1. Introduction
Let
be a locally square integrable martingale on
. The predictable quadratic variation of
is given by
Many authors studied the upper bound of
The celebrated Freedman inequality is as follows.
Theorem 1 (Freedman [
1]).
Let be a locally square integrable martingale on .If for each , then This result can be regarded as an extension of Hoeffding [
2]. Fan, Grama and Liu [
3,
4], and Rio [
5] obtained a series of remarkable extensions of Freedman inequality [
1]. See also Bercu et al. [
6] for a recent review in this field.
De la Peña [
7] establishes a nice exponential inequality for discrete time locally square integrable martingales.
Theorem 2 (De la Peña [
7]).
Let be a locally square integrable and conditionally symmetric martingale on . Then, This result is quite different from the classical Freedman’s inequality. The challenge for obtaining Theorem 1 is to find an approach based on the use of the exponential Markov’s inequality. De la Peña constructed a supermartingale to get Theorem 1. Furthermore, Bercu and Touati [
8] established the following result for self-normalized martingales, which are similar to Theorem 1.
Theorem 3 (Bercu and Touati [
8]).
Let be a locally square integrable martingale on . Then, for all , and , It is natural to ask what will happen when we study the continuous-time processes for the above cases? Let
be a stochastic basis.
is a continuous locally square integrable martingale. The predictable quadratic variation of
M,
, is a continuous increasing process, such that
is a local martingale. However, we cannot define an analogy for
M like
in Theorems 1 and 3. Since
has jumps, we can replace
by
. It is an interesting problem to consider De la Peña type inequalities for continuous-time local square integrable martingale with jumps. Some authors obtained the concentration inequalities for continuous-time stochastic processes. Bernstein’s inequality for local martingales with jumps was given by van der Geer [
9]. Khoshnevisan [
10] found some concentration inequalities for continuous martingales. Dzhaparidze and van Zanten [
11] extended Khoshnevisan’s results to martingales with jumps.
This paper focuses on the De la Peña type inequalities for stochastic integrals of multivariate point processes. Stochastic integrals of multivariate point processes are an essential example of purely discontinuous local martingales. Some useful facts and results essential for this paper’s proofs will be collected in
Section 2.
Section 3 will present our main results and give their proofs, while
Section 4 will derive an exponential inequality for block counting process in
coalescent as applications. Usually,
denote positive constants, which very often may be different at each occurrence.
2. Preliminaries
Let
be a stochastic basis. A stochastic process
is called a purely discontinuous local martingale if
and
M is orthogonal to all continuous local martingales. The reader is referred to the classic book [
12] due to Jacod and Shiryayev for more information. We shall restrict ourselves to the integer-valued random measure
on
induced by a
-valued multivariate point process. In particular, let
be a multivariate point process, and define
where
is the delta measure at point
. Then
for all
. Let
,
, where
is a Borel
-field on
and
a
-field generated by all left continuous adapted processes on
. The predictable function is a
-measurable function on
. Let
be the unique predictable compensator of
(up to a
-null set). Namely,
is a predictable random measure such that for any predictable function
W,
is a local martingale, where the
is defined by
Note the
admits the disintegration
where
is a transition kernel from
into
, and
is an increasing càdlág predictable process. For
in (
1), which is defined through multivariate point process,
admits
where
is a regular version of the conditional distribution of
with respect to
. In particular, if
, the point process
has the compensator
, which satisfies
Now, we define the stochastic integrals of multivariate point processes. For a stopping time
T,
is the graph of
T. For
in (
1), define
. With any measurable function
W on
, we define
, and
We denote by the set of all measurable real-valued functions W such that is local integrable variation process, where .
Definition 1. If , the stochastic integral of W with respect to is defined as a purely discontinuous local martingales, the jump process of which is indistinguishable from .
We denote the stochastic integral of W with respect to by . For a given predictable function W, is a purely discontinuous local martingale, which is defined through jump process. It is easy to prove that . Denote .
Itô’s formula for a purely discontinuous local martingale is essential for our proofs. Now, we present Itô’s formula for M.
Lemma 1 (Itô’s formula, Jacod and Shiryaev [
12]).
Let μ be a multivariate point process, ν be the predictable compensator of μ, W be a given predictable function on , and . Let f be a differentiable function, for and , Under some conditions, Wang, Lin and Su [
13] obtained
where
is the predictable quadratic variation process of
,
When
M is a purely discontinuous local martingale,
is a local martingale. There will be an interesting problem when the predictable quadratic variation
in (
3) is replaced by the quadratic variation
. In this paper, we will estimate the upper bound of two types of tail probabilities:
and
It is important to note that the continuity of
A implies the quasi-left continuity of
M. However, the quasi-left continuity of
M can be destroyed easily by changing the filtration in the underlying space. For example, let
N be a homogeneous Poisson process with respect to
. Let
be the sequence of the jump-times of
N. The process
N is not quasi-left continuous in the filtration
obtained by enlarging
initially with the
field
. (
is a sequence of
-stopping times announcing
). The main purpose of this paper consists in estimating (
4) and (
5) when
M is not quasi-left continuous.
