On the Exponential Process associated with a CARMA-type Process
Muneya Matsui†and Narn-Rueih Shieh‡
Abstract
We study the correlation decay and the expected maximal increments of the exponential processes determined by continuous-time autoregressive moving average type processes of order (p, q). We consider two background driving processes, namely fractional Brownian motions and L´evy processes with exponential moments. The results presented in this paper are significant extensions of those very recent works on the Ornstein-Uhlenbeck-type case (p = 1, q = 0), and we develop more refined techniques to meet the general (p, q). In a concluding section, we discuss the perspective role of exponential CARMA-type processes in stochastic modeling of the burst phenomena in telecommunications and the leverage effect in financial econometrics.
Keywords: Continuous-time ARMA process, Fractional Brownian motion, L´evy process, Expo- nential process, Correlation decay, Maximal increments.
AMS 2000 Subject classifications. Primary 60G17, 60G15; Secondly 62M10, 60G10.
1 Introduction
Empirically observed data from financial markets usually are of high-frequency, and those from envi- ronmental sciences (hydrology and geology in particular) usually are irregularly spaced. Moreover, they exhibit characteristics of long-memory and/or non-Gaussianity. Therefore, the traditional ARMA or ARIMA time-series models must be treated in the continuous-time setting and to reflect the above characteristics, and various modelings have been proposed and studied in detail; see re- cent papers by [26, 27, 28, 29] for FBM driven models and [7, 8, 9] for L´evy driven models, and the references therein. In this paper, our model is based on Continuous-time Autoregressive-Moving- Average type (CARMA-type, for brevity) processes driven by fractional Brownian Motions and by L´evy processes, as those papers cited above. We shall study the exponential processes associated with these CARMA-type processes.
We mention that, in recent years, it has been of great interest to study the exponential func- tionals and the exponential processes determined by Brownian motion and by L´evy processes, see [10, 11, 17], with the view toward application in financial economics. There also appear papers by [2, 21] to study the exponential processes associated with Ornstein-Uhlenbeck-type (OU-type for short) processes driven by fractional Brownian Motions and by L´evy processes, which are motivated
†Correspondence to Department of Business Administration, Nanzan University, 18 Yamazato-cho, Showa-ku, Nagoya 466-8673, Japan. E-mail:mmuneya@nanzan-u.ac.jp
‡Correspondence to Department of Mathematics and Statistics, York University, 4700 Keele Street, Toronto ON M3J 1P3, Canada. On leave from National Taiwan University. E-mail:shiehnr@ntu.edu.tw
by the connection to Kahane-Mandelbrot theory of turbulence. Furthermore, these exponential pro- cesses play roles in the context of stochastic volatility modelling (see, for example, [22, Chapter 11]). Indeed, the exponential CARMA-type processes driven by L´evy processes have been used very recently to study the leverage effect between the stock return and the volatility of continuous-time financial data, [16].
We study two fundamentally important properties of such an exponential processes Z. We study the correlation decay
Cov (Z(t), Z(t + s)) as s↑ ∞, and estimate the upper bound for the expected maximal increments
E
·
t≤s≤t+rmax |Z(s) − Z(t)|
¸
as r ↓ 0.
The first result is useful to understand the spectral structure of the process. The second result is of intrinsic importance to the path variation (hence toward various applications) of the process.
In Section 2, we present some preliminaries. In Sections 3 and 4, we state the main results, respectively, on the CARMA processes driven by a fractional Brownian motion and by a L´evy process. We present all proofs of our results in Section 5. In a concluding Section 6, we discuss the advantage of our exponential CARMA-type process on stochastic modeling of the burst phenomena in telecommunications and the leverage effect in financial econometrics.
Finally, we should mention that, though the CARMA-type process given in Section 2 below may lead one to feel that it could be a superposition of OU-type processes. This is not true, since the superpositions proposed in, say, [3, 5] mean to a sum of independent OU-type processes, while the expression in Section 2 below for a CARMA-type process is a sum of the same driving force and the parameters λi in the expression are complex-valued. Therefore, the existing techniques for superposed OU-type processes are not applicable to the CARMA-type processes discussed in this paper, and we develop more refined techniques to prove the main results of this paper.
2 Preliminaries
2.1 Fractional Brownian motions and L´evy processes
We begin with a review on the definitions and properties of two background driving processes, fractional Brownian motion (FBM for short) and L´evy process (LP for short).
Definition 2.1 Let 0 < H < 1. A fractional Brownian motion{BH(t)}t∈Ris a real-valued centered Gaussian process, with BH(0) = 0 and Cov¡
BH(s), BH(t)¢
= 12¡
|t|2H+|s|2H− |t − s|2H¢
, (t, s)∈ R2.
It is well known that FBM has stationary increments and self-similarity with index H. We may and will assume that the sample path of a FBM is everywhere continuous. For more on FBM we refer to e.g. [15].
