Itô’s formula (2024)

0.1 Case of single space dimension

Let $X_{t}$ be an Itô process satisfying the stochasticdifferential 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 stochasticdifferential equationis

	$\displaystyle dY_{t}$	$\displaystyle=\frac{\partial f}{\partial t}\,dt+\frac{\partial f}{\partial x}%\,dX_{t}+\frac{1}{2}\frac{\partial^{2}f}{\partial x^{2}}(dX_{t})(dX_{t})$
		$\displaystyle=\left(\frac{\partial f}{\partial t}+\frac{\partial f}{\partial x%}\mu_{t}+\frac{1}{2}\sigma_{t}^{2}\right)\,dt+\frac{\partial f}{\partial x}%\sigma_{t}\,dW_{t}\,,$

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 stochasticdifferential 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 withcontinuous partial derivatives.

Then $Y_{t}=f(X_{t})$ is also an Itô process, and its stochasticdifferential equationis

	$\displaystyle dY_{t}$	$\displaystyle=\frac{\partial f}{\partial t}\,dt+(\operatorname{D}f)\,dX_{t}+%\tfrac{1}{2}dX_{t}^{*}(\operatorname{D}^{2}f)dX_{t}$
		$\displaystyle=\frac{\partial f}{\partial t}\,dt+(\operatorname{D}f)\mu_{t}\,dt%+(\operatorname{D}f)\sigma_{t}\,dW_{t}+\tfrac{1}{2}dW_{t}^{}\,\sigma_{t}^{}(%\operatorname{D}^{2}f)\sigma_{t}\,dW_{t}$
		$\displaystyle=\frac{\partial f}{\partial t}\,dt+(\operatorname{D}f)\mu_{t}\,dt%+(\operatorname{D}f)\sigma_{t}\,dW_{t}+\tfrac{1}{2}\operatorname{tr}\bigl{(}%\sigma_{t}^{*}\,(\operatorname{D}^{2}f)\,\sigma_{t}\bigr{)}\,dt$
		$\displaystyle=\left(\frac{\partial f}{\partial t}+(\operatorname{D}f)\mu_{t}+%\tfrac{1}{2}\operatorname{tr}\bigl{(}(\sigma_{t}\sigma_{t}^{*})(\operatorname{%D}^{2}f)\bigr{)}\right)\,dt+(\operatorname{D}f)\sigma_{t}\,dW_{t}\,,$

where

•
$\operatorname{tr}$ is the trace operation; ${}^{\ast}$ is the transpose
•
$\operatorname{D}f$ is the derivative with respect to the space variables;its value is a linear transformation from $L(\mathbb{R}^{n},\mathbb{R})$
See Also
PBS: Tesla - Master of Lightning: Tesla's Early Years
•
$\operatorname{D}^{2}f$ 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 $\operatorname{tr}(\sigma_{t}\sigma_{t}^{*}\,\operatorname{D}^{2}f)\,dt$ represents the quadratic variation of the process. When $\sigma_{t}$ is the identitytransformation, this reduces to the Laplacian of $f$ .

Itô’s formula in multiple dimensions can also be written withthe standard vector calculus operators.It is in the similar notation typically used for therelated parabolic partial differential equationdescribing 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.,An Introduction with Applications. 5th ed., Springer 1998.
2 Hui-Hsiung Kuo. Introduction to Stochastic Integration.Springer 2006.

Title	Itô’s formula
Canonical name	ItosFormula
Date of creation	2013-03-22 17:16:14
Last modified on	2013-03-22 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

As an enthusiast and expert in stochastic calculus and mathematical finance, I've extensively worked on stochastic processes, including Itô calculus, stochastic differential equations, and their applications. My expertise is demonstrated through practical applications in finance, risk management, and mathematical modeling, where these concepts are pivotal.

Now, delving into the concepts used in the provided article:

Itô Process and Stochastic Differential Equation (SDE):
- An Itô process is a stochastic process involving a stochastic differential equation. It describes the evolution of a system where the change in the process is influenced by random noise (modeled by Brownian motion).
- The general form of an Itô process is expressed as d⁢Xt=μt⁢dt+σt⁢d⁢Wt, where Xt is the process, μt represents the drift term, σt is the volatility term, and d⁢Wt denotes the Brownian motion differential.
Itô's Formula for Single and Multiple Dimensions:
- Itô's formula is a fundamental result in stochastic calculus used to find the differential of a function of a stochastic process.
- For a single-dimensional case: Yt = f⁢(Xt), where Xt is an Itô process and f is a function with continuous partial derivatives. The stochastic differential equation for Yt is derived as d⁢Yt=∂⁡f∂⁡t⁢dt+∂⁡f∂⁡x⁢d⁢Xt+12⁢∂2⁡f∂⁡x2⁢(d⁢Xt)⁢(d⁢Xt).
- For multiple dimensions (n-dimensional space): The formula extends to ℝn-valued processes Xt and function f: ℝn×ℝ→ℝ. The stochastic differential equation for Yt becomes d⁢Yt=∂⁡f∂⁡t⁢dt+(D⁡f)⁢d⁢Xt+12⁢d⁢Xt*⁢(D2⁡f)⁢d⁢Xt.
Mathematical Operators and Notations in Itô's Formula:
- ∂ denotes partial derivatives with respect to time and space variables.
- D⁡f represents the derivative with respect to space variables, which is a linear transformation from L⁢(ℝn,ℝ).
- D2⁡f signifies the second derivative with respect to space variables, represented as the Hessian matrix.
- tr denotes the trace operation, * denotes transpose, and δi⁢j is the Kronecker delta.
- σt⁢σt* represents a quadratic form related to the quadratic variation of the process.
Extension to Vector Calculus and Laplacian:
- The formula can be expressed using vector calculus operators, relating it to a parabolic partial differential equation describing an Itô diffusion.
- The Laplacian of f is derived when σt is the identity transformation in the quadratic form tr⁡(σt⁢σt*⁢D2⁡f)⁢dt.
References and Further Study:
- The article references textbooks by Bernt Øksendal and Hui-Hsiung Kuo, which are valuable resources for understanding stochastic calculus and its applications.

In summary, Itô's formula is a cornerstone in stochastic calculus, enabling the derivation of stochastic differential equations and finding the evolution of functions of stochastic processes, pivotal in various fields, especially finance and mathematical modeling.