Test $\int_{0}^{\infty} \frac{1}{x-1} \mathrm{d}x$ [quotetheorem:32] [theorem:Dominated Convergence] [/theorem] [proof] First we do this [claim] aha [/claim] [proof] proof me [/proof] [/proof] # Itô's Lemma in $\mathbb{R}^n$ [motivation] In stochastic calculus, sample paths of Brownian motion are almost surely nowhere differentiable, so classical chain rules do not apply. Nevertheless, many stochastic models involve compositions of smooth functions with stochastic processes. Itô's lemma provides the precise differential rule that replaces the classical chain rule when a function is evaluated along an Itô process. The additional second-order term arises from the quadratic variation of Brownian motion and is fundamental in stochastic differential equations and mathematical finance. [/motivation] ## Formal Definition [definition:ItoProcess] Let $(\Omega, \mathcal{F}, (\mathcal{F}_t)_{t \ge 0}, \mathbb{P})$ be a filtered probability space satisfying the usual conditions. Let $T > 0$ and $n,m \in \mathbb{N}$. Let \begin{align*} W : [0,T] \times \Omega &\to \mathbb{R}^m \\ (t,\omega) &\mapsto W(t,\omega) \end{align*} be an $m$-dimensional Brownian motion adapted to $(\mathcal{F}_t)_{t \ge 0}$. Let \begin{align*} b : [0,T] \times \mathbb{R}^n &\to \mathbb{R}^n \end{align*} and \begin{align*} \sigma : [0,T] \times \mathbb{R}^n &\to \mathbb{R}^{n \times m} \end{align*} be measurable functions such that, for each $i \in \{1,\dots,n\}$ and $j \in \{1,\dots,m\}$, the component functions \begin{align*} b_i : [0,T] \times \mathbb{R}^n &\to \mathbb{R} \\ \sigma_{ij} : [0,T] \times \mathbb{R}^n &\to \mathbb{R} \end{align*} are jointly continuous and locally Lipschitz in the spatial variable. An $\mathbb{R}^n$-valued stochastic process \begin{align*} X : [0,T] \times \Omega &\to \mathbb{R}^n \end{align*} is called an Itô process if, for each $i \in \{1,\dots,n\}$, \begin{align*} X_i(t) = X_i(0) + \int_0^t b_i(s,X(s)) \, ds + \sum_{j=1}^m \int_0^t \sigma_{ij}(s,X(s)) \, dW_j(s), \end{align*} where the first integral is a Lebesgue integral with respect to $ds$ and the second integral is an Itô integral with respect to $W_j$. [/definition] ## Examples ### Example 1: One-Dimensional Geometric Brownian Motion Let $n=m=1$, constants $\mu, \sigma \in \mathbb{R}$, and define \begin{align*} b : [0,T] \times \mathbb{R} &\to \mathbb{R} \\ (t,x) &\mapsto \mu x, \end{align*} \begin{align*} \sigma : [0,T] \times \mathbb{R} &\to \mathbb{R} \\ (t,x) &\mapsto \sigma x. \end{align*} The associated Itô process $X$ satisfies \begin{align*} X(t) = X(0) + \int_0^t \mu X(s)\, ds + \int_0^t \sigma X(s)\, dW(s). \end{align*} ### Example 2: Multidimensional Diffusion For $n=2$, $m=2$, let $b_i$ and $\sigma_{ij}$ be smooth bounded functions. The resulting process \begin{align*} X : [0,T] \times \Omega \to \mathbb{R}^2 \end{align*} is an Itô diffusion in $\mathbb{R}^2$. ## Key Results [theorem:ItoLemma] Let $X : [0,T] \times \Omega \to \mathbb{R}^n$ be an Itô process as in Definition [ItoProcess]. Let \begin{align*} f : [0,T] \times \mathbb{R}^n &\to \mathbb{R} \end{align*} be a function of class $C^{1,2}$, meaning that $f$ is continuously differentiable in $t$ and twice continuously differentiable in $x$. Then the real-valued process \begin{align*} Y : [0,T] \times \Omega &\to \mathbb{R} \\ (t,\omega) &\mapsto f(t,X(t,\omega)) \end{align*} satisfies, for all $t \in [0,T]$, \begin{align*} Y(t) &= Y(0) + \int_0^t \partial_t f(s,X(s))\, ds + \sum_{i=1}^n \int_0^t \partial_{x_i} f(s,X(s))\, b_i(s,X(s))\, ds \\ &\quad + \frac{1}{2} \sum_{i=1}^n \sum_{k=1}^n \int_0^t \partial_{x_i x_k} f(s,X(s)) \left( \sum_{j=1}^m \sigma_{ij}(s,X(s)) \sigma_{kj}(s,X(s)) \right) ds \\ &\quad + \sum_{i=1}^n \sum_{j=1}^m \int_0^t \partial_{x_i} f(s,X(s)) \sigma_{ij}(s,X(s)) \, dW_j(s). \end{align*} [/theorem] The additional second-order term arises from the quadratic variation identity \begin{align*} dW_j(t)\, dW_\ell(t) = \delta_{j\ell}\, dt, \end{align*} where $\delta_{j\ell}$ denotes the Kronecker symbol. ## References K. Itô, *On Stochastic Differential Equations* (1951). B. Øksendal, *Stochastic Differential Equations* (2003). I. Karatzas, S. Shreve, *Brownian Motion and Stochastic Calculus* (1991).