The Kolmogorov forward equation, also known as the Fokker-Planck equation, describes the time evolution of the probability density function
\\( p(x,t) \\) of a stochastic process. It is particularly useful for continuous-time and continuous-state-space Markov processes.<\/p>\n\n\n\n
The Kolmogorov forward equation is given by:<\/p>\n\n\n\n
$$ \\frac{\\partial p(x,t)}{\\partial t} = -\\frac{\\partial}{\\partial x} \\left[ A(x,t) p(x,t) \\right] + \\frac{1}{2} \\frac{\\partial^2}{\\partial x^2} \\left[ B(x,t) p(x,t) \\right] $$<\/p>\n\n\n\n
where:<\/p>\n\n\n\n
Consider a stochastic process \\( X(t) \\) governed by the following It\u00f4 stochastic differential equation (SDE):<\/p>\n\n\n\n
$$ dX(t) = A(X(t), t) dt + \\sqrt{B(X(t), t)} dW(t) $$<\/p>\n\n\n\n
where \\( W(t) \\) is a Wiener process (or standard Brownian motion).<\/p>\n\n\n\n
To derive the Fokker-Planck equation, we start with the Chapman-Kolmogorov equation for the transition probability \\( p(x,t | x_0, t_0) \\):<\/p>\n\n\n\n
$$ p(x, t + \\Delta t | x_0, t_0) = \\int_{-\\infty}^{\\infty} p(x, t + \\Delta t | x’, t) p(x’, t | x_0, t_0) dx’ $$<\/p>\n\n\n\n
Assuming \\( \\Delta t \\) is infinitesimally small, we can expand \\( p(x, t + \\Delta t | x’, t) \\) using a Taylor series:<\/p>\n\n\n\n
$$ p(x, t + \\Delta t | x’, t) \\approx p(x, t | x’, t) + \\Delta t \\frac{\\partial}{\\partial t} p(x, t | x’, t) $$<\/p>\n\n\n\n
Similarly, expand \\( p(x’, t | x_0, t_0) \\) around \\( x \\):<\/p>\n\n\n\n
$$ p(x’, t | x_0, t_0) \\approx p(x, t | x_0, t_0) + (x’ – x) \\frac{\\partial}{\\partial x} p(x, t | x_0, t_0) + \\frac{1}{2} (x’ – x)^2 \\frac{\\partial^2}{\\partial x^2} p(x, t | x_0, t_0) $$<\/p>\n\n\n\n
Substituting these expansions into the Chapman-Kolmogorov equation and integrating over \\( x’ \\), we obtain:<\/p>\n\n\n\n
$$ p(x, t + \\Delta t | x_0, t_0) – p(x, t | x_0, t_0) \\approx \\Delta t \\frac{\\partial}{\\partial t} p(x, t | x_0, t_0) $$<\/p>\n\n\n\n
$$ = \\int_{-\\infty}^{\\infty} \\left[ p(x, t | x’, t) + \\Delta t \\frac{\\partial}{\\partial t} p(x, t | x’, t) \\right] \\left[ p(x’, t | x_0, t_0) \\right] dx’ $$<\/p>\n\n\n\n
Considering the contributions from the drift and diffusion terms:<\/p>\n\n\n\n
$$ \\begin{align} \\mathbb{E}[(x’ – x)] &= A(x, t) \\Delta t, \\ \\mathbb{E}[(x’ – x)^2] &= B(x, t) \\Delta t, \\end{align} $$<\/p>\n\n\n\n
and higher-order terms can be neglected as \\( \\Delta t \\to 0 \\). This yields:<\/p>\n\n\n\n
$$ \\frac{\\partial}{\\partial t} p(x, t) = – \\frac{\\partial}{\\partial x} \\left[ A(x, t) p(x, t) \\right] + \\frac{1}{2} \\frac{\\partial^2}{\\partial x^2} \\left[ B(x, t) p(x, t) \\right] $$<\/p>\n\n\n\n
Thus, we have derived the Kolmogorov forward equation.<\/p>\n\n\n\n
For an Ornstein-Uhlenbeck process, which is a type of Gaussian process, the SDE is given by:<\/p>\n\n\n\n
$$ dX(t) = \\theta (\\mu – X(t)) dt + \\sigma dW(t) $$<\/p>\n\n\n\n
where \\( \\theta \\), \\( \\mu \\), and \\( \\sigma \\) are constants. In this case:<\/p>\n\n\n\n
The Kolmogorov forward equation becomes:<\/p>\n\n\n\n
$$ \\frac{\\partial p(x,t)}{\\partial t} = \\frac{\\partial}{\\partial x} \\left[ \\theta (\\mu – x) p(x,t) \\right] + \\frac{\\sigma^2}{2} \\frac{\\partial^2}{\\partial x^2} p(x,t) $$<\/p>\n\n\n\n
This equation describes how the probability density function of the Ornstein-Uhlenbeck process evolves over time.<\/p>\n","protected":false},"excerpt":{"rendered":"
The Kolmogorov forward equation, also known as the Fokker-Planck equation, describes the time evolution of the<\/p>\n","protected":false},"author":1,"featured_media":458,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-456","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-differential-equation"],"_links":{"self":[{"href":"https:\/\/mp-superkler.com\/index.php?rest_route=\/wp\/v2\/posts\/456","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/mp-superkler.com\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mp-superkler.com\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mp-superkler.com\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/mp-superkler.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=456"}],"version-history":[{"count":3,"href":"https:\/\/mp-superkler.com\/index.php?rest_route=\/wp\/v2\/posts\/456\/revisions"}],"predecessor-version":[{"id":461,"href":"https:\/\/mp-superkler.com\/index.php?rest_route=\/wp\/v2\/posts\/456\/revisions\/461"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/mp-superkler.com\/index.php?rest_route=\/wp\/v2\/media\/458"}],"wp:attachment":[{"href":"https:\/\/mp-superkler.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=456"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mp-superkler.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=456"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mp-superkler.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=456"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}