Definition 2.2 A real-valued process L :={L(t)}t≥0 with L(0) = 0 is a L´evy process on R, if (i) For any choice of n≥ 1 and 0 ≤ t0 < t1· · · < tn, the random variables L(t0), L(t1)−L(t0), L(t2)− L(t1), . . . , L(tn)− L(tn−1) are independent.
(ii) L(t)− L(s) has the same distribution as L(t − s), 0 ≤ s < t < ∞, and (iii) L is continuous in probability.
We refer to [25] for intensive study on L´evy processes. We may and will assume that the sample path of a LP is everywhere right-continuous (L(t+) = L(t)) with left-limits L(t−), and we may also assume that it is quasi-left-continuous (see [25, p.279]).
It is well-known that a LP is characterized by its generating triplet (b, a2, ν), where b ∈ R is the drift part, a2 ≥ 0 is the variance of the Gaussian part, and ν is the L´evy measure of the jump part, i.e. it is a σ-finite measure onR \ {0} satisfying
Z
R\{0}(1∧ |x|2)ν(dx) <∞.
The characteristic function of L(1) is determined as ξ(θ) := log E[eiθL(1)] = ibθ−a2
2 θ2+ Z
R\{0}(eiθx− 1 − iθx1{|x|≤1})ν(dx), θ∈ R, (2.1) and that for L(t) is given as etξ(θ). Hence we know the distribution of L(t) for any t≥ 0.
The following form of the L´evy-Itˆo decomposition is written as [1, p.108], L(t) = bt + aB(t) +
Z
|x|<1x ˜N ([0, t], dx) + Z
|x|≥1xN ([0, t], dx),
where B is the standard Brownian motion on R, and N(ds, dx) (resp. ˜N (ds, dx)) is the Poisson (resp. compensated Poisson) random measure onR+
N(R\{0}) with intensity measure dsN ν(dx).
In this paper, since the second moment is essential, we assume that E[(L(1))2] < ∞ and is scaled so that Var(L(1)) = 1.
In order to obtain causal stationarity CARMA processes we define a two-sided (in time) L´evy process
L(t) =
½ L1(t) if t≥ 0
L2((−t)−) if t < 0, (2.2)
where{L1(t)}t≥0 and{L2(t)}t≥0 are independent copies of{L(t)}t≥0. 2.2 CARMA processes driven by FBM and by LP
Now we define a CARMA-type process Y := Y (p, q) as follows. We use similar notations as those in [8] and in [26]. The following framework of CARMA processes are the same for both two cases.
Let 0≤ q < p, the process Y (p, q) is represented by observation and state equations;
Y (t) = β0X(t), t≥ 0, (2.3)
dX(t) = AX(t)dt + δpdW (t), (2.4)
where
A =
0 1 0 · · · 0 0 0 1 · · · 0 ... ... ... . .. ...
0 0 0 · · · 1 α1 α2 α3 · · · αp
, X(t) =
X(0)(t) X(1)(t)
... X(p−2)(t) X(p−1)(t)
, δp=
0 0 ... 0 1
, β =
1 β1
... βp−2
βp−1
and the coefficients βj satisfy βj = 0 for j > q.
In the above, W := {W (t)}t∈R is the driving process, which is BM σB (σ > 0 and B is the standard BM) in the classical context. In this work, we will firstly consider W to be FBM and secondly consider W to be LP. The first one is Gaussian with long-range dependence, in case 1/2 < H, and the second one is non-Gaussian Markovian.
When we take the driving process to be FBM σBH (σ > 0 and BH is the FBM defined above), we write Y := Y (p, H, q) to be the associated CARMA process, which has appeared in [26], yet without loss of essential significance we skip the drift term appeared in [26, (3)]. The solution of (2.4) can be written as
X(t) = eAtX(0) + σ Z t
0
eA(t−u)δpdBH(u), where eAt = I +P∞
n=1
©(At)n(n!)−1ª
and I is the identity matrix. Here we confine to the causal stationary solution ([26, p.182]). If the eigenvalues of A all have negative real parts, a strictly stationary solution of (2.4) is given by the form;
X(t) = σ Z t
−∞eA(t−u)δpdBH(u), t≥ 0.
This solution can be extended over all real t in natural way. Thus the CARMA-type process Y := Y (p, H, q) has the moving-average form:
Y (t) = σ Z t
−∞g(t− u)dBH(u), t∈ R, (2.5)
where
g(t) =
½ β0eAtδp if t≥ 0
0 otherwise. (2.6)
The kernel g here is given in the closed form and for our purpose it is desirable to get a more tractable expression. In [8, p.483] it is shown that, when all eigenvalues λ = (λ1, . . . , λp) of A have negative real parts and are all distinct, g has the form
g(t) = Xp
i=1
β(λi)
α(1)(λi)eλit, (2.7)
where α(z) = zp− αpzp−1− · · · − α1, α(1)(·) denotes its first derivative, β(z) = 1 + β1z + β2z2+
· · · + βqzq. Consequently, Y (t) can be written as
Y (t) = σ Xp
i=1
β(λi) α(1)(λi)
Z t
−∞eλi(t−u)dBH(u). (2.8) In this form one can see the clear difference between CARMA-type processes and superposition of OU-type processes. A more general expression for (2.6) is obtained in [29] (see also [8]), and (2.7) is the most explicit case. In this paper we work in the form (2.8), namely we always assume the condition below.
