Itô’s formula

0.1 Case of single space dimension

Let $X_t$ be an Itô process satisfying the stochasticdifferential 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 stochasticdifferential equationis

$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 ${ℝ}^n$-valued Itô process satisfying the stochasticdifferential 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 ${ℝ}^n$-valued and ${\sigma }_t$ is $L({ℝ}^m,{ℝ}^n)$-valued.

Let $f:{ℝ}^n\times ℝ\to ℝ$ be a function withcontinuous partial derivatives.

Then $Y_t=f(X_t)$ is also an Itô process, and its stochasticdifferential equationis

$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

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

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

• ${\mathrm{D}}^{2}f$ 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 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:

$$d{Y}_t=\left(\frac{\partial f}{\partial t}+{\mu }_t\cdot \nabla f+\frac{1}{2}\left(\nabla \cdot ({\sigma }_t{\sigma }_t^{*})\nabla \right)f\right)dt+({\sigma }_{t}d{W}_t)\cdot \nabla f.$$