1. Introduction
Let
${\bf X}=(X_1,\ldots, X_n)$
be an n-dimensional Lévy process and let
${\bf T}=(T_1,\ldots,T_n)$
be an n-dimensional subordinator independent of X. The operation that evaluates the process X at times given by the subordinator T is defined by

and known as strong subordination. This creates a ‘time-changed’ process. The study of the multivariate subordination of Lévy processes originated with the work of Barndorff-Nielsen, Pedersen, and Sato [Reference Barndorff-Nielsen, Pedersen and Sato1]. It is well known that strong subordination produces a Lévy process in the following cases.
-
(C1) T has indistinguishable components.
-
(C2) X has independent components.
-
(C3) T and X satisfy the stacked univariate subordination condition: for some
$1\leq d\leq n$ and
$n_1+\cdots+n_d=n$ ,
(1)where\begin{align} {\bf X}=({\bf Y}_1,\ldots,{\bf Y}_d),\quad {\bf T} = (R_1{\bf e}_1,\ldots ,R_d{\bf e}_d), \end{align}
${\bf Y}_1,\ldots,{\bf Y}_d$ are independent Lévy processes,
${\bf Y}_m$ is
$n_m$ -dimensional,
$(R_1,\ldots, R_d)$ is a d-dimensional subordinator, and
${\bf e}_m=(1,\ldots,1)\in\mathbb{R}^{n_m}$ ,
$1\leq m \leq d$ .
For the proof of sufficiency under conditions () and (), see [Reference Sato14, Theorem 30.1] and [Reference Barndorff-Nielsen, Pedersen and Sato1, Theorem 3.3], respectively, though the origin of the former case goes back to [Reference Zolotarev16]. Both conditions () and () are implied by condition (). It is condition (), as opposed to condition (), that more commonly appears in financial applications [Reference Buchmann, Kaehler, Maller and Szimayer3,Reference Luciano and Semeraro10,Reference Semeraro15], since it has a more intuitive interpretation. Outside of these sufficient conditions, strong subordination does not necessarily produce a Lévy process [Reference Buchmann, Lu and Madan5, Proposition 3.9].
These restrictions on X and T for
${\bf X}\circ{\bf T}$
to remain in the well-understood Lévy process framework are problematic in applications because they severely limit the dependence structure of
${\bf X}\circ{\bf T}$
. To address this shortcoming, Buchmann, Lu, and Madan [Reference Buchmann, Lu and Madan5] introduced a new operation for constructing general time-changed multivariate Lévy processes
${\bf X}\odot{\bf T}$
, known as weak subordination, without any restriction on the subordinate X or the subordinator T.
Weak subordination is based on the idea, roughly speaking, of constructing the Lévy process that has the distribution of
${\bf X}({\bf t})\;:\!=\; (X_1(t_1),\ldots,X_n(t_n))$
conditional on
${\bf T}(t) = {\bf t}\;:\!=\; (t_1,\ldots,t_n)$
,
$t\geq0$
. To be more precise, the idea is to decompose the subordinator
${\bf T}(t) = {\bf d} t + {\bf S}(t)$
,
$t\geq0$
, into deterministic and pure-jump parts. For the deterministic subordinator part,
${\bf X}({\bf d})$
is infinitely divisible and hence associated with a Lévy process, while for the pure-jump subordinator part, a marked Poisson point process can be constructed such that it jumps with the distribution of
${\bf X}({\bf t})$
when the subordinator jumps by
$\Delta {\bf T}(t) = {\bf t}$
and then associated with a Lévy process by the Lévy–Itô decomposition, and finally the two Lévy processes are combined by convolution. This allows for more flexible dependence modelling while remaining in the class of Lévy processes, a closure property not enjoyed by strong subordination. Weak subordination coincides with strong subordination under conditions () or () in the sense that
$({\bf T},{\bf X}\circ {\bf T})\stackrel{D}{=} ({\bf T},{\bf X}\odot {\bf T})$
[Reference Buchmann, Lu and Madan5, Proposition 3.3], and it also reproduces many analogous properties [Reference Buchmann, Lu and Madan5, Propositions 3.3, 3.7].
In this paper we show that the more general case () of stacked univariate subordination considered in [Reference Barndorff-Nielsen, Pedersen and Sato1] also satisfies
$({\bf T},{\bf X}\circ {\bf T})\stackrel{D}{=} ({\bf T},{\bf X}\odot {\bf T})$
, which unifies it under weak subordination (see Theorem 1). This raises the question of whether there are alternative definitions of weak subordination that are also consistent with strong subordination in this way. We partially address this by showing that if
$({\bf T}, {\bf X}\circ {\bf T})$
is a Lévy process, then
$({\bf T}, {\bf X}\circ {\bf T})\stackrel{D}{=}({\bf T}, {\bf X}\odot {\bf T})$
in two cases: T is deterministic (see Proposition 1), or T is a pure-jump subordinator with finite activity (see Theorem 2). In the former case we can weaken the assumption to
${\bf X}\circ {\bf T}$
being a Lévy process. Our proof tracks the construction of weak subordination in the deterministic subordinator and pure-jump subordinator cases mentioned above, and in the latter case the theory of marked Poisson point processes is used to verify that the relevant characteristics coincide.
We briefly mention some applications. The subordination of Lévy processes is used in mathematical finance to create time-changed models of stock prices. This idea began with the work of Madan and Seneta [Reference Madan and Seneta12], who introduced the variance gamma (VG) process for modelling stock prices, created by subordinating a Brownian motion with a gamma subordinator. Subordination can also be applied to model dependence in multivariate price processes. The multivariate VG process in [Reference Madan and Seneta12] was created by subordinating multivariate Brownian motion with a univariate gamma subordinator, so the components cannot have idiosyncratic time changes and must have equal kurtosis when there is no skewness. These deficiencies were addressed by the use of an alpha-gamma subordinator, resulting in the variance alpha-gamma process which was introduced in [Reference Semeraro15] and also considered in [Reference Guillaume7] and [Reference Luciano and Semeraro10]. However, in this case the Brownian motion subordinate must have independent components. In both models, the use of strong subordination to create a Lévy process restricts the dependence structure. By using weak subordination instead, and a general Brownian motion, Buchmann, Lu, and Madan [Reference Buchmann, Lu and Madan4,Reference Buchmann, Lu and Madan5] introduced the weak variance alpha-gamma (WVAG) process to provide additional flexibility in dependence modelling while remaining tractable. The WVAG process exhibits a wider range of dependence while remaining parsimoniously parametrised; the marginal components have both common and idiosyncratic time changes, and are VG processes with possibly different levels of kurtosis. These weakly subordinated processes have been applied to option pricing [Reference Michaelsen and Szimayer13] and instantaneous portfolio theory [Reference Madan11].
The paper is structured as follows. In Section 2 we review some notations, definitions, and preliminary results relating to Lévy processes, weak subordination, and Poisson random measures. In Section 3 we state and prove the main results, namely that weak subordination is consistent with strong subordination under condition (), and that if strong subordination produces a Lévy process it is necessarily equal in law to weak subordination when the subordinator is deterministic or pure-jump with finite activity. We conclude in Section 4 with a brief discussion placing this work in the context of open questions relating to the subordination of multivariate Lévy processes.
2. Preliminaries
2.1 Lévy processes
We write
${\bf x}=(x_1,\ldots,x_n)\in\mathbb{R}^n$
as a row vector. For
$A\subseteq\mathbb{R}^n$
, let
$A_*\;:\!=\; A\backslash\{{\bf 0}\}$
and let
${\bf 1}_A$
denote the indicator function for A. Let
$\mathbb{D}\;:\!=\; \{{\bf x}\in\mathbb{R}^n\colon \|{\bf x}\|\le 1\}$
be the Euclidean unit ball centred at the origin. Let
$\|{\bf x}\|^2_\Sigma\;:\!=\; {\bf x}\Sigma{\bf x}'$
, where
${\bf x}\in\mathbb{R}^n$
and
$\Sigma\in\mathbb{R}^{n\times n}$
. Let
$I\colon [0,\infty)\to[0,\infty)$
be the identity function.
For references on Lévy processes, see [Reference Bertoin2] and [Reference Sato14]. The law of an n-dimensional Lévy process
${\bf X}=(X_1,\ldots,X_n)=({\bf X}(t))_{t\ge 0}$
is determined by its characteristic function
$\Phi_{\bf X}\;:\!=\; \Phi_{{\bf X}(1)}$
with

