Let $(\Omega, \mathcal F,\mathbb P)$ be a probability space equipped with a filtration $\{\mathcal F_t:t\in\mathbb R_+\}$ satisfying the usual conditions of completeness and right-continuity. Suppose the process $\{A_t:t\in\mathbb R_+\}$ is càdlàg, adapted to $\{\mathcal F_t\}$ and for all $T\in\mathbb R_+$, $$\sup_{\{t_i\}} \left(\sum_i \left|A_{t_{i+1}}(\omega) - A_{t_i}(\omega)) \right|\right)<\infty,\ \mathrm{ w.p. }\ 1 $$ with the supremum taken over increasing sequences $\{t_i:i\in\mathbb N\}$ in $[0,T]$. Define for $t\in\mathbb R_+$ the random variable $C_t$ by $$C_t(\omega) = \inf\{s:A_s(\omega)>t\}. $$ In the text I am studying (Stochastic Calculus and Applications by Cohen and Elliot), the process $C$ is called the time change associated with $A$.
I am not sure why this is called a "time change" of $\{A_t\}$, or how to determine the process $\{C_t\}$ from $\{A_t\}$. I've only ever heard of a "time change" (or "time inversion") in the following two cases:
- $\{N(t):t\in\mathbb R_+\}$ is a nonhomogeneous Poisson process with intensity function $\lambda(t)$ and $\mathbb E[N(t)] = \int_0^t\ \lambda(s)\ \mathsf ds:=\Lambda(t)$. Then $\tilde N(u) := N(\Lambda^{-1}(u))$ is a homogeneous Poisson process with unit intensity obtained by a time change.
- $\{B_t:t\in\mathbb R_+\}$ is a standard Brownian motion. Then $B^{(1)}_t:= tB_{\frac1t}\mathsf 1_{(0,\infty)}(t)$ is again a standard Brownian motion, obtained by a time inversion.
But this seems to be something entirely different and some motivation/intuition behind the definition would be greatly appreciated.