Feeds:
Posts

## THE INVERSE STABLE SUBORDINATOR

In a continous time random walk when the random times $P(W_n>t) \sim Ct^{-\beta}$ for $0<\beta<1$ and therefore $E(W_n) = + \infty$ the extended central limit theorem says $n^{1/\beta} T_{[nu]} \Rightarrow D_{u}$, a stable subordinator whose probability density $g(s,u)$ has Laplace transform

$\tilde{g}(s,u) = \exp(-u s^\beta)$

The inverse process has an inverse limit $n^{-\beta} N_{nt} \Rightarrow E_$ where the inverse stable subordinator
$E_t = \inf\{ u>0: D_u > t\}$

This is extremely useful for all financial time series because this is the time-change from Brownian motion to which continuous-time random walks converge.

More details here.