Feller trial

In the theory of stochastic processes, Feller processes are homogeneous Markov processes in continuous time with general state spaces, the transition probabilities of which meet certain continuity requirements , the so-called Feller continuity , and which can be accessed via the functional analytical Hille-Yosida semigroup theory .

Basics

The basic idea of ​​the theory is the Chapman-Kolmogorow equations of the transition probabilities ${\displaystyle P_{s}(\cdot ,\cdot )}$

${\displaystyle P_{s+t}(x,A)=\int P_{t}(y,A)P_{s}(x,dy)\quad s,t\geq 0,A{\text{ messbar}}}$

for by point by point on by ${\displaystyle C_{0}}$

${\displaystyle Tf(x):=\int f(y)\,dP(x,dy),\quad f\in C_{0}}$

to write defined operators as semigroup properties

${\displaystyle T_{t+s}=T_{t}\circ T_{s}\;\,\forall s,t\geq 0,\quad T_{0}=\mathbf {I} .}$

Here denotes the space of continuous functions that vanish in infinity and the identical mapping . The semigroup property suggests the existence of an operator which records and transfers the change behavior in infinitesimal time ${\displaystyle C_{0}}$ ${\displaystyle \mathbf {I} }$ ${\displaystyle A}$

${\displaystyle T_{t}=\mathrm {e} ^{tA}}$

enables the reconstruction of long-term behavior. The difficulty is that the operator is restricted only for pure jump processes and thus the generalized exponential function is defined. ${\displaystyle A}$

In order for the long-term behavior to be reconstructed, the Feller property

${\displaystyle T_{t}C_{0}\subset C_{0},}$

${\displaystyle T_{t}f(x)\to f(x),\quad t\to 0}$

must be assumed, from which it then follows that a positive contractive strongly continuous semigroup and the theorem of Hille and Yosida can be applied. Such a semigroup is called a Feller semigroup. Depending on the context, different variants of this Feller continuity property are considered. ${\displaystyle T}$

A persistent difficulty is that Hille and Yosida's theorem relates the semigroup to the closure of and this is not always easy to determine. In (Feller) diffusion processes, for example, restrictions of the generator have the simpler form of differential operators of the second order. ${\displaystyle A}$

history

The access to Markov processes via semigroups made in the theory of the Feller processes can be found implicitly in the core of Andrei Nikolajewitsch Kolmogorov's pioneering work on analytical methods in probability theory from 1931.

William Feller , who had already dealt with the topic before, took up Kolmogorov's analytical approach in works from 1952 and 1954, in which the dynamics of the process were recorded using differential equations of the density functions of the transition probabilities, and was able to go back to Hille and Yosida with the help of that Half- group theory fully characterize one-dimensional diffusion processes (Feller processes with continuous paths) via the infinitesimal producers of the corresponding half-groups.

EB Dynkin initiated and was the driving force behind the systematic expansion of the theory from 1954 on for general state spaces and transition probabilities, which represent a strongly continuous semi-group of positive contraction operators. Rogers and Williams therefore propose the designation Feller-Dynkin process for such processes, whereby the strong continuity of the semigroup follows from the Feller continuity of the transition probabilities under suitable conditions.

Kallenberg gives an outline of the story.

Important results

For a given Feller semigroup, a stochastic process with càdlàg paths (JR Kinney, 1953) can be constructed which fulfills a form of the strong Markov property (EB Dynkin, AA Juschkewitsch, RM Blumenthal), and the Dynkin formula is the consequence for stop times. Feller processes with continuous paths, the generator of which is defined on the smooth functions with compact support, are (Feller) diffusion processes whose behavior can be described by the local properties drift, scattering and loss rate ( killing ) in the form of a second-order elliptical differential operator .

literature

• Olav Kallenberg: Foundations of Modern Probability , 2nd edition. Springer, New York 2002, ISBN 0-387-95313-2 , p. 585.

Web links

• SN Smirnov: Feller process in Encyclopaedia of Mathematics (Springer Online Reference Works, engl.)

Individual evidence

1. ^ LCG Rogers and David Williams: Diffusions, Markov Processes and Martingales. Vol. 1 . Cambridge University Press, Cambridge, 2000.
2. ↑ AN Kolmogorow: About the analytical methods in the calculation of probability . In: Mathematische Annalen No. 104 , 1936, pp. 415–458.
3. ^ Olav Kallenberg: Foundations of Modern Probability . 2nd edition, Springer, New York 2002, ISBN 0-387-95313-2 , p. 585.
4. ^ William Feller: Diffusion processes in one dimension. In: Trans. Amer. Math. Soc. No. 77, 1954, pp. 1-13.
5. ↑ Kallenberg, ibid. P. 585f.