Semi-Levy driven continuous-time GARCH process
M. Mohammadi S. Rezakhah N. Modarresi



Semi-Lévy driven continuous-time GARCH process
    M. Mohammadi
Faculty of Mathematics and Computer Science, Amirkabir University of Technology, 
Tehran, Iran. E-mail: m.mohammadiche@aut.ac.ir(M. Mohammadi) and rezakhah@aut.ac.ir(S. 
Rezakhah).
S. Rezakhah^* N. Modarresi
Department of Mathematics and computer science, Allameh Tabataba'i University, Tehran, 
Iran. E-mail: n.modarresi@atu.ac.ir(N. Modarresi).
    December 30, 2023
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In this paper, we study the class of semi-Lévy driven continuous-time GARCH, denoted by SLD-COGARCH, process. The statistical properties of 
this process are characterized. We show that the state process of such process can be described by a random recurrence equation with the periodic 
random coefficients. We establish sufficient conditions for the existence of a strictly periodically stationary solution of the state process which causes
the volatility process to be strictly periodically stationary. Furthermore, it is shown that the increments with constant length of such SLD-COGARCH
 process are themselves a discrete-time periodically correlated (PC) process. We apply some tests to verify the PC behavior of these increments by 
 the simulation studies. Finally, we show that how well this model fits a set of high-frequency financial data.

 
AMS 2010 Subject Classification: 60G12, 62M10, 60G51, 60J75, 91B70, 93E15.

 
Keywords: continuous-time GARCH process; semi-Lévy process; strictly periodically stationary; periodically correlated.





§ INTRODUCTION


The discrete-time ARCH and GARCH processes are often applied to model financial time series <cit.>. Kluppelberg et al. <cit.> introduced a continuous-time version of the GARCH(1,1), called COGARCH(1,1), process which has the potential to have a better description of changes by determining the underling continuous structure. The examples of such underling structure are Lévy processes like Brownian motion and Poisson process. They studied Lévy driven COGARCH(1,1) processes where noises are considered as increments of some Lévy process and proved the stationarity and also second order properties, under some regularity conditions. Brockwell et al. <cit.> generalized the Lévy driven COGARCH(1,1) process to the higher order. They defined the COGARCH(p,q) process (G_t)_t≥0 driven by a Lévy process (L_t)_t≥0 as the solution to the stochastic differential equation

    dG_t=√(V_t)dL_t,     t>0,     G_0=0,

where the volatility process (V_t)_t≥0 is a continuous-time ARMA(q,p-1), called CARMA (q,p-1), process. They presented the state-space representation of the volatility process as (see e.g. <cit.>)

    V_t=α_0+𝐚^'𝐘_t^-,     t>0,     
    V_0=α_0+𝐚^'𝐘_0,

and

    d𝐘_t=B𝐘_t^-dt+𝐞(α_0+𝐚^'Y_t^-)d[L,L]_t^(d),
         t>0,

where [L,L]_t^(d)=∑_0<u≤ t(Δ L_u)^2 is the discrete part of the quadratic variation process ([L,L]_t)_t≥0 (see Protter <cit.>), α_0>0 and 

    B = [      0      1      0      ⋯      0;      0      0      1      ⋯      0;      ⋮      ⋮      ⋮      ⋱      ⋮;      0      0      0      ⋯      1;   -β_q -β_q-1 -β_q-2      ⋯   -β_1;        ],  𝐚=[ α_1; α_2;   ⋮; α_q;     ],  𝐞=[ 0; 0; ⋮; 0; 1;   ],

with α_1,⋯,α_p∈ℝ, β_1,⋯, β_q∈ℝ, α_p≠0, β_q≠0,  and α_p+1=⋯=α_q=0. They showed that the state process (𝐘_t)_t≥0 can be expressed as a stochastic recurrence equation with the random coefficients. Under some sufficient conditions, they proved that the volatility process (V_t)_t≥0 is strictly stationary which causes the increments with constant length of the process (G_t)_t≥0 to be stationary.

The Lévy driven COGARCH processes have the restriction that the relevant underling process has the stationary increments. In contrast with such stationary increment processes, processes with periodically stationary increments have a wider application. For example, the accumulated compensation claims from an insurance company in many cases have periodic increments. Specifically, car insurance companies are dealt with more accidental claims at the starting of the new year and in summer than the other parts of the year. As another example, the precipitation in almost all parts of the world have periodic behavior and so the volume of underground waters are higher in some months and lower in some months in each year. The observations of such time series have significant dependency to the previous periods as well. Motivated by these observations, the underling processes with periodically stationary increments, say semi-Lévy processes, are more prominent than the Lévy processes. So, we study semi-Lévy driven COGARCH, denoted by SLD-COGARCH, process in this paper. The semi-Lévy processes have been extensively studied by Maejima and sato <cit.>. Recognizing such semi-Lévy processes often can be followed by considering some fixed partitions of the corresponding period where inside successive partitions the process has different Lévy type structure and has similar structure to the corresponding element of the previous periods.

