= Subordination of a Lévy process
{title2=$Y_t=X_{T_t}$}
If $X$ is a <Lévy process> and $T$ an independent <subordinator>, then $Y_t=X_{T_t}$ is a <Lévy process>. Conditional on the clock, increments of $X$ over disjoint clock intervals are independent; averaging over the independent <stationary increments> of the clock gives the same properties for $Y$. If $\mathbb Ee^{iuX_s}=e^{-s\Psi(u)}$, its <characteristic function> is $\mathbb E e^{-T_t\Psi(u)}$. For standard <Brownian motion>, $\Psi(u)=u^2/2$, so this is the clock's <Laplace transform> at $u^2/2$.
Back to article page
