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

• 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

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

노트

말뭉치

1. Vlad Gheorghiu (CMU) Ito calculus in a nutshell April 7, 2011 4 / 23 Elementary random processes If we now calculate expectations of Si it does matter what information we have.
2. Itos lemma is often used in Ito calculus to nd the dierentials of a stochastic process that depends on time.
3. The Ito calculus is about systems driven by white noise.
4. This test of survival under the limit dt=0 and sum determines the rules ( Ito calculus ) at the beginning of this section.
5. 1. First Contact with Ito Calculus From the practitioners point of view, the Ito calculus is a tool for manip- ulating those stochastic processes which are most closely related to Brow- nian motion.
6. Abstract The Functional Ito calculus is a non-anticipative functional calculus which extends the Ito calculus to path-dependent functionals of stochas- tic processes.
7. The Functional Ito calculus has led to various applications in the study of path-dependent functionals of stochastic processes.

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• [{'LOWER': 'ito'}, {'LEMMA': 'calculus'}]
• [{'LOWER': 'itō'}, {'LEMMA': 'calculus'}]