Ito calculus

introduction

• start with Brownian motion
• http://www.mathematica-journal.com/issue/v9i4/contents/StochasticIntegrals/StochasticIntegrals_1.html
• http://www-stat.wharton.upenn.edu/~steele/Publications/PDF/AoMtSC.pdf

basic probability theory

• Let $(\Omega, \mathcal{F}, P)$ be probability space
• A real-valued function $X : \Omega\to \mathbb{R}$ is called a random variable
• let $A\subseteq \mathbb{R}$ be the range of $X$, $A=\{s|X(s)=x,s\in S\}$. We call $A$ the space of $X$
• $\{X=x\}$ denote the subset $\{s|X(s)=x\}$ of $\mathbb{R}$
• the induced probability measure $P_X : \mathbb{R}\to [0,1]$
• probability density function $f : \mathbb{R}\to [0,\infty)$ of $X$ satisfies

$$ P_{X}(X\in A)=\int_A f(x)\, dx=1 $$ and $$ P_{X}(X\in B)=\int_B f(x)\, dx $$ for $B\subseteq A$.

Ito SDE

def

A stochastic process $X(t)$ is said to satisfy an Ito SDE, written as, $$ dX(t)= \alpha(X(t),t)dt+\beta(X(t),t)dW(t)\label{ito} $$ if for $t\ge 0$ it satisfies the integral equation, $$ X(t)= X(0) +\int_{0}^{t}\alpha(X(\tau),\tau)\,d\tau+\int_{0}^{t}\beta(X(\tau),\tau)dW(\tau) $$

Kolmogorov equation

• Fokker-Planck equations, also known as Fokker-Planck-Kolmogorov equations or forward Kolmogorov equations, are deterministic equations describing how probability density functions evolve
• let $p(x,t)$ be the p.d.f. of the stochastic process $X(t)$ satisfying \ref{ito}. Then

$$ \frac{\partial p}{\partial t} = -\frac{\partial(\alpha(x,t)p)}{\partial x}+\frac{1}{2}\frac{\partial^2 (\beta^2(x,t)p)}{\partial x^2} $$

multi-dimensional version

• see Klebaner2005
• consider the following Ito SDE

\begin{equation}\label{s1_000} {\rm d} X(t) = f(X(t)){\rm d}t + \sigma(X(t)) {\rm d} B(t), \qquad X(0) = x_0\in \mathbb{R}^{d}, \end{equation} where $X(t)=(X_1(t),X_2(t),\cdots,X_d(t))^T \in \mathbb{R}^d$, $f=(f_1, f_2,\cdots,f_d)^T: \mathbb{R}^d\to \mathbb{R}^d$, $\sigma=(\sigma_{ij})_{d\times n}: \mathbb{R}^d \to \mathbb{R}^{d\times n}$. $B(t)$ is an $n$-dimensional Brownian motion, and $f$ and $g$ satisfy certain smoothness conditions. The probability density function $p(x,t)$ for the solution $X(t)$ in (\ref{s1_000}) can be expressed as \begin{align}\label{s1_001} \frac{\partial p(x, t)}{\partial t} &= - \sum^{d}_{i=1}\frac{\partial }{\partial x_{i}} \left[f_{i}(x) p(x, t)\right] + \sum^{d}_{i,j=1} \frac{\partial ^2}{\partial x_{i}\partial x_{j}}\left[D_{ij}(x)p(x, t) \right], \end{align} where $D_{ij}(x)= \sum_{k=1}^n \sigma_{ik}(x)\sigma_{kj}(x)$.

example

• Loewner equantion

 

related items

• Stochastic PDE
• Brownian motion

computational resource

• http://mathematica.stackexchange.com/questions/30558/solving-a-stochastic-differential-equation?rq=1
• http://mathematica.stackexchange.com/questions/83645/martingale-pricing-simulation-random-walk-stock-price