and characteristic exponent

where
$\boldsymbol{\mu}\in\mathbb{R}^n$
,
$\Sigma\in\mathbb{R}^{n\times n}$
is a covariance matrix, and
$\mathcal{X}$
is a Lévy measure, that is, a non-negative Borel measure on
$\mathbb{R}^n_*$
such that
$\int_{\mathbb{R}^n_*}(1\wedge \|{\bf x}\|^2)\,\mathcal{X}({\textrm{d}} {\bf x})<\infty$
. We write
${\bf X}\sim L^n(\boldsymbol{\mu},\Sigma,\mathcal{X})$
(or
${\bf X}\sim L^n$
for short) to mean that X is an n-dimensional Lévy process with characteristic triplet
$(\boldsymbol{\mu},\Sigma,\mathcal{X})$
.
An n-dimensional Lévy process T with almost surely non-decreasing sample paths is called a subordinator, and it is denoted by
${\bf T} \sim S^n({\bf d},\mathcal{T}) \;:\!=\; L^n(\boldsymbol{\mu},0,\mathcal{T})$
(or
${\bf T}\sim S^n$
for short), where
${\bf d} \;:\!=\; \boldsymbol{\mu} - \int_{\mathbb{D}_*} {\bf t} \,\mathcal{T}({\textrm{d}} {\bf t})\in[0,\infty)^n$
is the drift, and
$\mathcal{T}$
is the Lévy measure. The law of T is characterised by its Laplace exponent
$\Lambda_{\bf T}$
, which satisfies
$\mathbb{E}[\!\exp(-\langle {\boldsymbol{\lambda}},{{\bf T}(1)}\rangle)]=\exp(-\Lambda_{\bf T}(\boldsymbol{\lambda}))$
,
$\boldsymbol{\lambda}\in[0,\infty)^n$
. For
${\bf w},{\bf z} \in\mathbb{C}^n$
, let
$\langle {{\bf w}},{{\bf z}}\rangle \;:\!=\; \sum_{k=1}^n w_kz_k$
, noting that there is no conjugation. The domain of
$\Lambda_{\bf T}$
can be extended, giving

