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
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