Assumption 2.1 All eigenvalues of λ = (λ1, . . . , λp) of A have negative real parts and are all distinct, and hence the kernel g in (2.6) has expression (2.7).
The autocovariance function of Y , which characterizes the memory property of the process, is shown in [26, Theorem 2]. Under Assumption 2.1, for h≥ 0, the autocovariance function of Y is equal to
γY(h) := Cov(Y (t + h), Y (t))
= σ2
2 Γ(2H + 1) Xp i=1
β(λi)β(−λi)
α(1)(λi)α(−λi)u(H, λi, h), (2.9) where
u(H, λ, h) = 2(−λ)1−2Hcosh(λh) + λ1−2HeλhP (2H, λh)− (−λ)1−2He−λhP (2H,−λh) and P (a, z) =Rz
0 e−uua−1du/Γ(a), z∈ C. For H 6= 1/2 as h → ∞, we have the asymptotic behavior γY(h) = σ2H(2H− 1)β2(0)
α2(0)h2H−2+ O(h2H−3). (2.10) This shows that Y (p, H, q) with H ∈ (12, 1) exhibits long-range dependence. When H ∈ (12, 1) the autocovariance function of Y has another expression
γY(h) = σ2H(2H− 1) Z 0
−∞
Z h
−∞g(h− u)g(−v)|u − v|2H−2dudv, (2.11) in which we easily see the sign of γY(h). Finally, we note that the Y (1, H, 0) process is the fractional Ornstein-Uhlenbeck (FOU for short) process,
Y (t) = σ Z t
−∞eα1(t−u)dBH(u), and its autocovariance function is previously known (e.g. [12]) to be
γY(h) = σ2cH Z ∞
−∞eihx|x|1−2H
α21+ x2dx, cH = Γ(2H + 1) sin(πH)
2π .
Thus we see again that CARMA process Y (p, H, q) is a natural extension of FOU process.
Next, we replace the driving FBM σBH in the above construction by a LP L to obtain a corresponding CARMA-type process Y := Y (p, L, q) driven by L. Recall we suppose the second moment for L and Var(L(1)) = 1. We remark that a L´evy driven CARMA-type process has been studied in [7, 8].
We again suppose Assumption 2.1. Then, similar to the FBM case, the process {X(t)}t∈R
defined by X(t) = Rt
−∞eA(t−u)δpdL(u), t ∈ R, is the causal strictly stationary solution of (2.4) with W now taken to be L, and the corresponding CARMA process is
Y (t) = Z t
−∞g(t− u)dL(u), (2.12)
where g(t) is given by (2.6). The following expression which we use later is useful, since X(0) is independent of{L(t)}t≥0,
Y (t) = β0eAtX(0) + Z t
0
β0eA(t−u)δpdL(u). (2.13)
Under the Assumption 2.1 we have the expression (2.7) for the kernel g, which yields a corresponding expression, as that in the F BM -driven case, we have
Y (t) = Xp
i=1
β(λi) α(1)(λi)
Z t
−∞eλi(t−u)dL(u).
It is immediate that EY (t) = −β0A−1δpE[L(1)], and due to [8, Remark 5] the autocovariance function has a form,
γY(h) = Xp
i=1
β(λi)β(−λi)
α(1)(λi)α(−λi)eλi|h|. (2.14) We thus see that the L´evy driven CARMA process exhibits short memory. Note that, despite a L´evy driven CARMA process is different from the superposition of p independent L´evy driven OU processes, as seen in (2.14) the correlation structures nevertheless look similar.
Since Y is an integral with a L´evy process, the distribution of Y (t) is given by the cumulant generating function;
log E[eiθY (t)] = Z ∞
0
ξ(θg(u))du, where ξ is given in (2.1).
The purpose of this paper is to study the exponential process associated with Y , Z :={Z(t)}t∈R:={eY (t)}t∈R,
where Y is Y (p, H, q) and Y (p, L, q). For the OU-type case, i.e. Y (1, H, 0) and Y (1, L, 0), the associated process is investigated very recently in [21] and, respectively, in [2]. The present work is a certain continuation of these two papers, yet we need to develop more refined techniques to meet the general (p, q).
Finally, we mention that, upon various modelling needs, for example [7, 22, 28, 29], we are led to consider the situation: the kernel g is non-negative, and/or the correlations for the exponential process Z are non-negative. For instance, in stochastic volatility modelling, the volatility processes are known to be clustering, which suggests positivity of their autocorrelation functions. Thus we also impose
Assumption 2.2 Assume that the kernel g defined in (2.6) is non-negative.
A condition for g to be non-negative is given in [28, 29]. In our case, Assumption 2.2 implies non- negativity for the correlations of our exponential processes, as we show in the following sections.
3 The main result I: the FBM-driven case
Recall that we assume Assumptions 2.1 throughout. Firstly we study the correlation decay of the exponential processes.
