Jump process

A jump process is a special stochastic process and thus an object of investigation of probability theory , a branch of mathematics . Jump processes are clearly characterized by the fact that their value remains constant for a certain (random) time in order to then make a jump to another value, on which they remain for a while. In the simplest case of a jump process with the index set and the state set , the paths of a jump process form a step function . ${\ displaystyle \ mathbb {R}}$

definition

Two paths of a Poisson process , a typical hop process

A stochastic process with index set and values in . ${\ displaystyle X = (X_ {t}) _ {t \ in T}}$ ${\ displaystyle T}$ ${\ displaystyle E}$

Then a jump process is called if the paths of the process, i.e. the mappings ${\ displaystyle X}$

{\ displaystyle W _ {\ omega} \ colon T \ to E}

,

defined by

{\ displaystyle W _ {\ omega} (t): = X_ {t} (\ omega)}

are piecewise constant.

Examples

The counting processes , to which the Poisson process also belongs, are a large class of jump processes . These clearly count the number of events that have occurred up to a certain point in time, similar to a Geiger counter . Each time an event occurs, they jump up by one.

comment

Degenerate cases are also possible with jump processes and must be explicitly excluded in case of doubt. One of these special cases is a so-called explosion . The jump process has an infinite number of jumps (upwards) in a finite time.

A possible path for such an explosion would be given by

{\ displaystyle W (t): = n \;}

For

{\ displaystyle \; t \ in [- {\ tfrac {1} {n}}; - {\ tfrac {1} {n + 1}})}

with . Such explosions occur, for example, when modeling patient admissions in a hospital when an epidemic breaks out. Patients are admitted to the hospital at ever shorter intervals. In the example above, the time interval between patient and patient would be exactly time units. The number of occupied beds at the time (before the explosion) is given by. ${\ displaystyle t \ in [-1,0)}$ ${\ displaystyle n-1}$ ${\ displaystyle n}$ ${\ displaystyle {\ tfrac {1} {n}}}$ ${\ displaystyle t}$ ${\ displaystyle W (t)}$

Individual evidence

↑ Yu.M. Kabanov: Jump Process . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).
↑ David Meintrup, Stefan Schäffler: Stochastics . Theory and applications. Springer-Verlag, Berlin Heidelberg New York 2005, ISBN 3-540-21676-6 , pp. 273-274 , doi : 10.1007 / b137972 .

[1] Yu.M. Kabanov: Jump Process . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).

[2] David Meintrup, Stefan Schäffler: Stochastics . Theory and applications. Springer-Verlag, Berlin Heidelberg New York 2005, ISBN 3-540-21676-6 , pp. 273-274 , doi : 10.1007 / b137972 .