(see the proof of [Reference Barndorff-Nielsen, Pedersen and Sato1, Theorem 3.3]).
We have the following characterisation of piecewise constant Lévy processes from [Reference Sato14, Theorem 21.2].
Lemma 1. Let
${\bf X}\sim L^n(\boldsymbol{\mu},\Sigma,\mathcal{X})$
. The following are equivalent:
-
(i) X is piecewise constant a.s.,
-
(ii) X is driftless,
$\Sigma=0$ and
$\mathcal{X}(\mathbb{R}^n_*)<\infty$ ,
-
(iii) X is a compound Poisson process or X is the zero process.
2.2. Weak subordination
Let
${\bf X}\sim L^n$
and
${\bf T}\sim S^n({\bf d},\mathcal{T})$
. Let
${\bf t}=(t_1,\ldots,t_n)\in[0,\infty)^n$
and
$\langle{(1),\ldots,(n)}\rangle$
be a permutation of
$\{1,\ldots,n\}$
such that
$t_{(1)} \leq \cdots \leq t_{(n)}$
, and define
$\Delta t_{(k)}\;:\!=\; t_{(k)}-t_{(k-1)}$
,
$1\leq k \leq n$
, with
$t_{(0)}\;:\!=\; 0$
. Further, let
$\boldsymbol{\pi}_{J}\colon \mathbb{R}^n\to\mathbb{R}^n$
be the projection onto the coordinate axes in
$J\subseteq\{1,\ldots,n\}$
. For all
${\bf t}\in[0,\infty)^n$
, by [Reference Buchmann, Lu and Madan5, Proposition 2.1], the random vector
${\bf X}({\bf t})\;:\!=\; (X_1(t_1),\ldots, X_n(t_n))$
is infinitely divisible with characteristic exponent

and characteristic function

It is convenient to consider both the subordinator and the subordinated process together as a joint 2n-dimensional Lévy process. In this form, we can define weak subordination as follows (see [Reference Buchmann, Lu and Madan5, Proposition 3.1]).
Definition 1. The weak subordination of
${\bf X}\sim L^n$
and
${\bf T}\sim S^n({\bf d},\mathcal{T})$
is the joint Lévy process
${\bf Z}\stackrel{D}{=}({\bf T},{\bf X}\odot {\bf T})$
with characteristic exponent

where
$\boldsymbol{\theta}=(\boldsymbol{\theta}_1,\boldsymbol{\theta}_2)$
,
$\boldsymbol{\theta}_1,\boldsymbol{\theta}_2\in\mathbb{R}^n$
.
This is a valid characteristic exponent for a Lévy process. Weak subordination can equivalently be defined in terms of a characteristic triplet. From [Reference Buchmann, Lu and Madan5, Definition 2.1], if
${\bf d} ={\bf 0}$
, then
${\bf Z}\sim L^{2n}({\bf m},\Theta,\mathcal{Z})$
, where



For additional details on weak subordination, see [Reference Buchmann, Lu and Madan5].
2.3. Poisson random measures
For references on Poisson random measures and their relationship to the jumps of Lévy processes, see [Reference Bertoin2], [Reference Çinlar6], and [Reference Kingman9]. A Poisson random measure (PRM)
$\mathbb{Z}$
with intensity measure
$\mu$
on a measurable space
$(E,\mathcal{E})$
is a random measure such that
$\mathbb{Z}(A)\sim \operatorname{Poisson}(\mu(A))$
for all
$A\in\mathcal{E}$
, and
$\mathbb{Z}(A_1),\ldots,\mathbb{Z}(A_m)$
are independent for all disjoint
$A_1,\ldots,A_m\in\mathcal{E}$
. In general, a PRM has the form
$\mathbb{Z}=\sum_{i=1}^\infty \boldsymbol{\delta}_{{\bf Z}_i}$
, where
$\boldsymbol{\delta}_{{\bf Z}_i}$
,
$i\in\mathbb{N}$
, is the Dirac measure at the random vector
${\bf Z}_i$
taking values in
$(E,\mathcal{E})$
. Define the random variable
$Z_f\;:\!=\; \int_{E} f({\bf x})\,\mathbb{Z}({\textrm{d}} {\bf x})$
, where f is a non-negative,
$\mathcal{E}$
-measurable real function. The Laplace functional of
$\mathbb{Z}$
is