Lemma 3.1 Let Z(t) := eY (t) be the exponential process determined by a stationary Gaussian process Y (t). Then
Cov(Z(0), Z(s))T 0 if and only if Cov(Y (0), Y (s))T 0.
Moreover, assume that as s→ ∞, Cov(Y (0), Y (s)) → 0. Then
Cov(Z(0), Z(s)) = exp [Var(Y (0))]{Cov(Y (0), Y (s)) + o(Cov(Y (0), Y (s)))} . (3.1)
Due to the stationarity the same result holds for Cov(Z(t), Z(t + s)), t ∈ R in Lemma 3.1. Now one can see that Assumption 2.2 implies non-negative correlation of the process Z since by (2.11) the correlation of Y is non-negative under this assumption. The following corollary is immediate from Lemma 3.1 with the covariance decay (2.10) for Y (p, H, q).
Corollary 3.1 Let Y := Y (p, H, q) process with H ∈ (12, 1). Assume 2.1 and 2.2, then the station- ary process Z(t) := eY (t) has the following covariance structure as s→ ∞;
0≤ Cov(Z(t), Z(t + s)) = exp(γY(0))©
γY(s) + O((γY(s))2)ª
= exp(γY(0))
½
σ2H(2H− 1)β2(0)
α2(0)s2H−2+ O(s2H−3)
¾ , where γY(·) is the autocovariance function of Y .
Remark 3.1 Assumption 2.2 is only used to assure the non-negativity for the autocorrelation function of Z; indeed, if we only concern the correlation decay in Lemma 3.1 and Corollary 3.1, then this assumption can be dropped.
Next, we estimate the expected maximum increments of the process Z associated with Y (p, H, q).
We need two lemmas. The first one is Lemma 2.3 of [21], which is based on Statement 4.8 of [23].
The second one requires new proof.
Lemma 3.2 Let H ∈ [12, 1). Then for any r≥ 0 and t ∈ R we have
E
·µ
t≤s≤t+rmax ¯¯BH(s)¯¯¶m¸
≤ (
rHm 2
√2
√π(m− 1)!! if m is odd
rHm2(m− 1)!! if m is even. (3.2) Lemma 3.3 Let H ∈ [12, 1), m = 1, 2, . . . and Y := Y (p, H, q), and assume 2.1. Then the station- ary process Y has the following bound for m-th moment of maximal increments for all r≥ 0
E[max0≤s≤r|Y (s) − Y (0)|m]
m! ≤ C(λ)rHm, (3.3)
where λ = (λ1, λ2, . . . , λp) are eigenvalues of A and C(λ) is a constant, which depends on λ and does not depend on any m or H.
Now, we state the harder result in this section as follows.
Theorem 3.1 Let H ∈ [12, 1) and Y := Y (p, H, q). Define the stationary process Z(t) := eY (t). Then there exists a constant C(H, λ) such that for all r with rH < 1/2 and all t≥ 0,
E
·
t≤s≤t+rmax |Z(s) − Z(t)|
¸
≤ C(H, λ)rH. (3.4)
Here λ = (λ1, λ2, . . . , λp) are eigenvalues of A and C(H, λ) depends on H and λ.
Remark 3.2 To our knowledge, the maximal inequality in the above is new even when the driving process is Brownian motion(H = 1/2).
4 The main result II: the L´ evy-driven case
We begin with the correlation decay. Under Assumption 2.1, the CARMA process Y := Y (p, L, q) is strictly stationary and has causal stationary representations (2.12), as that mentioned in Section 2.
We assume that the L´evy process L given in (2.1) has no Gaussian part, i.e. a2 = 0, and that we denote the drift part by bt and the Poisson random measure of the jump part by N (ds, dx) for which the intensity measure is dsN
ν(dx). From now on, we always use γ to denote the following finite and positive quantity,
γ := p sup
i∈{1,...,p}
¯¯¯¯ β(λi) α(1)(λi)
¯¯¯¯, by which we bound g as 0≤ g(u) ≤ γ, ∀u ≥ 0.
We start with two lemmas; the second one is the moment generating function of Y , namely the moment of Z = eY.
Lemma 4.1 Let g the kernel given in (2.7). For every t > 0 it follows that Z 0
−∞g(t− u)du ≤ M1e−M2t,
Z 0
−∞g2(t− u)du ≤ M3e−M4t, Z t
0
g(t− u)du ≤ M5, Z t
0
g2(t− u)du ≤ M6, where Mi, i = 1, . . . , 6 are some positive constants.
Lemma 4.2 Let Y := Y (p, L, q) and the kernel g as in (2.7). Assume 2.1 and 2.2, and suppose, for a given θ > 0, that
Z
x>1
eθγxxν(dx) <∞ and Z
x<−1|x|ν(dx) < ∞. (4.1) Then, for each t > 0 we have
E[eθY (t)] = E[(Z(t))θ]
= exp
½ θb
Z t
−∞g(t− s)ds
¾
× exp
½Z t
−∞
Z
R
£exp(θg(t− s)x) − 1 − θg(t − s)x1{|x|≤1}¤
ν(dx)ds
¾
. (4.2)
We state the covariance decay of the exponential process Z as follows.
