Cumulative-Rate Time Change: Regularity, Substitution, and Crossing Times

Let $T>0$ and $B\ge0$ be real numbers and let $a:[0,T]\to[0,B]$ be measurable with respect to the trace Borel $\sigma$-algebra $\mathcal{B}_{[0,T]}$. Define the cumulative rate $A:[0,T]\to[0,\infty)$ by $A(t)=\int_{[0,T]}a\,\mathbf{1}_{[0,t]}\,d\lambda_{[0,T]},$ the Lebesgue integral on $[0,T]$ of $a$ multiplied by the indicator function $\mathbf{1}_{[0,t]}$. Then:

1. (Regularity) $A(0)=0$, $A$ is nondecreasing, $A(T)\le BT$, and $0\le A(t)-A(s)\le B\,(t-s)$ for $0\le s\le t\le T$; in particular $A$ is continuous.

2. (Substitution) For every measurable $f:\mathbb{R}\to[0,\infty]$ (with respect to the Borel $\sigma$-algebra), the map $t\mapsto f(A(t))\,a(t)$ is $\mathcal{B}_{[0,T]}$-measurable and $\int_{[0,T]}f(A(t))\,a(t)\,d\lambda_{[0,T]}(t)=\int_{\mathbb{R}}f\,\mathbf{1}_{[0,A(T)]}\,d\lambda,$ where $\lambda$ is Lebesgue measure.

3. (Crossing times) For a real $u$, write $L(u)=\{t\in[0,T]:A(t)\ge u\}$. For $0<u\le A(T)$, the set $L(u)$ is nonempty, and its greatest lower bound, the crossing time $\kappa(u)$, satisfies $\kappa(u)>0$, $A(\kappa(u))=u$, and $L(u)=\{t\in[0,T]:t\ge\kappa(u)\}$. For $u>A(T)$, the set $L(u)$ is empty.

4. (Inverse substitution) For every $\mathcal{B}_{[0,T]}$-measurable $\Phi:[0,T]\to[0,\infty]$, the map $u\mapsto\Phi(\kappa(u))$ on $(0,A(T)]$ is measurable with respect to the trace of the Borel $\sigma$-algebra, and $\int_{\mathbb{R}}\Phi(\kappa(u))\,\mathbf{1}_{(0,A(T)]}(u)\,d\lambda(u)=\int_{[0,T]}\Phi(t)\,a(t)\,d\lambda_{[0,T]}(t),$ the integrand on the left being extended by $0$ off $(0,A(T)]$, and the left side read as $0$ when $A(T)=0$.