Funzione càdlàg

In matematica, una funzione càdlàg (acronimo dal francese continue à droite, limitée à gauche, che significa continua a destra, limitata a sinistra; in italiano scritto talvolta cadlag) è una funzione di variabile reale che sia in ogni punto continua da destra e possegga limite sinistro finito.

Funzioni càdlàg emergono naturalmente come funzioni di ripartizione. Compaiono quindi nello studio dei processi stocastici che ammettono traiettorie con discontinuità di prima specie.

Esempi

Tutte le funzioni continue sono naturalmente càdlàg.
Tutte le funzioni di ripartizione sono càdlàg.^[1]

Spazio di Skorokhod

Lo spazio di tutte le funzioni càdlàg su un certo dominio $E$ a valori nello spazio metrico $M$ viene detto spazio di Skorokhod. Esso si denota con $D(E;M)$ . Tale spazio può essere munito di una topologia. Per semplicità, consideriamo come dominio l'intervallo $[0,T]$ con $T$ finito e come codominio lo spazio euclideo reale.

Dobbiamo prima definire un analogo del modulo di continuità. Per ogni $F\subseteq E$ , sia

w_{f}(F):=\sup _{s,t\in F}|f(s)-f(t)|

l'oscillazione di $f$ su $F$ ; per $\delta >0$ , definiamo allora il modulo càdlàg come

\varpi '_{f}(\delta ):=\inf _{\Pi }\max _{1\leq i\leq k}w_{f}([t_{i-1},t_{i})),

dove l'estremo inferiore è fatto su tutte le partizioni $\Pi$ dell'intervallo $E$ con mesh minore di $\delta$ . Si può provare che $f$ è càdlàg se e solo se $\varpi '_{f}(\delta )\to 0$ quando $\delta \to 0$ .

Definiamo dunque la distanza di Skorokhod come

\sigma (f,g):=\inf _{\lambda }\max\{\|\lambda -id_{E}\|_{\infty },\|f-g\circ \lambda \|_{\infty }\}

dove $id_{E}$ è l'identità di $E$ , $\|\cdot \|_{\infty }$ è la norma uniforme e $\lambda$ varia sull'insieme di tutte le biiezioni continue strettamente monotone su $E$ . Si dimostra che effettivamente $\sigma$ è una metrica. La topologia indotta è detta topologia di Skorokhod.

Intuitivamente, il termine $\|\lambda -id_{E}\|_{\infty }$ misura la "distorsione nel tempo" e il termine $\|f-g\circ \lambda \|_{\infty }$ la "distorsione nello spazio".

Proprietà

Lo spazio $D(E;M)$ contiene lo spazio $C(E;M)$ delle funzioni continue. Su tale sottospazio la topologia di Skorokhod e la topologia uniforme coincidono.

La metrica $\sigma$ non rende lo spazio di Skorokhod completo; tuttavia esiste una metrica equivalente a $\sigma$ per cui ciò è vero. Tale metrica (e dunque anche $\sigma$ ) rende inoltre $D(E;M)$ uno spazio separabile e quindi uno spazio polacco.

Come applicazione del teorema di Ascoli, si può mostrare che una successione di misure di probabilità su $D$ è tight se solo se sono verificate le seguenti due condizioni:

$\lim _{a\to \infty }\limsup _{n\to \infty }\mu _{n}\{f\in D|\|f\|\geq a\}=0,$
$\lim _{\delta \to 0}\limsup _{n\to \infty }\mu _{n}\{f\in D|\varpi '_{f}(\delta )\geq \varepsilon \}=0$

con la seconda valida per ogni $\varepsilon >0$ .

Note

^ Questo vale se, come largamente in uso, si definisce una funzione di ripartizione mediante la formula $F(x)=P(x\leq x)$ . La proprietà cade se si definisce $F(x)=P(X<x)$ , in quanto essa risulta essere una funzione continua a sinistra e con limite finito a destra.

Bibliografia

Billingsley, Patrick (1995). Probability and Measure. New York, NY: John Wiley & Sons, Inc.. ISBN 0-471-00710-2.

Voci correlate

Collegamenti esterni

(EN) Eric W. Weisstein, Funzione càdlàg, su MathWorld, Wolfram Research.

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