Proposition 4.1 Let Z :={Z(t)}t≥0 := {eY (t)}t≥0 be the exponential process associated with the CARMA process Y (p, L, q). Assume 2.1 and 2.2, and suppose that the L´evy measure ν of L satisfies
Z
x>1
e2γxxν(dx) <∞ and Z
x<−1x2ν(dx) <∞. (4.3) Then, there exist positive constants c0 and c1 such that
0≤ Cov(Z(0), Z(s)) ≤ c0e−c1s (4.4)
for all s > 0.
Due to the stationarity the same result holds for Cov(Z(t), Z(t + s)), t∈ R. At here the assumption on the L´evy measure is stronger than that in Lemma 4.2; yet, this is needed for the existence of E[Z(0)Z(s)].
Remark 4.1 (1). Proposition 4.1 extends the previous OU-type case (p = 1, q = 0) [2, Proposition 1] to the general CARMA-type, and its proof in Section 5 also corrects the previous flaw. Moreover, it now only requires the exponential decay of the L´evy measure ν(dx) on x > 1 and the second moment on x <−1.
(2). In [16, Remark 5.1], the authors also obtained the exponential decay of the covariance function.
However, their condition on the L´evy measure (in their Propositions 4.3 and 4.4) is stronger than ours, and we are able to show the positivity of the covariance function which is not achieved by their method. The positivity of the covariance function is important in both the applications to the telecommunications (see Section 6 below) and to the volatility clustering (see, for example, [22]).
(3). If we replace (4.1) and (4.3) with a stronger condition, namely suitable exponential decay of the L´evy measure ν(dx) as x tends to both ±∞, then Proposition 4.1 can be valid without Assumption 2.2.
(4). The covariance decay of Z = eY is closely related with quantity
U (θ1, θ2; t) = E[ei(θ1Y (t)+θ2Y (0))]− E[eiθ1Y (t)]E[eiθ1Y (0)], −∞ < θ1, θ2 <∞,
in [24, p.212 and pp.580-581], which characterizes the mixing property of any stationary infinitely divisible process. Thus, we can apply and assert that Y (p, L, q) is mixing, suppose that the ex- ponential integrability imposed on the L´evy measure ν(dx) is valid for whole range θ. In this aspect, we mention that: in [16, Theorem 3.1(i)] the authors prove, by a suitable application of [20, Theorem 4.3], that both Y and Z can be strongly mixing, under somewhat different conditions.
Now we present the harder estimate, the expected maximum increments of the exponential process Z := eY.
Theorem 4.1 Let Y := Y (p, L, q) and the kernel g be as in (2.6). Assume for a given θ > 0, that Z
x>1
e2θγxxν(dx) <∞ and Z
x<−1x2ν(dx) <∞. (4.5) Then, for t0> 0 small enough, there exists a positive constant C such that
E
·
0max≤t≤t0
|(Z(t))θ− (Z(0))θ|
¸
≤ C√ t0.
Here C depends on the kernel g and L´evy measure ν(dx) and drift parameter b.
Remark 4.2 1. As one may see from the proof given in the next Section 5, what we require on the kernel g for Theorem 4.1 is that |g(u)| ≤ γ, ∀u ≥ 0 (may also allow g to be ±). This can be achieved, for example, when Re(λj) < 0 (yet not necessarily all λj are distinct), as one may see in a general expression for g given in [29]).
2. Theorem 4.1 is still valid when the L´evy triplet has Gaussian part; we simply combine Theorems 3.1 (with H = 1/2) and 4.1 and use the L´evy-Itˆo decomposition theorem. We leave it to readers for the detail.
5 Proofs
Proof of Lemma 3.1.
Since the distribution of (Y (0), Y (s)) is a bivariate Gaussian, its moment generating function taken at 1 yields
E[Z(0)Z(s)] = E[eY (0)+Y (s)]
= exp
½1
2(1, 1)Σ(1, 1)0
¾ , where
Σ =
µ Var (Y (0)) Cov (Y (0), Y (s)) Cov (Y (0), Y (s)) Var (Y (s))
¶ . Then by using Cov (Y (0), Y (s))→ 0 as s → ∞ and ex =P∞
n=0xn/n! for|x| < ∞, we have Cov (Z(0), Z(s))
= exp{Var (Y (0))} [exp {Cov(Y (0), Y (s))} − 1] (5.1)
= exp{Var (Y (0))}
"
Cov(Y (0), Y (s)) + (Cov (Y (0), Y (s)))2
2! + (Cov (Y (0), Y (s)))3 3! +· · ·
#
= exp{Var (Y (0))}£
Cov(Y (0), Y (s)) + o¡
{Cov(Y (0), Y (s))}2¢¤
.
2 Proof of Lemma 3.3.
Without loss of generality set σ = 1. By Assumption 2.1 we use the fact Re(λi) < 0 for all i = 1, 2, . . . , p without mention. The increment has the following form via the integral by parts formula (see Theorem 2.21 of [30]).
