Subordinators and Time-Changed Processes

Ordinary calendar time ticks forward at a constant rate, but markets don't experience information or trading activity at a constant rate — some hours are frantic, others are dead quiet. A subordinator is a nondecreasing random process (it only ever goes up or stays flat, never backward) used precisely to model this: think of it as a random, uneven "business clock" that runs faster during active periods and slower during quiet ones.

The key application is "time-changing" another process by running it on the subordinator's clock instead of calendar time. If $X_t$ is an ordinary process (like Brownian motion) and $\tau_t$ is a subordinator, the time-changed process $X_{\tau_t}$ behaves like $X$ but experienced through the lens of the random business clock — during a burst where $\tau_t$ jumps forward quickly, $X_{\tau_t}$ moves a lot in a short calendar interval; during a lull where $\tau_t$ barely advances, $X_{\tau_t}$ barely moves at all. This is the standard way to build models where volatility clusters and price moves happen in bursts tied to trading activity, rather than smoothly and independently of it.

A concrete example is the variance-gamma process, built by running Brownian motion on a gamma-distributed subordinator: the resulting price process has fatter tails and more realistic clustering of large moves than plain Brownian motion, because trading activity itself is randomly bursty rather than constant.