In this paper, we study the statistical properties of the SLD-COGARCH(p,q) process. Providing a random recurrence equation with periodic random coefficients for the state process (𝐘_t)_t≥0 defined by (<ref>), we show that it converges almost surely under some regular conditions. We establish sufficient conditions for the existence of a strictly periodically stationary solution of the state process (𝐘_t)_t≥0 which causes the volatility process (V_t)_t≥0 defined by (<ref>) to be strictly periodically stationary. Furthermore, the increments with constant length of the SLD-COGARCH(p,q) process are a discrete-time periodically correlated (PC) process. We use some tests to verify the PC behavior of the increments. Finally, we apply the introduced model to some real data set.

This paper is organized as follows. In section 2, we introduce the SLD-COGARCH processes. We also present the structure of the semi-Lévy process and obtain its characteristic function. Section 3 is devoted to some sufficient conditions that cause the volatility process to be strictly periodically stationary. We also show that the increments with constant length of such SLD-COGARCH process form a PC process. In section 4, we illustrate the results with simulations. In section 5, we apply the model to a set of high-frequency financial data and show that this process has the consistent properties. All proofs are presented in Section 6.



§ SEMI-LÉVY DRIVEN COGARCH PROCESS


In this section, we describe the structure of semi-Lévy compound Poisson process in subsection 2.1. Following the method of Brockwell et al. <cit.>, we present the structure of corresponding semi-Lévy compound Poisson process as the underling process of the COGARCH process in subsection 2.2.


 §.§ Semi-Lévy process

We present the definition of semi-Lévy Poisson and semi-Lévy compound Poisson processes. Then, we drive the characteristic function of the semi-Lévy compound Poisson process. 
 (Semi-Lévy Poisson process)

Let A_j=(s__j-1,s__j]  where 0=s__0<s__1<⋯ constitutes a partition of positive real line 
and |A_j|=|A_j+d|=s__j-s__j-1
 for some integer d 
and all j∈ℕ
 and also τ=∑_j=1^d|A_j|.
 Then, a non-homogeneous Poisson process (N(t))_t≥0
 is called a semi-Lévy Poisson process with period τ>0 and intensity function λ(u) 
if E(N(t))=Λ(t)
where 

    Λ(t)=∫_0^tλ(u)du

and λ(t)=∑_j=1^dλ_jI_𝔇_j(t) for t,λ_j≥0 and 𝔇_j=⋃_k=0^∞A_j+kd.

One can easily verify that a semi-Lévy Poisson process has periodically stationary increments.
 (Semi-Lévy compound Poisson process)

Let (N(t))_t≥0 be a semi-Lévy Poisson process with some period τ>0. Then, the semi-Lévy compound Poisson, denoted by SLCP, process (S_t)_t≥0 is defined as 

    S_t=δ t+∑_n=1^N(t)Z_n

where δ∈ℝ and Z_n=∑_j=1^dZ_n^(j)I_{Υ__n∈𝔇_j} in which Υ__n is the arrival time of n^th jump Z_n, 𝔇_j=⋃_k=0^∞A_j+kd and Z_n^(j) are independent and identically distributed (i.i.d.) random variables with distribution F_j, j=1, ⋯, d, such that ∫_ℝz^2F_j(dz)<∞.

The characteristic function of this process is derived from the following Lemma.

Let (S_t)_t≥0 be the SLCP process introduced by Definition 2.2. Then its characteristic function has the following Lévy-Khintchine representation

    E[e^iuS_t]=exp[iuγ(t)+∫_ℝ(e^iu𝑧-1-iu𝑧 
    I_{|z|≤1})ν_t(dz)],

where 

    γ(t)=δ t+∫_|z|≤1zν_t(dz)

and

    ν_t(dz)=∑_j=1^r-1(m+1)λ_j|A_j|F_j(dz)+∑_j=r^dmλ_j|A_j|F_j(dz)+
    λ_r(t-s__r-1+md)F_r(dz),

in which m=[t/τ] and r∈{1,⋯, d}.

Proof: see Appendix, P1.

(i) By the fact that the numbers of jumps of the SLCP process (S_t)_t≥0 in the time interval [0,t] are finite and the size of each jump is finite as well, we deduce that the process (S_t)_t≥0 is a finite variation process, see <cit.>. Hence, the process (S_t)_t≥0 is a semimartingale <cit.>.

