Talk:Kolmogorov backward equations (diffusion)

Dear 194.129.108.69 (with no talk page)

I do not think that your change of 8:43 1 September 2006 is correct. You moved Mu and Sigma inside the differentiation operator. In the two sources that I used

www.cs.cmu.edu/~chal/Shreve/chap16.pdf (see page 180)

www.maths.ox.ac.uk/~hambly/PDF/O10/lecture7.pdf

they are outside. Please give a source (book or link) for your version of the equation. Encyclops 22:45, 1 September 2006 (UTC)

Duplication ?

In reference to the tag added by 129.234.252.66, I don't understand exactly what the duplicate information is. Can you please be specific ? Are you saying the KFE should not be mentioned, only the KBE? Encyclops (talk) 16:27, 21 October 2010 (UTC)

Hgsolari writes

KFE and KBE refer to the four highlighted equations in Kolmogorov 1931 seminal paper:

Über die analytisehen Methoden in der Wahrseheinliehkeitsreehnung. 

http://www.springerlink.com/content/v724507673277262/fulltext.pdf

Two pairs KFE-KBE, but "forward" or "backward" is not used, the real difference is whether the time-differentiation is with respect to the final or initial time.

Hence, there is some ambiguity in the origin.

There are at least three serious problems with this stub:

1. The equation presented is not precisely Kolmogorov's equation as written in the 1931 paper, which is more general. The generality loss relates to the 3rd problem.

2. Ambiguity. Kolmogorov Backward (Forward) Equations refer to different mathematical objects in different mathematical communities. Within Markov Jumps theory, KFE are the mathematical version of physicists' Master equations. Historically, the name KBE appeared apparently in W Feller's 1957 paper: On Boundaries and Lateral Conditions for the Kolmogorov Differential Equations Author(s): William Feller Source: The Annals of Mathematics, Second Series, Vol. 65, No. 3 (May, 1957), pp. 527-570 Published by: Annals of Mathematics Stable URL: http://www.jstor.org/stable/1970064

For the Stochastic Differential Equations community, KFE and KBE refer to the Kolmogorv Differential Equations (Feller) related to the diffusion (Feller) or dispersion (Kolmogorov) process represented in physics by the Fokker-Plack equation. Such use of KFE was apparently pioneered in Some Problems of Stochastic Processes in Genetics Author(s): Motoo Kimura Source: The Annals of Mathematical Statistics, Vol. 28, No. 4 (Dec., 1957), pp. 882-901 Published by: Institute of Mathematical Statistics Stable URL: http://www.jstor.org/stable/2237051

3. The page referes to KFE as "the adjoint" of the KBE, this is true ONLY in the simplest cases. This matter was discussed by Feller: DIFFUSION PROCESSES IN ONE DIMENSION, 1954 (there is a free copy in the WeB) Incidentally, he calls the equation in the page: Backward Equation of Diffusion Theory.

In conclusion, I find this page misleading and lacking fundamental references. I propose to withdraw it for the time being and later to produce a page on Kolmogorov Equations including motivation, historical facts, links to other pages, and disambiguation.

Hgsolari (talk) 12:19, 7 December 2010 (UTC)

You've created quite a spaghetti mess of oddly named and oddly styled articles and redirects. I've made some attempt to straighten them out some. There's a bot that will try to fix the multiple-redirects shortly, I think. Dicklyon (talk) 01:37, 17 December 2010 (UTC)

Incorrect sign?

I'm coming at this as a beginner and find it hard to reconcile the Kolmogorov backward equation (diffusion) given

${\frac {\partial }{\partial t}}p(x,t)=\mu (x,t){\frac {\partial }{\partial x}}p(x,t)+{\frac {1}{2}}\sigma ^{2}{\frac {\partial ^{2}}{\partial x^{2}}}p(x,t)$

with intuition and with the Feynman–Kac formula. It seems to me that it should instead be given as

${\frac {\partial }{\partial t}}p(x,t)+\mu (x,t){\frac {\partial }{\partial x}}p(x,t)+{\frac {1}{2}}\sigma ^{2}{\frac {\partial ^{2}}{\partial x^{2}}}p(x,t)=0$

i.e., the sign for the partial derivative with respect to time needs to be flipped.

Intuitively, if at time $s>t$, $u(x,s)$ is an indicator function, which has infinite curvature on the boundary of $B$, then by stepping backwards in time we would expect $u(x,s-\Delta s)$ to be smoother than $u(x,s)$ and to still be bounded between 0 and 1. The equation provided leads to increasing smoothness as we step forwards in time, i.e., decreasing smoothness as we step backwards.

When using the Feynman–Kac formula (as suggested in the current article) using $\psi (x)=u_{s}(x)$, and $V(x,t)=f(x,t)=0$, we obtain the second formula I give above (with the opposite sign to that used in the KBE diffusion article).

Do I have a misunderstanding or is a correction needed? Ts4079 (talk) 08:52, 23 July 2024 (UTC)

Your intuition gave the right signs for a dispersive process. However, without the derivative on the outside of your products, your fluxes in the limit sigma to zero are not locally conservative. The form of this equation is the (incomplete) adjoint. 129.93.161.205 (talk) 20:03, 29 August 2024 (UTC)

@129.93.161.205 thanks I again confess to being a beginner and not really understanding your reply. If we focus on the pure matter of the sign in the PDE, does the article need correction? Ts4079 (talk) 14:13, 30 August 2024 (UTC)

The sign has now been changed by @MissPlacement - thanks Ts4079 (talk) 09:27, 4 September 2024 (UTC)

I think there is still a sign error. Ts4079 gave the correct sign in their equation above (from July 23, 2024), but the current version is not correct. 73.25.114.33 (talk) 17:36, 23 October 2024 (UTC)

The sign has been fixed by @Ludwigwinkler 73.25.114.33 (talk) 17:39, 28 October 2024 (UTC)