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==basic probability theory==
 
==basic probability theory==
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* Let $(\Omega, \mathcal{F}, P)$ be probability space
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* [[Basic probability theory]]
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* A real-valued function $X : \Omega\to \mathbb{R}$ is called a random variable
 
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* let $A\subseteq \mathbb{R}$ be the range of $X$, $A=\{s|X(s)=x,s\in S\}$. We call $A$ the space of $X$
 
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* $\{X=x\}$ denote the subset $\{s|X(s)=x\}$ of $\mathbb{R}$
 
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* the induced probability measure $P_X : \mathbb{R}\to [0,1]$
 
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* probability density function $f : \mathbb{R}\to [0,\infty)$ of $X$ satisfies
 
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$$
 
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P_{X}(X\in A)=\int_A f(x)\, dx=1
 
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$$
 
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and
 
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$$
 
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P_{X}(X\in B)=\int_B f(x)\, dx
 
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$$
 
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for $B\subseteq A$.
 
  
 
==Ito SDE==
 
==Ito SDE==
 
;def
 
;def
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A stochastic process $X(t)$ is said to satisfy an Ito SDE, written as,
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A stochastic process <math>X(t)</math> is said to satisfy an Ito SDE, written as,
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$$
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:<math>
 
dX(t)= \alpha(X(t),t)dt+\beta(X(t),t)dW(t)\label{ito}
 
dX(t)= \alpha(X(t),t)dt+\beta(X(t),t)dW(t)\label{ito}
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$$
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</math>
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if for $t\ge 0$ it satisfies the integral equation,
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if for <math>t\ge 0</math> it satisfies the integral equation,
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$$
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:<math>
 
X(t)= X(0) +\int_{0}^{t}\alpha(X(\tau),\tau)\,d\tau+\int_{0}^{t}\beta(X(\tau),\tau)dW(\tau)
 
X(t)= X(0) +\int_{0}^{t}\alpha(X(\tau),\tau)\,d\tau+\int_{0}^{t}\beta(X(\tau),\tau)dW(\tau)
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$$
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</math>
 
===Kolmogorov equation===
 
===Kolmogorov equation===
 
* [[Fokker-Planck equation|Fokker-Planck equations]], also known as Fokker-Planck-Kolmogorov equations or forward Kolmogorov equations, are deterministic equations describing how probability density functions evolve
 
* [[Fokker-Planck equation|Fokker-Planck equations]], also known as Fokker-Planck-Kolmogorov equations or forward Kolmogorov equations, are deterministic equations describing how probability density functions evolve
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* let $p(x,t)$ be the p.d.f. of the stochastic process $X(t)$ satisfying \ref{ito}. Then
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* let <math>p(x,t)</math> be the p.d.f. of the stochastic process <math>X(t)</math> satisfying \ref{ito}. Then
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$$
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:<math>
 
\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}
 
\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}
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$$
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</math>
 
===multi-dimensional version===
 
===multi-dimensional version===
 
* see Klebaner2005
 
* see Klebaner2005
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   {\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},
 
   {\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}
 
\end{equation}
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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  
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where <math>X(t)=(X_1(t),X_2(t),\cdots,X_d(t))^T \in \mathbb{R}^d</math>, <math>f=(f_1, f_2,\cdots,f_d)^T: \mathbb{R}^d\to \mathbb{R}^d</math>, <math>\sigma=(\sigma_{ij})_{d\times n}: \mathbb{R}^d \to \mathbb{R}^{d\times n}</math>. <math>B(t)</math> is an <math>n</math>-dimensional Brownian motion,  and <math>f</math> and <math>g</math> satisfy certain smoothness conditions.    The probability density function <math>p(x,t)</math> for the solution <math>X(t)</math> in (\ref{s1_000}) can be expressed as  
 
\begin{align}\label{s1_001}
 
\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],
 
   \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}
 
   \end{align}
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where $D_{ij}(x)= \sum_{k=1}^n \sigma_{ik}(x)\sigma_{kj}(x)$.
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where <math>D_{ij}(x)= \sum_{k=1}^n \sigma_{ik}(x)\sigma_{kj}(x)</math>.
  
 
==example==
 
==example==
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* Loewner equantion
 
* Loewner equantion
  
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==related items==
 
==related items==
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[[분류:math and physics]]
 
[[분류:math and physics]]
 
[[분류:probability]]
 
[[분류:probability]]
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[[분류:migrate]]
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== 노트 ==
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===말뭉치===
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# 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.<ref name="ref_680e2caa">[https://quantum.phys.cmu.edu/QIP/ito_calculus.pdf Itˆo calculus in a nutshell]</ref>
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# Itos lemma is often used in Ito calculus to nd the dierentials of a stochastic process that depends on time.<ref name="ref_1f963234">[https://math.uchicago.edu/~may/REU2017/REUPapers/Yoo.pdf Stochastic calculus and black-scholes model]</ref>
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# The Ito calculus is about systems driven by white noise.<ref name="ref_fc563a79">[https://www.math.nyu.edu/~goodman/teaching/StochCalc2007/notes/l7.pdf Stochastic calculus notes, lecture 7]</ref>
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# This test of survival under the limit dt=0 and sum determines the rules ( Ito calculus ) at the beginning of this section.<ref name="ref_458abe97">[http://www.lukoe.com/finance/quantNotes/Ito_calculus_.html Ito calculus.]</ref>
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# 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.<ref name="ref_507f2020">[http://www-stat.wharton.upenn.edu/~steele/Publications/PDF/EASItoCalculus.pdf It ˆo calculus]</ref>
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# Abstract The Functional Ito calculus is a non-anticipative functional calculus which extends the Ito calculus to path-dependent functionals of stochas- tic processes.<ref name="ref_6196a20d">[http://rama.cont.perso.math.cnrs.fr/BarcelonaLectureNotes.pdf Functional ito calculus]</ref>
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# The Functional Ito calculus has led to various applications in the study of path-dependent functionals of stochastic processes.<ref name="ref_6196a20d" />
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===소스===
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<references />
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== 메타데이터 ==
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===위키데이터===
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* ID :  [https://www.wikidata.org/wiki/Q947053 Q947053]
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===Spacy 패턴 목록===
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* [{'LOWER': 'ito'}, {'LEMMA': 'calculus'}]
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* [{'LOWER': 'itō'}, {'LEMMA': 'calculus'}]

2022년 8월 21일 (일) 19:13 기준 최신판

introduction

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


computational resource

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.[1]
  2. Itos lemma is often used in Ito calculus to nd the dierentials of a stochastic process that depends on time.[2]
  3. The Ito calculus is about systems driven by white noise.[3]
  4. This test of survival under the limit dt=0 and sum determines the rules ( Ito calculus ) at the beginning of this section.[4]
  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.[5]
  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.[6]
  7. The Functional Ito calculus has led to various applications in the study of path-dependent functionals of stochastic processes.[6]

소스

  1. ↑ Itˆo calculus in a nutshell
  2. ↑ Stochastic calculus and black-scholes model
  3. ↑ Stochastic calculus notes, lecture 7
  4. ↑ Ito calculus.
  5. ↑ It ˆo calculus
  6. ↑ 6.0 6.1 Functional ito calculus

메타데이터

위키데이터

Spacy 패턴 목록

  • [{'LOWER': 'ito'}, {'LEMMA': 'calculus'}]
  • [{'LOWER': 'itō'}, {'LEMMA': 'calculus'}]
원본 주소 "https://wiki.mathnt.net/index.php?title=Ito_calculus&oldid=53072"