In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a Feller-continuous process is a continuous-time
stochastic process
In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables. Stochastic processes are widely used as mathematical models of systems and phenomena that appea ...
for which the
expected value
In probability theory, the expected value (also called expectation, expectancy, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a l ...
of suitable statistics of the process at a given time in the future depend continuously on the initial condition of the process. The concept is named after
Croatia
, image_flag = Flag of Croatia.svg
, image_coat = Coat of arms of Croatia.svg
, anthem = "Lijepa naša domovino"("Our Beautiful Homeland")
, image_map =
, map_caption =
, capit ...
n-American
mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems.
Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change.
History
On ...
William Feller
William "Vilim" Feller (July 7, 1906 – January 14, 1970), born Vilibald Srećko Feller, was a Croatian-American mathematician specializing in probability theory.
Early life and education
Feller was born in Zagreb to Ida Oemichen-Perc, a Croa ...
.
Definition
Let ''X'' :
, +∞) × Ω → R''n'', defined on a probability space (Ω, Σ, P), be a stochastic process. For a point ''x'' ∈ R
''n'', let P
''x'' denote the
law
Law is a set of rules that are created and are enforceable by social or governmental institutions to regulate behavior,Robertson, ''Crimes against humanity'', 90. with its precise definition a matter of longstanding debate. It has been vario ...
of ''X'' given initial value ''X''
0 = ''x'', and let E
''x'' denote expectation with respect to P
''x''. Then ''X'' is said to be a Feller-continuous process if, for any fixed ''t'' ≥ 0 and any bounded function">bounded
Boundedness or bounded may refer to:
Economics
* Bounded rationality, the idea that human rationality in decision-making is bounded by the available information, the cognitive limitations, and the time available to make the decision
* Bounded e ...
is simply ''g''(''x''), which, by hypothesis, depends continuously upon ''x''.
* Every