3. The Main Results and Their Proofs
Now, we present our first main result.
Theorem 4. Let μ be a multivariate point process, ν be the predictable compensator of μ, , W be a given predictable function on , and . . Assume . Then, for , , Proof of Theorem 4. For simplicity of notation, put
where
.
Furthermore,
In addition, it is easy to see by noting
,
and
In combination, we have for all
where
. For any semimartingale
, the Doléans–Dade exponential is
We shall first show that the process
is a local martingale. Denote
,
.
The Itô formula yields
For
X, the jump part of
X is
where
D is the thin set, which is exhausted by
. Thus,
is a local martingale. Denote
, we have
Thus,
is a local martingale. Following the similar arguments in Wang Lin and Su [
13], we have
is a local martingale. In fact, set
,
,
and
The Itô formula yields
Since
, we have
Noting that
,
, we have
where
A is a predictable process, and
N is a local martingale. By the property of the Stieltjes integral, we have
Thus,
is a local martingale.
Let
and
Note by (4.12) in [
4], for
and
,
This implies
because
for any
, where
is the predictable compensate jump measure of
M. Inequality (
8) implies
. Since
and
,
for any stopping time
T. Thus,
is a supermartingale, where
Thus, on
We have
□
Put
where
for
. We have the following proposition from the proof of Theorem 4.
Proposition 1. Let μ be a multivariate point process, ν be the predictable compensator of μ, , W be a given predictable function on . . Denote , for . Then, is a local martingale.
In Theorem 4, the condition plays an important role. In the following theorem, we will present another result, which is the analogy of Theorem 1 in continuous time case.
Theorem 5. Let μ be a multivariate point process, ν be the predictable compensator of μ, , W be a given predictable function on , and . , In addition, defineand assume that for any and , . Then, for , , Proof of Theorem 5. By Proposition 1,
V is a local martingale. Note
for any
and
. We have
for any stopping time
T.
□
Remark 1. For integrable random variable ξ and a positive number , defineIf , and for all , . Then, ξ is called heavy on left. Bercu and Touati [14] extended Theorem 1 to general case. Let be a locally square integrable on . Iffor all and , Bercu and Touati [14] obtained In fact, our condition, , is analogy of (13) in continuous time case. Let be a homogeneous Poisson point process with parameter κ, and let be a sequence of i.i.d. r.v.’s with a common distribution function . Assume N is independent of . DefineThis is a so-called compound Poisson process. The jump measure of Y is given byand the predictable compensator isThus, is a purely discontinuous local martingale. For ,If for any , implies thatBercu and Touati [14] found that if is heavy on the left, then (17) holds. Thus, our condition is an analogy of (13) in continuous time case. In [
7,
15], there were obtained a series of exponential inequalities for events involving ratios in the context of continuous martingales, which in turn extended the results in [
10]. Su and Wang [
16] extended a similar problem for purely discontinuous local martingales in quasi-left continuous case. In this subsection, we obtained the similar inequality for stochastic integrals of a multivariate point process.
Theorem 6. Let μ be a multivariate point process, ν be the predictable compensator of μ, , W be a given predictable function on , and . Denote . Then, for all , Proof of Theorem 6. Recall that
is a local martingale, where
For any stopping time
T,
By Markov’s inequality, we obtain that for all
,
where
In fact,
Taking
, we get
Thus
□
From the proof of Theorem 6, we can obtain the following results.
Theorem 7. Let μ be a multivariate point process, ν be the predictable compensator of μ, , W be a given predictable function on , and . Denote . In addition, defineand assume that for any and , . Then for all , , 4. Application
In this section, we will derive exponential inequalities for block counting process in
coalescent. The
coalescent was introduced independently by Pitman [
17] and Sagitov [
18]. In this paper, the notation and details of
coalescent are from Limic and Talarczyk [
19].
Let
be an probability measure on
,
is a Markov jump process.
takes values in the set of partition of
. For any
, the restriction
of
to
is a continuous time Markov chain with the following transitions: when
has
b blocks, any given
tuples of blocks coalesces at rate
where
. Let
be the number of blocks of
at
t. In fact,
is a point process. Limic and Talarczyk [
19] presented integral equation for
N. Define
where
is an independent array of i.i.d. random variables
, where
have uniform distribution on
. The multivariate point processes
have the compensator
.
Limic and Talarczyk [
19] found that
for all
, where
Define
and
plays important role in the study of
coalescent. Limic and Talarczyk [
19] obtained that
M is a square integrable martingale. It is not difficult to see that
,
and
We have the following result.
Theorem 8. Let M be defined as above, we haveandwhere , .