(see [Reference Kingman9, equation (3.35)]). The Laplace functional is well-defined with
$L(f)\in[0,1]$
, where this equality can be interpreted as 0 if
$Z_f<\infty$
a.s. fails. Two PRMs are equal if their Laplace functionals are equal [Reference Çinlar6, Chapter VI, Proposition 1.4].
Let
${\bf X}\sim L^n(\boldsymbol{\mu},\Sigma,\mathcal{X})$
with
$\mathcal{X}\neq 0$
. For a fixed sample path, a time t is a jumping time of X if the jump
$\Delta {\bf X}(t)\;:\!=\; {\bf X}(t)-{\bf X}(t-)\neq {\bf 0}$
. The following result is [Reference Sato14, Theorem 21.3].
Lemma 2. Let
${\bf X}\sim L^n(\boldsymbol{\mu},\Sigma,\mathcal{X})$
with
$\mathcal{X}\neq 0$
; then its jumping times are countably infinite. Denoting these jumping times as
${\bf S}= (S_i)_{i\in\mathbb{N}}$
, we have in addition:
-
(i) if
$\mathcal{X}(\mathbb{R}^n_*)<\infty$ , then S is countable in increasing order, or
-
(ii) if
$\mathcal{X}(\mathbb{R}^n_*)=\infty$ , then S is dense in
$[0,\infty)$ .
For the Lévy process X, the countable sequence of random vectors giving the time and size of the jumps,
$({\bf Z}_{i})_{i\in\mathbb{N}} \;:\!=\; (t,\Delta {\bf X}(t))_{t>0,\Delta {\bf X}(t)\neq {\bf 0}}$
, is a Poisson point process. Consequently,
$\mathbb{Z}=\sum_{i=1}^\infty \boldsymbol{\delta}_{{\bf Z}_i}$
is the PRM of X (or of the jumps of X), defined on the Borel space
$([0,\infty)\times\mathbb{R}^n_*,\mathcal{B}([0,\infty)\times\mathbb{R}^n_*))$
with intensity measure
${\textrm{d}} t \otimes \mathcal{X}$
[Reference Bertoin2, Chapter I, Theorem 1].
3. Main results
3.1. Consistency of weak subordination for stacked univariate subordination
Here we show that the laws of weak and strong subordination coincide when the latter satisfies the stacked univariate subordination property in condition (). The proof follows along the lines of [Reference Buchmann, Lu and Madan5, Proposition 3.3].
Theorem 1. Let
${\bf T}\sim S^n$
and
${\bf X}\sim L^n$
be independent. If
${\bf T}$
and
${\bf X}$
satisfy the stacked univariate subordination condition in (1), then
$({\bf T},{\bf X}\circ {\bf T})\stackrel{D}{=}({\bf T},{\bf X}\odot {\bf T})$
.
Proof. Recall that
$n_1+\cdots+n_d=n$
and let
$\boldsymbol{\theta}=(\boldsymbol{\theta}_1,\boldsymbol{\theta}_2) = (\boldsymbol{\theta}_{11},\ldots, \boldsymbol{\theta}_{1d},\boldsymbol{\theta}_{21},\ldots, \boldsymbol{\theta}_{2d})$
,
$\boldsymbol{\theta}_1,\boldsymbol{\theta}_2\in\mathbb{R}^n$
,
$\boldsymbol{\theta}_{1m},\boldsymbol{\theta}_{2m}\in\mathbb{R}^{n_m}$
for all
$1\leq m \leq d$
. Since T and X are independent processes, using (4) and conditioning on T, we get

Let
${\bf e}=(1,\ldots,1)\in\mathbb{R}^n$
. Since T and X satisfy the stacked univariate subordination condition for n-dimensional processes, the subordinator
$({\bf T},{\bf T})$
and the subordinate
$(I{\bf e},{\bf X})$
satisfy this condition for 2n-dimensional processes. Thus
$(I{\bf e},{\bf X})\circ ({\bf T},{\bf T})= ({\bf T},{\bf X}\circ {\bf T})$
is a Lévy process by [Reference Barndorff-Nielsen, Pedersen and Sato1, Theorem 3.3], so it suffices to show that
$\Psi_{({\bf T},{\bf X}\circ {\bf T})}=\Psi_{({\bf T},{\bf X}\odot {\bf T})}$
.
Noting that
${\bf X} = ({\bf Y}_1,\ldots, {\bf Y}_d)$
, where
${\bf Y}_1\sim L^{n_1},\ldots, {\bf Y}_d\sim L^{n_d}$
are independent Lévy processes, Kac’s theorem gives