Y (s)− Y (0) = Xp
i=1
β(λi) α(1)(λi)
µZ s
−∞eλi(s−u)dBH(u)− Z 0
−∞e−λiudBH(u)
¶
= Xp
i=1
β(λi) α(1)(λi)
µ
BH(s) + λi
Z s
0
eλi(s−u)BH(u)du + (eλis− 1)Yi(0)
¶ ,
where for simplicity we denote R0
−∞e−λiudBH(u) by Yi(0). Take absolute value of this to obtain
|Y (s) − Y (0)| ≤ Xp i=1
¯¯¯¯ β(λi) α(1)(λi)
¯¯¯¯
Ã
|BH(s)| + max
0≤u≤s|BH(u)| |λi|
Re(λi)(eRe(λi)s− 1) +(1− eRe(λi)s)|Yi(0)|
!
≤ C Ã
|BH(s)| + max
0≤u≤s|BH(u)|(1 − eµs) + (1− eµs) Xp
i=1
|Yi(0)|
! ,
where C = p supi¯¯¯αβ(λ(1)(λi)i) λi
Re(λi)¯¯¯ and µ = infiRe(λi) < 0. Take m-th product of this to get
|Y (s) − Y (0)|m
≤ Cm X
j+k+`=m;j,k,`≥0
m!
j! k! `!|BH(s)|j
½
0max≤u≤s|BH(u)|(1 − eµs)
¾k(
(1− eµs) Xp
i=1
|Yi(0)| )`
≤ Cm X
j+k+`=m;j,k,`≥0
m!
j! k! `!
µ
0max≤u≤s|BH(u)|
¶j+k
(1− eµs)k+`
à p X
i=1
|Yi(0)|
!`
.
Moreover, taking expectation of maxima of this, we have E
·
0max≤s≤r|Y (s) − Y (0)|m
¸
≤ Cm X
j+k+`=m;j,k,`≥0
m!
j! k! `!E
max
0≤u≤s|BH(u)|j+k à p
X
i=1
|Yi(0)|
!`
(1 − eµs)k+`. (5.2)
Let Y (0) :=R0
−∞e−ηsdBH(s) with η = supiRe(λi) < 0. To analyze expectation is (5.2) we start to bound E[|Yn(0)|2`], n = 1, . . . , p by E[|Y (0)|2`]. Define the real and the imaginary part of Yn(0) as
Yn(0) = Z 0
−∞e−(iIm(λn)+Re(λn))udBH(u)
= Z 0
−∞e−Re(λn)ucos(Im(λn)u)dBH(u)− i Z 0
−∞e−Re(λn)usin(Im(λn)u)dBH(u)
=: RYn(0)− iIYn(0), n = 1, . . . , p.
Noticing |Yn(0)|2 = (RYn(0))2+ (IYn(0))2 we have E[|Yn(0)|2`]≤ 22`−1³
E[(RYn(0))2`] + E[(IRn(0))2`]
´ .
Since RYn(0) and IYn(0) are both Gaussian and their variances are smaller than E[|Y (0)|2], we further bound E[|Yn(0)|2`] as
E[|Yn(0)|2`]≤ 22`E[|Y (0)|2`]≤ 22`(2`− 1)!!M12`, (5.3) where M1 =
q
Var(Y (0)). Due to this together with Lemma 3.2, the expectation in each term of the sum is bounded as
E
max
0≤u≤s|BH(u)|j+k à p
X
i=1
|Yi(0)|
!`
≤ s
E
·
0max≤u≤s|BH(u)|2(j+k)
¸vuu tE
"
p2`−1 Xp
i=1
|Yi(0)|2`
#
≤ s
E
·
0max≤u≤s|BH(u)|2(j+k)¸q E£
(2p)2`|Y (0)|2`¤
≤q
2(2(j + k)− 1)!!r2H(j+k)q
(2`− 1)!!(2pM1)2`
≤q
2j+k(j + k)!r2H(j+k) q
2``!(2pM1)2`
=p
(j + k)!`!(√
2rH)j+k(2√
2pM1)`. (5.4)
Here M1 depends on the parameter H and η by the definition. However since H ∈ [12, 1), we can make M1 to attain a certain bound regardless of H. In fact
Var(Y (0)) = 2σ2cH Z ∞
−∞
|x|1−2H
η2+ x2dx, cH = Γ(2H + 1) sin(πH)
2π .
Note that as H ↑ 1 the integral diverges, but simultaneously cH ↓ 0 in the same rate. Therefore, we conclude that Var(Y (0)) is bounded regardless of H ∈ [12, 1).
Now substituting (5.4) into (5.2) and dividing by m!, we have E [max0≤s≤r|Y (s) − Y (0)|m]
m!
≤ Cm
√m!
