subordination of a process

Subordination is the operation of running one Lévy process not on clock time but on a random, increasing clock supplied by an independent subordinator. It is a remarkably productive machine: starting from Brownian motion alone and time-changing it by various subordinators yields the variance-gamma, the normal-inverse-Gaussian, and the symmetric stable processes — all the famous jump models of mathematical finance arise this way.

Formally, let X_t be a Lévy process and T_t an independent subordinator with Laplace exponent Phi. The subordinated process is Y_t = X_{T_t}, the original process evaluated at the random time T_t. The key computation (Bochner): Y is again a Lévy process, and its characteristic exponent is the composition psi_Y(theta) = -Phi(-psi_X(theta)), where psi_X is the exponent of X. In words, subordination composes exponents through the subordinator's Laplace exponent. Because T_t is increasing and jumps, even a continuous X (like Brownian motion) acquires jumps when subordinated — the random clock leaps over intervals, and the value jumps accordingly. When X = Brownian motion the recipe is especially clean: Y = B_{T_t} has exponent -Phi(theta^2/2).

Subordination is how the field manufactures the heavy-tailed, skewed jump processes used to fit asset returns: variance-gamma is Brownian motion subordinated by a gamma process, normal-inverse-Gaussian is Brownian motion subordinated by an inverse-Gaussian process, and the symmetric alpha-stable process is Brownian motion subordinated by the (alpha/2)-stable subordinator. The honest caveat: the subordinator must be INDEPENDENT of the process being time-changed for the Bochner formula to hold; correlated time changes (as in stochastic-volatility leverage) are not subordination and the clean exponent composition fails.