Form the partition
$\{1,\ldots,n\}= J_1\cup\cdots \cup J_d$
, where

Let
${\bf r}=(r_1,\ldots,r_d)\in[0,\infty)^d$
and
$\langle(1),\ldots,(d)\rangle$
be a permutation of
$\{1,\ldots, d\}$
such that
$r_{(1)}\leq \cdots \leq r_{(d)}$
. Define the projections
$\boldsymbol{\pi}_m\;:\!=\; \boldsymbol{\pi}_{J_{(m)} \cup\cdots \cup J_{(d)}}$
,
$1\leq m \leq d$
. Thus, for all
$1\leq m\leq d$
,

Next, due to (1), we can write
${\bf T}= {\bf R} A$
for some
$A\in\mathbb{R}^{d\times n}$
, where
${\bf R}= (R_1,\ldots,R_d)\sim S^d({\bf d},\mathcal{R})$
. Then (3) and (10) give

Let
${\bf z}\;:\!=\; (z_1,\ldots,z_d)\in\mathbb{C}^d$
, where
$z_m = -{\textrm{i}}\langle {\boldsymbol{\theta}_{1m}},{{\bf e}_m}\rangle-\Psi_{{\bf Y}_m}(\boldsymbol{\theta}_{2m})$
,
$1 \leq m \leq d$
. Using (11), we have

Thus (9) becomes
$\Phi_{({\bf T},{\bf X}\circ {\bf T})}(\boldsymbol{\theta})=\mathbb{E}[\!\exp(-\langle {{\bf z}},{{\bf R}(1)}\rangle)]$
. By noting that
$\Re{\bf z}\in[0,\infty)^d$
and using (2) to obtain the Laplace exponent of R, we have
$\Psi_{({\bf T},{\bf X}\circ {\bf T})}(\boldsymbol{\theta})=-\Lambda_{\bf R}({\bf z})$
, where

Using (4) and (12), we have
${\textrm{e}}^{-\langle {{\bf z}},{\bf r}\rangle }=\Phi_{({\bf r} A,{\bf X}({\bf r} A))}(\boldsymbol{\theta})$
for
${\bf r}\in[0,\infty)^d_*$
. Thus

by the transformation theorem. This matches the right-hand side of (5) because
${\bf T} \sim S^n({\bf d} A, \mathcal{R}\circ A^{-1})$
. Therefore
$({\bf T},{\bf X}\circ {\bf T})\stackrel{D}{=}({\bf T},{\bf X}\odot {\bf T})$
.
3.2. Necessity of weak subordination for deterministic and pure-jump, finite activity subordinators
In this subsection we assume that
$({\bf T},{\bf X}\circ{\bf T})\sim L^{2n}$
, and under some conditions we show that it is equal in law to
$({\bf T},{\bf X}\odot{\bf T})$
. This is in contrast to Theorem 1, where we proved, under a condition for which it is known that
$({\bf T},{\bf X}\circ{\bf T})\sim L^{2n}$
, that it is equal in law to
$({\bf T},{\bf X}\odot{\bf T})$
. The conditions we consider in this subsection are that the subordinator T is deterministic, or that T is pure-jump with finite activity, which are dealt with in Proposition 1 and Theorem 2, respectively. In the former case we use the weaker assumption
${\bf X}\circ {\bf T}\sim L^{n}$
.
Proposition 1. Let
${\bf T}\sim S^n({\bf d},0)$
and
${\bf X} \sim L^{n}$
be independent, with
${\bf d}\in[0,\infty)^n$
. If
${\bf X}\circ {\bf T}\sim L^{n}$
, then
$({\bf T},{\bf X}\circ{\bf T})\stackrel{D}{=} ({\bf T},{\bf X}\odot{\bf T})$
.
Proof. Since T is deterministic,
${\bf T}={\bf d} I$
. The stationary and independent increment property of the Lévy process
${\bf X}\circ{\bf T}\sim L^{n}$
is equivalent to, for all
$0\le t_1<\cdots <t_{m+1}$
,
$m\geq1$
,

