"Ito calculus"의 두 판 사이의 차이
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==Ito SDE== | ==Ito SDE== | ||
;def | ;def | ||
| − | A stochastic process | + | A stochastic process <math>X(t)</math> is said to satisfy an Ito SDE, written as, |
| − | + | :<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} | ||
| − | + | </math> | |
| − | if for | + | if for <math>t\ge 0</math> it satisfies the integral equation, |
| − | + | :<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) | ||
| − | + | </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 | ||
| − | * let | + | * let <math>p(x,t)</math> be the p.d.f. of the stochastic process <math>X(t)</math> satisfying \ref{ito}. Then |
| − | + | :<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} | ||
| − | + | </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} | ||
| − | where | + | 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} | ||
| − | where | + | where <math>D_{ij}(x)= \sum_{k=1}^n \sigma_{ik}(x)\sigma_{kj}(x)</math>. |
==example== | ==example== | ||
2020년 11월 16일 (월) 05:26 판
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