X
j+k+`=m;j,k,`≥0
m!
j! k! `!
r(j + k)! `!
m! (√
2rH)j(√
2rH(1− eµs))k(2√
2pM1(1− eµs))`
≤ 1
√m!Cmn√
2rH +√
2rH(1− eµs) + 2√
2pM1(1− eµs) om
≤ rHm
√m!Cm½√
2 +√
2(1− eµs) + 2√ 2pM1
(1− eµs) rH
¾m .
Here we observe that (1− eµs)/rH ≤ |µ| for any 0 ≤ s ≤ r and any H ∈ [12, 1) and that cm/m!≤ ec for any c > 0 and any m = 1, 2, . . .. Hence we can take a universal constant C(λ) > 0 and obtain
E [max0≤s≤r|Y (s) − Y (0)|m]
m! ≤ C(λ)rHm.
2 Proof of Theorem 3.1.
Once we have Lemma 3.3, the proof of the theorem can be proceeded exactly as that of of Theorem
2.1 in [21]; thus we omit it. 2
Proof of Lemma 4.1.
The expression (2.7) of g yields
|g(t − u)| = ¯¯
¯¯¯ Xp i=1
β(λi)
α(1)(λi)eλi(t−u)
¯¯¯¯
¯ ≤ γ p
Xp i=1
eRe(λi)(t−u).
Then, since we assume that Re(λi) < 0, i = 1, . . . , p, the results are obtained by basic calculations.
2
Proof of Lemma 4.2.
Since g is a non-negative bounded continuous function with g ∈ L1 ∩ L2, due to approximation Lemma in [13, p.91], we may take a step function gn(x) =Pn−1
i=1 ai1(si,si+1](x), where ai ≥ 0, i = 1, . . . , n and −∞ < s1 < s2<· · · < sn<∞ such that
gn(x)≤ gn+1(x)≤ g(x),
nlim→∞gn(x) = g(x) a.e. and gn→ g ∈ L1∩ L2. Let Yn(t) = Rt
−∞gn(t− u)dL(u), then by [25, p.165], with the drift b and without the Gaussian part at there, we see that
E h
eθ
Rt
−∞gn(t−u)dL(u)i
= exp
½ θb
Z t
−∞gn(t− u)du + Z t
−∞
Z
R[eθgn(t−u)x− 1 − θgn(t− u)x1{|x|≤1}]ν(dx)du
¾
. (5.5) The integrand of the last integral is bounded by
θ2eθγ|x||gn(t− u)|2|x|21{|x|≤1}+ θ|gn(t− u)||x|³
eθγx1{x>1}+ 1{x<−1}
´
≤ θ2eθγ|x||g(t − u)|2|x|21{|x|≤1}+ θg(t− u)|x|³
eθγx1{x>1}+ 1{x<−1}
´ ,
which is integrable. Hence by the dominated convergence theorem the right hand side of E[eθR−∞t gn(t−u)dL(u)] converges to (4.2) and (4.2) is continuous at θ = 0. Then due to definition of stochastic integral,
(4.2) is turned out to be the moment generating function of Y . 2 Proof of Proposition 4.1.
Due to the expression (2.13), where X(0) and {L(t)}t≥0 are independent, it follows that Cov(Z(0), Z(t))
= E[eY (0)+Y (t)]− E[eY (0)]E[eY (t)]
= E[eβ0X(0)+β0eAtX(0)]E
· exp
½Z t 0
g(t− u)dL(u)
¾¸
−Eh eβ0X(0)
i E
h
eβ0eAtX(0) i
E
· exp
½Z t
0
g(t− u)dL(u)
¾¸
= E
· exp
½Z t 0
g(t− u)dL(u)
¾¸
| {z }
I
³ E
h
eβ0X(0)+β0eAtX(0)
i− Eh eβ0X(0)
i E
h
eβ0eAtX(0) i´
| {z }
II
.
We begin to see the calculation of the quantity I. By Lemma 4.2 and Taylor expansion it follows that
I ≤ exp
½ b
Z t
0
g(t− u)du + Z t
0
Z
R
¯¯¯eg(t−u)x− 1 − g(t − u)x1{|x|≤1}¯¯¯ ν(dx)du
¾
≤ exp (
b Z t
0
g(t− u)du + Z t
0
Z
|x|≤1eg(t−u)|x|g2(t− u)|x|2ν(dx)du +
Z t
0
Z
x>1
eg(t−u)xg(t− u)xν(dx)du + Z t
0
Z
x<−1g(t− u)|x|ν(dx)du
¾
≤ exp (
b Z t
0
g(t− u)du + Z t
0
g2(t− u)du Z
|x|≤1eγ|x|2ν(dx) +
Z t
0
g(t− u)du µZ
x>1
eγxxν(dx) + Z
x<−1|x|ν(dx)
¶¾ . Here in the second step we use the inequalities;
ex− 1 ≤ exx and 1− e−x ≤ x, x ≥ 0. (5.6) In view of Lemmas 4.1 and 4.2, the quantity I is proven to be bounded uniformly in t. Again by Lemma 4.2 we have
E[eβ0(I+eAt)X(0)]
= exp
½ b
Z 0
−∞(g(t− u) + g(−u))du
¾
× exp
½Z 0
−∞
Z
R[e(g(t−u)+g(−u))x− 1 − (g(t − u) + g(−u))x1{|x|≤1}]ν(dx)du
¾
and
E[eβ0X(0)]E[eβ0eAtX(0)]
= exp
½ b
Z 0
−∞(g(t− u) + g(−u))du
¾
× exp
½Z 0
−∞
Z
R[eg(−u)x− 1 − g(−u)x1{|x|≤1}]ν(dx)du
¾
× exp
½Z 0
−∞
Z
R[eg(t−u)x− 1 − g(t − u)x1{|x|≤1}]ν(dx)du
¾ . Thus we have
II = exp
½ b
Z 0
−∞(g(t− u) + g(−u))du
¾
| {z }
IIa
×
· exp
½Z 0
−∞
Z
R[e(g(t−u)+g(−u))x− eg(−u)x− eg(t−u)x+ 1]ν(dx)du
¾
− 1
¸
| {z }
IIb
× exp
½Z 0
−∞
Z
R[eg(−u)x− 1 − g(−u)x1{|x|≤1}]ν(dx)du
¾
| {z }
IIc
× exp
½Z 0
−∞
Z
R[eg(t−u)x− 1 − g(t − u)x1{|x|≤1}]ν(dx)du
¾ .