for
$\boldsymbol{\theta}_{21},\ldots, \boldsymbol{\theta}_{2m}\in\mathbb{R}^n.$
Multiplying both sides by
$\prod_{k=1}^{m} \exp({\textrm{i}}\langle {\boldsymbol{\theta}_{1k}},{(t_{k+1}-t_k){\bf d}}\rangle)$
,
$\boldsymbol{\theta}_{11},\ldots, \boldsymbol{\theta}_{1m}\in\mathbb{R}^n$
, shows that
$({\bf T},{\bf X}\circ{\bf T})$
also has stationary and independent increments. Further,
$({\bf T},{\bf X}\circ{\bf T})(0)={\bf 0}$
a.s. and the sample paths of
$({\bf T},{\bf X}\circ{\bf T})$
are a.s. càdlàg. Thus
$({\bf T},{\bf X}\circ{\bf T})\sim L^{2n}$
, so we just need to verify
$\Psi_{({\bf T},{\bf X}\circ {\bf T})}=\Psi_{({\bf T},{\bf X}\odot {\bf T})}$
.
Let
$\boldsymbol{\theta}=(\boldsymbol{\theta}_1,\boldsymbol{\theta}_2)$
,
$\boldsymbol{\theta}_1,\boldsymbol{\theta}_2\in\mathbb{R}^n$
. Noting that
${\bf T}={\bf d} I$
and using (4), we have

which is the same as
$\Psi_{({\bf T},{\bf X}\odot {\bf T})}(\boldsymbol{\theta})$
from (5).
We now consider the case where T is a pure-jump subordinator. The following lemma establishes the Laplace functional corresponding to the PRM of the weakly subordinated process
$({\bf T},{\bf X}\odot{\bf T})$
without the finite activity assumption on T. It is based on the marked Poisson point process of jumps of
$({\bf T},{\bf X}\odot{\bf T})$
given in the proof of [Reference Buchmann, Lu and Madan5, Theorem 2.1 (ii)], and closely follows the arguments in the proof of [Reference Çinlar6, Chapter VI, Theorem 3.2].
Throughout the rest of this section, we let
$E\;:\!=\; [0,\infty)\times[0,\infty)^n_*\times \mathbb{R}^n$
, and let Q be the mapping
$({\bf t}^*,B)\mapsto \mathbb{P}({\bf X}({\bf t})\in B)$
for
${\bf t}^*\;:\!=\; (t,{\bf t})\in [0,\infty)\times [0,\infty)^n_*$
and Borel sets
$B\subseteq\mathbb{R}^n$
.
Lemma 3. Assume
${\bf T}\sim S^n({\bf 0},\mathcal{T})$
,
$\mathcal{T}\neq 0$
,
${\bf X}\sim L^n$
. The PRM of the Lévy process
$({\bf T},{\bf X}\odot{\bf T})$
, denoted
$\mathbb{Z}_{\odot}$
, has Laplace functional

where f is a non-negative, measurable real function,

Proof. Noting that the Lévy process
$({\bf T},{\bf X}\odot{\bf T})$
has only countably many jumps by Lemma 2, the Poisson point process of jumps can be written as

Here
${\bf T}^*_i$
and
${\bf Y}_i$
are random vectors that take values in
$[0,\infty)\times [0,\infty)^n_*$
and
$\mathbb{R}^n$
, respectively. So the PRM of
$({\bf T},{\bf X}\odot{\bf T})$
is

Now Q is a probability kernel, and
$\mathbb{Z}_{\odot}$
corresponds to a marked Poisson point process with marks
$({\bf Y}_i)_{i\in\mathbb{N}}$
, and
${\bf Y}_i$
,
$i\in\mathbb{N}$
, are conditionally independent given
${\bf T}^*\;:\!=\; ({\bf T}^*_i)_{i\in\mathbb{N}}$
with probability distribution
$Q({\bf T}^*_i,{\textrm{d}} {\bf y})$
(see the proof of [Reference Buchmann, Lu and Madan5, Theorem 2.1 (ii)]). Consequently, the Laplace functional of
$\mathbb{Z}_{\odot}$
is

as required.
Theorem 2. Let
${\bf T}\sim S^n({\bf 0},\mathcal{T})$
and
${\bf X} \sim L^{n}$
be independent, with
$\mathcal{T}\neq 0$
and
$\mathcal{T}([0,\infty)^n_*)<\infty$
. If
$({\bf T},{\bf X}\circ{\bf T})\sim L^{2n}$
, then
$({\bf T},{\bf X}\circ{\bf T})\stackrel{D}{=} ({\bf T},{\bf X}\odot{\bf T})$
.
Proof. Let

with
${\bf T}_i^*\, :\!=(S_i,\Delta {\bf T}(S_i))$
and
${\bf S}^*\;:\!=\; (S_i)_{i\in\mathbb{N}}$
. The PRM of
$({\bf T},{\bf X}\circ{\bf T})$
is