(ii) By Definition 2.2, we have that if there is a jump at point u, 0≤ u≤ t, then 

    Δ S_u:=S_u-S_u^-   =[δ u+∑_n=1^N(u)Z_n]-[δ 
    u^-+∑_n=1^N(u^-)Z_n]=Z_N(u),

otherwise Δ S_u=0.  Thus, by the Theorem 22 of Protter <cit.>, if there is a jump at point u then Δ[S,S]_u=|Z_N(u)|^2 and otherwise Δ[S,S]_u=0. So, the quadratic variation of the process (S_t)_t≥0 is given by

    [S,S]_t=∑_0≤ u≤ tΔ[S,S]_u=∑_0≤ u≤ t|Z_N(u)|^2=∑_n=1^N(t)
    |Z_n|^2.





 §.§ Semi-Lévy driven COGARCH process

The representation of the Lévy driven COGARCH(p,q) process is based on the structure of the GARCH(p,q) process (ξ_n)_n∈ℕ^0 as

    ξ_n   =√(V_n)ε_n
    
    V_n   =α_0+α_1ξ_n-1^2+⋯+α_pξ_n-
    p^2+β_1V_n-1+⋯+β_qV_n-q,   n≥ s,

where s=max{ p, q} and (ε_n)_n∈ℕ^0 is a sequence of i.i.d. random variables. The process (ξ_n)_n∈ℕ^0 can be interpreted as the log return of the asset price process (G_n)_n∈ℕ^0, say ln(G_n+1/G_n). The volatility process (V_n)_n∈ℕ^0 is as an ARMA(q,p-1) process driven by the noise sequence (V_n-1ε_n-1^2)_n∈ℕ. Motivated by this, the dynamic of the Lévy driven COGARCH(p,q) process for the logarithm of the asset price process (G_t)_t≥0 is defined as

    dG_t=√(V_t)dL_t,      t>0,      G_0=0,

where (L_t)_t≥0 is a Lévy process and (V_t)_t≥0 is a continuous-time analog of the self-exciting ARMA(q,p-1) process (V_n)_n∈ℕ^0, see Brockwell et al. <cit.>.

We generalize the Lévy driven COGARCH(p,q) process by replacing the Lévy process (L_t)_t≥0 with the SLCP process (S_t)_t≥0.

(SLD-COGARCH(p,q))

Let (S_t)_t≥0 be the SLCP process with some period τ>0 defined by (<ref>). Then the left-continuous volatility process (V_t)_t≥0 is defined as 

    V_t=α_0+𝐚^'𝐘_t^-,     t>0,     
    V_0=α_0+𝐚^'𝐘_0,

where the state process (𝐘_t)_t≥0 is the unique càdlàg solution of the stochastic differential equation

    d𝐘_t=B𝐘_t^-dt+𝐞(α_0+𝐚^'Y_t^-)d[S,S]_t,
         t>0,

with an initial value 𝐘_0 which is independent of the driving process (S_t)_t≥0, in which [S,S]_t is the quadratic variation of the process (S_t)_t≥0 and the (q× q)-matrix B and vectors 𝐚 and 𝐞 are defined as

    B = [      0      1      0      ⋯      0;      0      0      1      ⋯      0;      ⋮      ⋮      ⋮      ⋱      ⋮;      0      0      0      ⋯      1;   -β_q -β_q-1 -β_q-2      ⋯   -β_1;        ],  𝐚=[ α_1; α_2;   ⋮; α_q;     ],  𝐞=[ 0; 0; ⋮; 0; 1;   ],

where p and q are integers such that q≥ p≥1, α_0>0, α_1,⋯,α_p∈ℝ, β_1,⋯,β_q∈ℝ, α_p≠0, β_q≠0, and α_p+1=⋯=α_q=0.

If the volatility process (V_t)_t≥0 defined by (<ref>) is almost surely non-negative, then (G_t)_t≥0 given by

    dG_t=√(V_t)dS_t,     t>0,   G_0=0,

is called the SLCP driven COGARCH(p,q), denoted by SLD-COGARCH(p,q), process with parameters B, 𝐚, α_0 and the driving process (S_t)_t≥0.


We present the definition of matrix norms and some notation that will be used throughout the paper.

Let ||·||_r denote the vector L^r-norm in ℂ^q, for r∈[1,∞]. Then the matrix L^r-norm of the (q× q)-matrix C is defined as

    ||C||_r=sup_𝐜∈ℂ^q∖{0}||C𝐜||_r/||𝐜||_r.


