You are here
HomeIt\^o's formula
Primary tabs
Itô’s formula
0.1 Case of single space dimension
Let $X_{t}$ be an Itô process satisfying the stochastic differential equation
$dX_{t}=\mu_{t}\,dt+\sigma_{t}\,dW_{t}\,,$ 
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
$\displaystyle dY_{t}$  $\displaystyle=\frac{\partial f}{\partial t}\,dt+\frac{\partial f}{\partial x}% \,dX_{t}+\frac{1}{2}\frac{\partial^{2}f}{\partial x^{2}}(dX_{t})(dX_{t})$  
$\displaystyle=\left(\frac{\partial f}{\partial t}+\frac{\partial f}{\partial x% }\mu_{t}+\frac{1}{2}\sigma_{t}^{2}\right)\,dt+\frac{\partial f}{\partial x}% \sigma_{t}\,dW_{t}\,,$ 
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
$dX_{t}=\mu_{t}\,dt+\sigma_{t}\,dW_{t}\,,$ 
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 $f\colon\mathbb{R}^{n}\times\mathbb{R}\to\mathbb{R}$ be a function with continuous partial derivatives.
Then $Y_{t}=f(X_{t})$ is also an Itô process, and its stochastic differential equation is
$\displaystyle dY_{t}$  $\displaystyle=\frac{\partial f}{\partial t}\,dt+(\D f)\,dX_{t}+\tfrac{1}{2}dX_% {t}^{*}(\D^{2}f)dX_{t}$  
$\displaystyle=\frac{\partial f}{\partial t}\,dt+(\D f)\mu_{t}\,dt+(\D f)\sigma% _{t}\,dW_{t}+\tfrac{1}{2}dW_{t}^{*}\,\sigma_{t}^{*}(\D^{2}f)\sigma_{t}\,dW_{t}$  
$\displaystyle=\frac{\partial f}{\partial t}\,dt+(\D f)\mu_{t}\,dt+(\D f)\sigma% _{t}\,dW_{t}+\tfrac{1}{2}\tr\bigl(\sigma_{t}^{*}\,(\D^{2}f)\,\sigma_{t}\bigr)% \,dt$  
$\displaystyle=\left(\frac{\partial f}{\partial t}+(\D f)\mu_{t}+\tfrac{1}{2}% \tr\bigl((\sigma_{t}\sigma_{t}^{*})(\D^{2}f)\bigr)\right)\,dt+(\D f)\sigma_{t}% \,dW_{t}\,,$ 
where

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

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

the third line follows because $dW_{t}^{i}\,dW_{t}^{j}=\delta_{{ij}}\,dt$.
The quadratic form $\tr(\sigma_{t}\sigma_{t}^{*}\,\D^{2}f)\,dt$ 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:
$dY_{t}=\left(\frac{\partial f}{\partial t}+\mu_{t}\cdot\nabla f+\tfrac{1}{2}% \bigl(\nabla\cdot(\sigma_{t}\sigma_{t}^{*})\nabla\bigr)f\right)dt+(\sigma_{t}% \,dW_{t})\cdot\nabla f\,.$ 
References
 1 Bernt Øksendal. Stochastic Differential Equations, An Introduction with Applications. 5th ed., Springer 1998.
 2 HuiHsiung Kuo. Introduction to Stochastic Integration. Springer 2006.
Mathematics Subject Classification
60H10 no label found60H05 no label found Forums
 Planetary Bugs
 HS/Secondary
 University/Tertiary
 Graduate/Advanced
 Industry/Practice
 Research Topics
 LaTeX help
 Math Comptetitions
 Math History
 Math Humor
 PlanetMath Comments
 PlanetMath System Updates and News
 PlanetMath help
 PlanetMath.ORG
 Strategic Communications Development
 The Math Pub
 Testing messages (ignore)
 Other useful stuff
Recent Activity
new question: Prove a formula is part of the Gentzen System by LadyAnne
Mar 30
new question: A problem about Euler's totient function by mbhatia
new problem: Problem: Show that phi(a^n1), (where phi is the Euler totient function), is divisible by n for any natural number n and any natural number a >1. by mbhatia
new problem: MSC browser just displays "No articles found. Up to ." by jaimeglz
Mar 26
new correction: Misspelled name by DavidSteinsaltz
Mar 21
new correction: underlinetypo by Filipe
Mar 19
new correction: cocycle pro cocyle by pahio
Mar 7
new image: plot W(t) = P(waiting time <= t) (2nd attempt) by robert_dodier
new image: expected waiting time by robert_dodier
new image: plot W(t) = P(waiting time <= t) by robert_dodier