Since
$\mathcal{T}([0,\infty)^n_*)<\infty$
by assumption, the jumps of T are countable in increasing order by Lemma 2 (i). Therefore the sample paths
$t\mapsto({\bf T},{\bf X}\circ{\bf T})(t)$
are piecewise constant a.s., which implies by Lemma 1 (ii) that the Lévy process
$({\bf T},{\bf X}\circ{\bf T})\sim L^{2n}({\bf m},\Theta,\mathcal{Z})$
must have
${\bf m} = \int_{\mathbb{D}_*} ({\bf t},{\bf x})\,\mathcal{Z}({\textrm{d}}{\bf t},{\textrm{d}}{\bf x})$
and
$\Theta = 0$
. These are the same m and
$\Theta$
as
$({\bf T},{\bf X}\odot{\bf T})$
in (6)–(7) provided that
$({\bf T},{\bf X}\circ{\bf T})$
and
$({\bf T},{\bf X}\odot{\bf T})$
have the same Lévy measure, which we now verify by showing they have the same PRM.
Let
$S_0\;:\!=\; 0$
. Recalling that
$({\bf T},{\bf X}\circ{\bf T})$
has càdlàg and piecewise constant sample paths a.s., and that
$(S_i)_{i\in\mathbb{N}}$
is countable in increasing order, we have

For
$i\in\mathbb{N}$
, let
$\mathbb{P}((\Delta {\bf T}(S_i),{\bf Y}_i^*)\in({\textrm{d}}{\bf t},{\textrm{d}}{\bf y}) {\,\vert\,} {\bf S}^*)$
denote the conditional distribution of
$(\Delta {\bf T}(S_i),{\bf Y}_i^*)$
given
${\bf S}^*$
. Since
$(\Delta {\bf T}(S_i),{\bf Y}_i^*)$
takes values on the Borel space
$([0,\infty)^n_*\times\mathbb{R}^n,\mathcal{B}([0,\infty)^n_*\times\mathbb{R}^n))$
, there exists a regular version of the conditional distribution (see [Reference Kallenberg8, Theorem 5.3]), so we can assume that
$\mathbb{P}((\Delta {\bf T}(S_i),{\bf Y}_i^*)\in({\textrm{d}}{\bf t},{\textrm{d}}{\bf y}) {\,\vert\,} {\bf S}^*)$
is a probability kernel.
Consequently,
$\mathbb{P}((\Delta {\bf T}(S_i),{\bf Y}_i^*)\in({\textrm{d}}{\bf t},{\textrm{d}}{\bf y}) {\,\vert\,} {\bf S}^*)$
is a probability measure for each value of
${\bf S}^*$
, so it is determined by its characteristic function, which by the disintegration theorem (see [Reference Kallenberg8, Theorem 5.4]) is

using the stationary increment property of
$({\bf T},{\bf X}\circ{\bf T})$
, (4), and the stationary increment property of T, where
$\boldsymbol{\theta}=(\boldsymbol{\theta}_1,\boldsymbol{\theta}_2)$
,
$\boldsymbol{\theta}_1, \boldsymbol{\theta}_2\in\mathbb{R}^n$
. Similarly, the conditional distribution
$\mathbb{P}((\Delta {\bf T}(S_i), {\bf X}(\Delta {\bf T}(S_i)))\in({\textrm{d}}{\bf t},{\textrm{d}}{\bf y}) {\,\vert\,} {\bf S}^*)$
is also a probability kernel, and using the tower law, its characteristic function is

which matches (15). Thus we have

where the third line is obtained by noting that the conditioning on
${\bf S}^*$
can be dropped because, by applying Lemma 1 (iii) to
$({\bf T},{\bf X}\circ {\bf T})$
, we see that
${\bf S}^*$
are the times of the jumps of a compound Poisson process, which are independent of the size of the jumps with distribution
$(\Delta {\bf T}(S_i),{\bf X}(\Delta {\bf T}(S_i)))$
, the fourth line is obtained by [Reference Kallenberg8, Chapter 5, equation (7)], and the fifth line is obtained by the independence of T and X.
By definition,
${\bf S}^*$
are the jumping times of
$({\bf T},{\bf X}\circ {\bf T})$
. Since T is a pure-jump subordinator with finite activity, by examining the sample paths,
${\bf X}\circ{\bf T}$
cannot jump unless T does. So almost surely,

which implies
${\bf S}^*=\{t>0\colon \Delta{\bf T}(t)\neq {\bf 0} \}$
. Thus
${\bf S}^*$
are also the jumping times of T.
Next, for any non-negative, measurable real function f, we have

where the first line follows from the disintegration theorem (see [Reference Kallenberg8, Theorem 5.4]) and using (16), the third line follows from a similar argument to that above, T being a compound Poisson process means the time and size of the jumps are independent, and the final line follows from another application of the disintegration theorem.
Putting this together, we compute the Laplace functional of
$\mathbb{Z}_{\circ}$
,

