Càdlàg function

A Càdlàg function (also Cadlag ) is a special real-valued function that is used in stochastics , for example . Càdlàg is a French acronym (in French, continue à droite, limite à gauche, " steady on the right , with limit values ​​from the left"). There is also the RCLL ( right continuous, left limits ) derived from the English . Analogously one speaks of Càglàd functions (or Làdcàg functions ) ( continue à gauche, limite à droite ).

definition

Distribution functions are examples of Càdlàg functions

Be a Polish room such as . One function ${\displaystyle E}$ ${\displaystyle \mathbb {R} ^{n}}$

${\displaystyle f\colon [0,\infty )\to E}$

called

• Càdlàg function, if for all the function in is continuous on the right and the limit on the left in exists and is finite. ${\displaystyle x\in [0,\infty )}$ ${\displaystyle f}$ ${\displaystyle x}$ ${\displaystyle x}$
• Càglàd function, if for all the function in is continuous on the left and the limit on the right in exists and is finite. ${\displaystyle x\in [0,\infty )}$ ${\displaystyle f}$ ${\displaystyle x}$ ${\displaystyle x}$

The space of all Càdlàg functions on an interval is often referred to as. ${\displaystyle f\colon I\to \mathbb {R} ^{d}}$ ${\displaystyle I=[a,b]}$ ${\displaystyle D([a,b])}$

Applications in stochastics

The distribution function of a real random variable is always a Càdlàg function. ${\displaystyle F(x)=P(X\leq x)}$ ${\displaystyle X}$

A stochastic process is called càdlàg when almost certainly every path is constant on the right at every point and the limit values ​​on the left exist there. Poisson processes are an example of this . ${\displaystyle X=(X_{t})_{t\geq 0}}$ ${\displaystyle t\rightarrow X_{t}}$ ${\displaystyle t}$

literature

• Achim Klenke: Probability Theory . 3. Edition. Springer-Verlag, Berlin / Heidelberg 2013, ISBN 978-3-642-36017-6 , doi : 10.1007 / 978-3-642-36018-3 .

Web links

• Eric W. Weisstein : Cadlag Function . In: MathWorld (English).