={\displaystyle \frac{\partial f}{\partial t}}dt+{\displaystyle \frac{\partial f}{\partial x}}d{X}_{t}+{\displaystyle \frac{1}{2}}{\displaystyle \frac{{\partial}^{2}f}{\partial {x}^{2}}}(d{X}_{t})(d{X}_{t})$  
$=\left({\displaystyle \frac{\partial f}{\partial t}}+{\displaystyle \frac{\partial f}{\partial x}}{\mu}_{t}+{\displaystyle \frac{1}{2}}{\sigma}_{t}^{2}\right)dt+{\displaystyle \frac{\partial f}{\partial x}}{\sigma}_{t}d{W}_{t},$ 
where all partial derivatives are to be taken at $(t,{X}_{t})$.
0.2 Case of multiple space dimensions
There is also an analogue for multiple space dimensions^{}.
Let ${X}_{t}$ be a ${\mathbb{R}}^{n}$valued Itô process satisfying the stochastic differential equation
$$d{X}_{t}={\mu}_{t}dt+{\sigma}_{t}d{W}_{t},$$ 
with ${\mu}_{t}$ and ${\sigma}_{t}$ being adapted processes, adapted to the same filtration as the $m$dimensional Brownian motion ${W}_{t}$. ${\mu}_{t}$ is ${\mathbb{R}}^{n}$valued and ${\sigma}_{t}$ is $L({\mathbb{R}}^{m},{\mathbb{R}}^{n})$valued.
Let $f:{\mathbb{R}}^{n}\times \mathbb{R}\to \mathbb{R}$ be a function with continuous partial derivatives.
Then ${Y}_{t}=f({X}_{t})$ is also an Itô process, and its stochastic differential equation is
$d{Y}_{t}$  $={\displaystyle \frac{\partial f}{\partial t}}dt+(\mathrm{D}f)d{X}_{t}+\frac{1}{2}d{X}_{t}^{*}({\mathrm{D}}^{2}f)d{X}_{t}$  
$={\displaystyle \frac{\partial f}{\partial t}}dt+(\mathrm{D}f){\mu}_{t}dt+(\mathrm{D}f){\sigma}_{t}d{W}_{t}+\frac{1}{2}d{W}_{t}^{*}{\sigma}_{t}^{*}({\mathrm{D}}^{2}f){\sigma}_{t}d{W}_{t}$  
$={\displaystyle \frac{\partial f}{\partial t}}dt+(\mathrm{D}f){\mu}_{t}dt+(\mathrm{D}f){\sigma}_{t}d{W}_{t}+\frac{1}{2}\mathrm{tr}\left({\sigma}_{t}^{*}({\mathrm{D}}^{2}f){\sigma}_{t}\right)dt$  
$=\left({\displaystyle \frac{\partial f}{\partial t}}+(\mathrm{D}f){\mu}_{t}+\frac{1}{2}\mathrm{tr}\left(({\sigma}_{t}{\sigma}_{t}^{*})({\mathrm{D}}^{2}f)\right)\right)dt+(\mathrm{D}f){\sigma}_{t}d{W}_{t},$ 
where

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$\mathrm{tr}$ is the trace operation; ${}^{\ast}$ is the transpose^{}

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$\mathrm{D}f$ is the derivative with respect to the space variables; its value is a linear transformation from $L({\mathbb{R}}^{n},\mathbb{R})$

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${\mathrm{D}}^{2}f$ is the second derivative with respect to space variables; represented as the Hessian matrix

•
the third line follows because $d{W}_{t}^{i}d{W}_{t}^{j}={\delta}_{ij}dt$.
The quadratic form $\mathrm{tr}({\sigma}_{t}{\sigma}_{t}^{*}{\mathrm{D}}^{2}f)dt$ represents the quadratic variation of the process. When ${\sigma}_{t}$ is the identity transformation, this reduces to the Laplacian of $f$.
Itô’s formula in multiple dimensions can also be written with the standard vector calculus operators. It is in the similar notation typically used for the related parabolic partial differential equation describing an Itô diffusion:
$$d{Y}_{t}=\left(\frac{\partial f}{\partial t}+{\mu}_{t}\cdot \nabla f+\frac{1}{2}\left(\nabla \cdot ({\sigma}_{t}{\sigma}_{t}^{*})\nabla \right)f\right)dt+({\sigma}_{t}d{W}_{t})\cdot \nabla f.$$ 
References
 1 Bernt Øksendal. , An Introduction with Applications. 5th ed., Springer 1998.
 2 HuiHsiung Kuo. Introduction to Stochastic Integration. Springer 2006.
Title  Itô’s formula 
Canonical name  ItosFormula 
Date of creation  20130322 17:16:14 
Last modified on  20130322 17:16:14 
Owner  stevecheng (10074) 
Last modified by  stevecheng (10074) 
Numerical id  13 
Author  stevecheng (10074) 
Entry type  Axiom 
Classification  msc 60H10 
Classification  msc 60H05 
Synonym  Itô’s formula 
Synonym  Itô’s chain rule^{} 
Synonym  Ito’s formula 
Synonym  Ito’s lemma 
Synonym  Ito’s chain rule 
Related topic  ItosLemma2 
Related topic  GeneralizedItoFormula 