where the second line follows since
$(\Delta {\bf T}(S_i),{\bf Y}_i^*)$
,
$i\in\mathbb{N}$
, are conditionally independent given
${\bf S}^*$
because of (14) and the independent increment property of the Lévy process
$({\bf T},{\bf X}\circ {\bf T})$
, the third line follows from (17), and the fourth line follows from a similar argument to the second line but applied to T. Thus we have proved
$\mathbb{Z}_{\circ}=\mathbb{Z}_{\odot}$
from (13), and hence
$({\bf T},{\bf X}\circ{\bf T})$
and
$({\bf T},{\bf X}\odot{\bf T})$
have the same Lévy measure and hence the same characteristic triplet.
Remark 1. It is difficult to extend Theorem 2 to the more general case of a finite activity subordinator with non-zero drift or to all pure-jump subordinators. In the former case the proof’s reliance on the properties of the compound Poisson process would fail. In the latter case, for any pure-jump subordinator T, we can create a finite activity subordinator
${\bf T}^{(k)}$
by truncating the size of the jumps to
$\{\|{\bf t}\|\in(1/k,\infty)\}$
,
$k>0$
, but it is not clear that the strongly subordinated process
$({\bf T}^{(k)},{\bf X}\circ{\bf T}^{(k)})$
is a Lévy process to which Theorem 2 can be applied.
Remark 2. It would be ideal if the assumption
$({\bf T},{\bf X}\circ {\bf T})\sim L^{2n}$
in Theorem 2 could be replaced by the weaker assumption
${\bf X}\circ {\bf T}\sim L^{n}$
. In the proof of Proposition 1 it is shown that
${\bf X}\circ {\bf T}\sim L^{n}$
implies
$({\bf T},{\bf X}\circ {\bf T})\sim L^{2n}$
under the assumption that T is deterministic. We conjecture that this result holds in general, although it is not clear how this can be proved. In Theorem 2, the assumption
$({\bf T},{\bf X}\circ {\bf T})\sim L^{2n}$
on the joint process is crucial to the proof.
4. Discussion
Let
${\bf T}\sim S^n$
and
${\bf X}\sim L^n$
be independent with
$n\geq 2$
. There are some simply stated but open questions on subordination of Lévy processes.
-
• If
${\bf X} \circ {\bf T}$ is a Lévy process, then can we conclude that
${\bf X}\circ {\bf T}\stackrel{D}{=} {\bf X}\odot {\bf T}$ ?
-
• What are the necessary and sufficient conditions on T and X such that
${\bf X}\circ{\bf T}$ is a Lévy process?
-
• If
${\bf X}\circ{\bf T}$ is not a Lévy process, what are the necessary and sufficient conditions on T and X such that it can be mimicked by some Lévy process Y in the sense that
$({\bf X}\circ{\bf T})(t) \stackrel{D}{=}{\bf Y}(t) $ for all
$t\geq 0$ ?
On the first question, if the answer is yes, then there cannot exist a different way to define the law of weak subordination for the class of subordinators
${\bf T}\sim S^n$
and subordinates
${\bf X}\sim L^n$
such that
${\bf X}\circ {\bf T}\sim L^n$
. Otherwise, if the answer is no, it would be interesting to determine the characteristics of
${\bf X}\circ {\bf T}$
. Here we have shown that the answer is yes if T is a deterministic subordinator, and it also holds under the stronger assumption
$({\bf T},{\bf X} \circ {\bf T})\sim L^{2n}$
if T is a pure-jump subordinator with finite activity.
On the second question, the sufficient conditions ()–() are well known. A partial converse has been given in [Reference Buchmann, Lu and Madan5, Proposition 3.9]. However, there are no known examples, outside of condition (), where
${\bf X}\circ {\bf T}\sim L^n$
(besides the trivial cases where some components of T are the zero process). If necessary and sufficient conditions were known, it might indeed turn out that there are no additional Lévy processes to which Proposition 1 and Theorem 2 are applicable besides those satisfying condition ().
We do not deal with the third question here, but [Reference Buchmann, Lu and Madan5, Proposition 3.4] shows that a sufficient condition for
$({\bf X}\circ{\bf T})(t) \stackrel{D}{=}({\bf X}\odot{\bf T})(t)$
, for all
$t\geq0$
, to hold is that
${\bf T}=(T_1,\ldots, T_n)$
has monotonic components, meaning that there exists a permutation
$\langle{(1),\ldots,(n)}\rangle$
such that
$T_{(1)}\leq \cdots \leq T_{(n)}$
. A partial converse is given in [Reference Buchmann, Lu and Madan5, Proposition 3.10], which suggests that outside of the monotonic assumption there may be no Lévy process mimicking
${\bf X}\circ {\bf T}$
. This raises the conjecture that if
${\bf X}\circ {\bf T}$
is not itself a Lévy process, then it can be mimicked by a Lévy process if and only if T has monotonic components. Furthermore, these results also suggest that the mimicking Lévy process, if it exists, may be
${\bf X}\odot{\bf T}$
.
Acknowledgements
This research was partially supported by ARC grant DP160104737. We thank the anonymous referees for their helpful comments.