| {z }
IId
We study IIa, IIb, and IIcand IIdin order. Due to Lemma 4.1, the quantity IIais bounded uniformly in t. Again by (5.6) it is easy that
IIb := exp
½Z 0
−∞
Z
R(eg(t−u)x− 1)(eg(−u)x− 1)ν(dx)du
¾
− 1
≤ exp (Z 0
−∞
Z
|x|≤1eg(t−u)xg(t− u) · eg(−u)xg(−u)|x|2ν(dx)du +
Z 0
−∞
Z
x>1
eg(t−u)x+g(−u)xg(t− u)xν(dx) + Z 0
−∞
Z
x<−1g(t− u)|x|ν(dx)du
¾
− 1
≤ exp
½Z 0
−∞g(t− u)du Z
R
³
γe2γ|x||x|21{|x|≤1}+ e2γ|x|x1{x>1}
´ ν(dx) +
Z 0
−∞g(t− u)du Z
x<−1|x|ν(dx)
¾
− 1.
Here in the second inequality we use (5.6). Then due to Lemma 4.1 and Taylor expansion we have IIb≤ exp
½ c
Z 0
−∞g(t− u)du
¾
− 1 ≤ c0e−c00t, (5.7)
where c, c0 and c00 are some positive constants. Next we see the boundedness of IIc and IId. Since proofs are similar we only show the boundedness of IIc. Again by Lemmas 4.1 and 4.2 with (5.6) we have
log IIc ≤ Z 0
−∞
Z
|x|≤1eg(−u)|x|(g(−u)x)2ν(dx)du +
Z 0
−∞
Z
x>1
eg(−u)xg(−u)xν(dx)du +
Z 0
−∞
Z
x<−1
³
1− eg(−u)x´
ν(dx)du
≤ Z 0
−∞g2(−u)du Z
|x|≤1eγ|x||x|2ν(dx) +
Z 0
−∞g(−u)du µZ
x>1
eγxxν(dx) + Z
x<−1|x|ν(dx)
¶
<∞.
The same calculation holds for IId, and IIc and IId are turned out to be bounded uniformly in t.
Hence with boundedness of I and IIa, the equation (5.7) yields the results. 2 Proof of Theorem 4.1.
In the following proof, the c, c0, c00, C, C0, C00, etc, will denote positive constants for which the exact values are irrelevant and may vary from line to line. By the L´evy-type stochastic integral of Y in Section 2 (recall that we have assumed there is no Gaussian part), for t > 0,
Y (t) = β0eAtX(0) + Z t
0
g(t− u)dL(u)
where g is the CARMA kernel defined in Section 2. The L´evy-Itˆo decomposition for L mentioned in Section 2, written in the differential form yields,
dL(s) = bds + Z
|x|<1x ˜N (ds, dx) + Z
|x|≥1xN (ds, dx).
This together with the state equation (2.4) gives dY (t) =
³
β0AeAtX(0) + Z t
0
g0(t− s)dL(s)´
dt + g(0)dL(t)
= β0AX(t)dt + g(0)dL(t)
= (β0AX(t) + g(0)b)dt + g(0) Z
|x|<1x ˜N (dt, dx) + g(0) Z
|x|≥1xN (dt, dx),
where g(0) = β0δp. We apply Itˆo’s formula for L´evy-type stochastic integrals, as that in [1, p.226, Theorem 4.4.7] with f (x) = eθx to obtain
eθY (t)− eθY (0) = Z t
0
θeθY (s−)(β0AX(s−) + g(0)b)ds +
Z t
0
Z
|x|≥1[eθ(Y (s−)+g(0)x)− eθY (s−)]N (ds, dx) +
Z t
0
Z
|x|<1[eθ(Y (s−)+g(0)x)− eθY (s−)] ˜N (ds, dx)