The matrix L^r-norms of the L^1, L^2 and L^∞  are called the maximum absolute column sum norm, spectral norm and maximum absolute row sum norm, respectively. 

Let P be a matrix such that P^-1BP is a diagonal matrix, where B is defined by (<ref>). Then the natural matrix norm of  C is

    ||C||_P,r=||P^-1CP|| _r.


The natural matrix norm ||·||_P,r  corresponding to the natural vector norm is defined as

    ||𝐜||_P,r=||P^-1𝐜|| _r,     𝐜∈ℂ^q.

The eigenvalues of the matrix B are denoted by η_1, ⋯, η_q and η:=max_i=1,⋯,qℛe(η_i), where ℛe(η_i) is the real part of the eigenvalue η_i. The (q× q)-identity matrix is denoted by I_q or simply I.


§ PERIODIC STATIONARITY OF THE INCREMENTS


In this section, we show that under some conditions, the volatility process (V_t)_t≥0 defined by (<ref>) is strictly periodically stationary with period τ. The increments with constant length of the SLD-COGARCH process are a periodically correlated (PC) process. Furthermore, under a necessary and sufficient condition, the volatility process (V_t)_t≥0 is non-negative.

The following theorem shows that the state process of the SLD-COGARCH process defined by (<ref>) satisfies a multivariate random recurrence equation with periodic random coefficients.

Let (𝐘_t)_t≥0 be the state process of the SLD-COGARCH process. Then the following results are hold.


  (a) There exists a random (q× q)-matrix J_s,t and a random vector 𝐊_s,t such that

    𝐘_t=J_s,t𝐘_s+𝐊_s,t,     0≤ s≤ t.


  (b) (J_s,t,𝐊_s,t) are periodic in both indices with the same period τ. So, the distribution of (J_s,t, 𝐊_s,t) is invariant under time changes of s,t to s+τ,t+τ.

  (c) (J_s,t,𝐊_s,t) and (J_r,u, 𝐊_r,u) are independent for 0≤ s≤ t≤ r≤ u. 𝐘_s is also independent of (J_s,t,𝐊_s,t).


Proof: see Appendix, P2.

In the next theorem, we consider some mild conditions to the state process that converges in distribution to a finite random vector.

Let (𝐘_t)_t≥0 be the state process of the SLD-COGARCH process that is satisfied in the followings


  (i) all the eigenvalues of the matrix B are distinct and have strictly negative real parts,

  (ii) there exists some r∈[1,∞] such that for t∈[0,τ] and j=1, ⋯, d

    ∫_ℝlog(1+||P^-1𝐞𝐚^'P||_rz^2)F_j(dz)
    <-ητ/Λ(t+τ)-Λ(t),

where P is a matrix which causes P^-1BP is diagonal.

Then, for the fixed t∈[0,τ), 𝐘_t+mτ converges in distribution to some vector 𝐔^(t), as m→∞. Furthermore, 𝐔^(t) has a unique distribution that is satisfied in equation

    𝐔^(t)d=J_t,t+τ𝐔^(t)+𝐊_t,t+τ,

where d= denotes the equality in 
distribution and 𝐔^(t) is independent of (J_t,t+τ,𝐊_t,t+τ). 

Proof: see Appendix, P3.

Under the conditions (i) and (ii) of Theorem 3.2, the following results are hold.


  (a) If 𝐘_td=𝐔^(t) for any t∈[0, τ), then 𝐘_t and V_t defined by (<ref>) are strictly periodically stationary with the period τ. In other words, for any n∈ℕ and t__1, t__2, ⋯, t__n≥0

    (𝐘_t__1, 𝐘_t__2, ⋯, 𝐘_t__n)d=(𝐘_t__1+τ, 𝐘_t__2+τ, ⋯, 𝐘_t__n+τ)

and

    (V_t__1, V_t__2, ⋯, V_t__n)d=(V_t__1+τ, V_t__2+τ, ⋯, 
    V_t__n+τ).


  (b) Increments with the constant length of the SLD-COGARCH process (G_t)_t≥0 is a PC process. In other words, G_t^(l):=∫_t^t+l√(V_u)dS_u for any t, h≥0 and fixed l>0 satisfies

    E(G_t^(l))=E(G_t+τ^(l))

and

    cov(G_t^(l),G_t+h^(l))=cov(G_t+τ^(l),G_t+h+τ^(l)).



Proof: see Appendix, P4.

One of the striking features of high-frequency financial data is that the returns have negligible correlation while the squared returns are significantly
correlated. These data typically show a PC structure in their squared logarithm returns (known as squared log return). 
The next corollary shows that under some conditions, the increments with constant length of the SLD-COGARCH 
which were defined in Corollary 3.3 (b) can capture these features.

