Subordinator (mathematics)
A subordinator is a one-dimensional Lévy process whose sample paths are almost surely nondecreasing. Equivalently, it is a stochastic process (S=(S_t)_{t\geq 0}) with stationary independent increments, càdlàg paths, (S_0=0), and
[ S_t\leq S_u\qquad\text{whenever }0\leq t\leq u ]
outside a single null event. Subordinators serve as random operational clocks: replacing deterministic time (t) by (S_t) in another process changes the rate and discontinuity structure of its evolution while preserving several important semigroup properties.
The term has no connection with grammatical subordination or institutional rank. The process is called a subordinator because it implements the probabilistic operation of Bochner subordination, under which one transition semigroup is obtained from another by randomizing its time parameter.
Lévy–Khintchine representation
The law of a subordinator is determined by its Laplace transform. There exists a function (\Phi:[0,\infty)\to[0,\infty)) such that
[ \mathbb E!\left[e^{-\lambda S_t}\right] =e^{-t\Phi(\lambda)}, \qquad \lambda\geq 0,\quad t\geq 0. ]
The function (\Phi) is the Laplace exponent of (S). Its Lévy–Khintchine representation is
[ \Phi(\lambda) =\kappa+d\lambda +\int_{(0,\infty)} \left(1-e^{-\lambda x}\right)\Pi(dx), ]
where (\kappa\geq 0) is a killing rate, (d\geq 0) is a deterministic drift coefficient, and (\Pi) is a measure on ((0,\infty)) satisfying
[ \int_{(0,\infty)}(1\wedge x),\Pi(dx)<\infty. ]
The measure (\Pi) is the Lévy measure. It records the intensity with which jumps of different positive sizes occur. The integrability condition ensures that the cumulative size of the small jumps is finite on each bounded time interval, even when infinitely many such jumps occur.
A subordinator has no Gaussian component, since a nonconstant Brownian motion cannot have monotone paths. Before killing, its pathwise decomposition is therefore
[ S_t=dt+\sum_{0<s\leq t}\Delta S_s, ]
where every jump (\Delta S_s=S_s-S_{s-}) is nonnegative. Killing is represented by sending the process to a cemetery state, conventionally written (+\infty), at an independent exponential time of rate (\kappa).
The exponent (\Phi) is a Bernstein function: it is nonnegative, infinitely differentiable on ((0,\infty)), and its derivative is completely monotone. Conversely, every Bernstein function has a unique representation of the displayed form and determines a possibly killed subordinator. This correspondence identifies subordinators with convolution semigroups of probability or subprobability measures on ([0,\infty)).
Path structure
The drift coefficient and Lévy measure separate the continuous and discontinuous contributions to the clock. When (\Pi(0,\infty)<\infty), the jump component is a compound Poisson process. Its path remains constant between isolated jump times, apart from any deterministic increase contributed by (d).
When (\Pi(0,\infty)=\infty), every nontrivial time interval contains infinitely many jumps almost surely. The integrability condition on (\Pi) nevertheless makes their total size finite over bounded intervals. Such a process has infinite jump activity but finite variation.
A non-killed subordinator is strictly increasing almost surely when (d>0), or when its Lévy measure has infinite total mass. If (d=0) and (\Pi) is finite, the process is constant between successive Poissonian jumps. Monotonicity consequently does not imply continuous or strictly increasing trajectories.
The mean, when finite, follows by differentiating the Laplace transform:
[ \mathbb E[S_t]=t\Phi'(0+) =t\left(d+\int_{(0,\infty)}x,\Pi(dx)\right). ]
The integral may diverge, in which case (S_t) remains finite before killing but has infinite expectation.
Random time change
Let (X=(X_t)_{t\geq0}) be a Lévy process independent of (S), and define
[ Y_t=X_{S_t}. ]
Then (Y) is again a Lévy process, subject to the usual cemetery-state convention when (S) is killed. If the characteristic exponent of (X) is (\psi), so that
[ \mathbb E[e^{i\xi X_t}]=e^{-t\psi(\xi)}, ]
then the characteristic exponent of the time-changed process is
[ \psi_Y(\xi)=\Phi(\psi(\xi)), ]
where the Bernstein function (\Phi) is evaluated through its extension to the right half-plane. The composition formula expresses the central algebraic feature of subordination: randomizing time corresponds to composing exponents.
