Kramers Moyal Development

In physics, the Kramers-Moyal development is a Taylor development of a master equation , which transforms the master equation as an integro differential equation into a partial differential equation. The following is developed according to the step size : ${\displaystyle \Delta x}$

${\displaystyle {\frac {\partial p(x,t)}{\partial t}}=\sum \limits _{n=1}^{\infty }{\frac {(-1)^{n}}{n!}}{\frac {\partial ^{n}}{\partial x^{n}}}{\Big [}a_{n}(x)p(x,t){\Big ]}}$

With

${\displaystyle a_{n}(x)=\int \limits _{-\infty }^{\infty }(\Delta x)^{n}W(x,\Delta x)\,\mathrm {d} (\Delta x)}$

It describes the development over time of a location- dependent probability of stay . Continuously distributed step sizes in space and time are considered. is the transition probability rate. Breaking off the series in the second order results in the Fokker-Planck equation . ${\displaystyle x}$ ${\displaystyle p}$ ${\displaystyle \Delta x=x-x'}$ ${\displaystyle \Delta t}$ ${\displaystyle W(x,\Delta x):=W(x'|x)}$

The development is named after Hendrik Anthony Kramers and José Enrique Moyal .

The Pawula theorem says that if the third term of the expansion vanishes, all higher terms vanish too. If the development does not break off with the third term, it contains an infinite number of contributions. .

Individual evidence

1. ↑ Wolfgang Paul, Jörg Baschnagel: Stochastic Processes: From Physics to Finance . Springer, 2013, ISBN 3-319-00327-5 , pp. 47 ( limited preview in Google Book search).
2. ↑ Jochen Veith: Evaluation of options under the Coherent Market Hypothesis . Springer, 2006, ISBN 3-8350-0419-0 , pp. 28 f . ( limited preview in Google Book search).
3. ↑ The Fokker-Planck Equation: Methods of Solution and Applications, Hannes Risken, page 70, https://books.google.de/books?id=dXvpCAAAQBAJ&lpg=PA70&ots=1IZwvn5hYJ&dq=Pawula-Theorem&hl=de&pg=PA70#v=onepage&q = Pawula theorem & f = false