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

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

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