The same construction applies to a strongly continuous Markov semigroup ((P_t)_{t\geq0}). If (\mu_t) denotes the distribution of (S_t), the subordinated semigroup is
[ Q_t f=\int_{[0,\infty)}P_s f,\mu_t(ds). ]
For a semigroup generator (A), the generator of the subordinated semigroup is formally
[ -\Phi(-A). ]
This identity underlies the probabilistic interpretation of fractional and nonlocal operators. For example, subordinating Brownian motion by a stable subordinator produces an isotropic stable process whose generator is a fractional power of the Laplacian.
Salomon Bochner formulated semigroup subordination as an analytic operation that connects convolution semigroups, stochastic time changes, and fractional powers of operators. Its later probabilistic formulation placed increasing Lévy processes at the center of the construction rather than treating the random clock only through its family of distributions.
Principal examples
An (\alpha)-stable subordinator, with (0<\alpha<1), has Laplace exponent
[ \Phi(\lambda)=c\lambda^\alpha ]
for a constant (c>0). Its Lévy measure has density
[ \Pi(dx)= \frac{c\alpha}{\Gamma(1-\alpha)} x^{-1-\alpha},dx. ]
The measure has infinite mass near zero, so the process has infinitely many jumps in every time interval. The scaling identity
[ (S_{rt}){t\geq0} \overset{d}{=} (r^{1/\alpha}S_t){t\geq0} ]
makes stable subordinators fundamental in the study of self-similar processes and fractional calculus.
A gamma subordinator has Laplace exponent
[ \Phi(\lambda)=a\log!\left(1+\frac{\lambda}{b}\right), \qquad a,b>0, ]
and Lévy density
[ \Pi(dx)=a x^{-1}e^{-bx},dx. ]
It has infinite jump activity, finite variation, and gamma-distributed increments. Subordination by a gamma process produces several variance-gamma constructions in mathematical finance and stochastic modelling.
An inverse Gaussian subordinator has an exponent of the form
[ \Phi(\lambda) =\delta\left(\sqrt{\gamma^2+2\lambda}-\gamma\right), \qquad \delta>0,\quad\gamma\geq0. ]
Its one-dimensional distributions belong to the inverse Gaussian distribution family. Despite the name, it is not the inverse process of a Gaussian motion; the terminology refers to its marginal probability law.
Potential measure and first passage
The potential measure of a non-killed subordinator is defined by
[ U(B)= \mathbb E!\left[ \int_0^\infty \mathbf 1_{{S_t\in B}},dt \right] ]
for Borel sets (B\subseteq[0,\infty)). Its Laplace transform satisfies
[ \int_{[0,\infty)}e^{-\lambda x},U(dx) =\frac{1}{\Phi(\lambda)}, \qquad \lambda>0, ]
whenever the right-hand side is finite. This relation connects the occupation structure of the process directly to its Laplace exponent.
In a 1956 treatment of renewal measures for increasing processes, You Watanabe derived the killed version of this transform identity and expressed the associated resolvent through ((q+\Phi(\lambda))^{-1}). That formulation integrated exponential killing into the potential theory without changing the underlying Lévy–Khintchine parametrization.
For a level (x\geq0), the first-passage time is
[ T_x=\inf{t\geq0:S_t>x}. ]
The family (T=(T_x)_{x\geq0}) is called the inverse subordinator. It is nondecreasing, but it is generally not a Lévy process because its increments are neither stationary nor independent. Flat portions of (S) become jumps of (T), while jumps of (S) generate intervals over which (T) is constant.
The overshoot at level (x) is (S_{T_x}-x). Its distribution is governed by the renewal measure and the Lévy measure, reflecting the possibility that a jump crosses the level without the process ever taking the value (x). The closed range
[ \mathcal R=\overline{{S_t:t\geq0}} ]
is a regenerative set: after the first point of the range beyond a fixed level, the translated future range has the same law as the original range and is independent of the preceding portion.
The systematic treatment of these range, passage, and overshoot structures was incorporated into modern fluctuation theory by probabilists including Jean Bertoin, particularly through the relation between subordinators and the ladder processes of general Lévy processes. Ascending and descending ladder times are themselves subordinators, possibly after killing, and encode new maxima or minima reached by the parent process.
Distinction from related time changes
An ordinary subordinator supplies a Lévy clock, and the resulting time-changed process retains stationary independent increments when the original process is Lévy and independent of the clock. An inverse subordinator instead records the physical time required for the operational clock to exceed a specified level. Time changes by inverse subordinators therefore produce non-Markovian waiting effects and commonly lead to evolution equations with fractional derivatives in time.
This distinction is visible in the transforms. Direct subordination composes a spatial or characteristic exponent with (\Phi), whereas inverse subordination introduces factors involving (\Phi(\lambda)) into Laplace-transformed time variables. The two constructions are related through first passage, but they do not define interchangeable stochastic processes.