Expected value function

An expected value function or mean value function is a real-valued function in the theory of stochastic processes , a branch of stochastics . It can be assigned to any integrable process and clearly shows which values ​​the process assumes at which point in time on average. Expected value functions are used, for example, in the theory of Gaussian processes , which are uniquely determined by the covariance function and the expected value function, or in the investigation of inhomogeneous Poisson processes .

definition

An integrable stochastic process is given . ${\displaystyle (X_{t})_{t\in T}}$

Then the function is called

${\displaystyle M\colon T\to \mathbb {R} }$

defined by

${\displaystyle M(t):=\operatorname {E} (X_{t})}$

the mean value function of the process. Here denotes the expected value of the random variable . ${\displaystyle \operatorname {E} (X)}$ ${\displaystyle X}$

If the index set is interpreted as time, the expected value function assigns the mean value of the stochastic process to each point in time. ${\displaystyle T}$ ${\displaystyle t}$

example

Be a martingale about filtration . By definition then applies to everyone ${\displaystyle (X_{t})_{t\geq 0}}$ ${\displaystyle \mathbb {F} =({\mathcal {F}}_{t})_{t\geq 0}}$ ${\displaystyle s>t}$

${\displaystyle \operatorname {E} (X_{s}|{\mathcal {F}}_{t})=X_{t}}$.

By forming the expected value and the calculation rules for the conditional expected value , one obtains

${\displaystyle \operatorname {E} (X_{s})=\operatorname {E} (\,\operatorname {E} (X_{s}|{\mathcal {F}}_{t}))=\operatorname {E} (X_{t})}$,

so

${\displaystyle M(t)\equiv \operatorname {E} (X_{0})}$.

Martingales thus have constant functions as expected value functions. Analogous considerations also show that submartingales have monotonically increasing expectation functions and supermartingales have monotonically decreasing expectation functions.

literature

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

Web links

• Yu.A. Rozanov, DV Anosov: Gaussian Process . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).

Individual evidence

1. ↑ Meintrup, Schäffler: Stochastics. 2005, p. 345.
2. ↑ Meintrup, Schäffler: Stochastics. 2005, p. 292.