Let the conditions of Theorem 3.2 hold and the driving process (S_t)_t≥0 defined by (2.2) has zero mean. Then increments with constant length G_t^(l) for t≥0 and h≥ l>0 satisfies 

    (a)   E(G_t^(l))=0,      cov(G_t^(l),G_t+h^(l))=0,
    
       (b)   E((G_t^(l))^2)=E((G_t+τ^(l))^2),     
    cov((G_t^(l))^2,(G_t+h^(l))^2)=cov((G_t+τ^(l))^2,(G_t+h+τ^(l))^2).


Proof: see Appendix, P5.

The following theorem represents the necessary and sufficient conditions for non-negativity of the volatility process V_t for any t≥0. 

Let (𝐘_t)_t≥0 be the state process of the SLD-COGARCH process with parameters B, 𝐚 and α_0>0. Suppose that γ≥-α_0 be a real constant that satisfies in 

    𝐚^'e^Bt𝐞≥0  ∀ t≥0,

and

    𝐚^'e^Bt𝐘_0≥γ   a.s.  ∀ t≥0.

Then, with probability one, V_t≥α_0+γ≥0 for t≥0.

Conversely, if either (<ref>) fails or (<ref>) holds with γ>-α_0 and (<ref>) fails, then there exists a SLCP process (S_t)_t≥0 and t≥0 such that P(V_t < 0)>0.

Proof: see Appendix, P6.


§ SIMULATION


In this section, we verify the theoretical results concerning the PC structure of the increments of the SLD-COGARCH process (G_t)_t≥0 is defined in (<ref>) through the simulation. For this, we first simulate the driving semi-Lévy process (S_t)_t≥0 with period τ introduced by Definition 2.2. Second, we present the discretized version of the state process (𝐘_t)_t≥0 defined by (<ref>) at the time of jump points. We generate the volatility process (V_t)_t≥0 defined by (<ref>) and the process (G_t)_t≥0 at the time of jump points. Then, we evaluate a sampled process from the simulated values of the process (G_t)_t≥0. Finally, we apply some tests to demonstrate the PC structure of the increments for the sampled process.

For simulating the driving process (S_t)_t≥0 with the underlying semi-Lévy Poisson process (N(t))_t≥0, we consider Υ__j as the time of j^th jump (N(t))_t≥0 and Υ__0=0. First, by Definition 2.1, we generate the arrival times Υ__1, Υ__2, ⋯ of the process (N(t))_t≥0 by the following algorithm.


  * Consider the positive value τ as the length of the period of the process (N_t)_t≥0 and some integer m as the number of period intervals for the simulation.

  * Choose the positive integer d as the number of elements of partition in each period interval. 

  * Consider positive real numbers l_1, ⋯, l_d such that τ=∑_j=1^dl_j and a partition of first period interval (0,τ] by A_1, A_2, ⋯, A_d where A_j=(s__j-1,s__j], s__0=0 and s__j=∑_i=1^jl_i, j=1, 2, ⋯, d. Elements of partition of k^th period interval ((k-1)τ,kτ] are A_1+(k-1)d, ⋯, A_kd where |A_j+(k-1)d|=|A_j|=l_j for k=2, ⋯, m and j=1, ⋯, d.

  * Let the positive real numbers λ_1, ⋯, λ_d be occurrence rates corresponding to the increments of (N(t))_t≥0 on A_1, ⋯, A_d. It is also followed from Definition 2.1 that the process (N(t))_t≥0 has the occurrence rate λ_j on A_j+(k-1)d for k=2, ⋯, m and j=1, ⋯, d.

  * Generate an independent sequence of Poisson random variables P_i with parameter λ_il_i on A_i for i=1, ⋯, md as p_1, ⋯, p_md.

  * Generate p_i independent samples from uniform distribution U(s__i-1,s__i] for i=1, ⋯, md. Then, sort these samples and denote these ordered samples by Υ__1, ⋯, Υ__md.

  * Generate the successive jump size Z_n independently and with distribution F_j if corresponding arrival time belongs to 𝔇_j=⋃_k=0^∞A_j+kd  for j=1, ⋯, d and evaluate S_t by (<ref>).


Now, by the following steps, we simulate the SLD-COGARH(p,q) process introduced by Definition 2.3.


  * Set integer valued p and q such that q≥ p≥1.

  * Choose the real parameters β_1, ⋯, β_q and α_1, ⋯, α_p and α_0>0 such that α_p≠0, β_q≠0, α_p+1=⋯=α_q=0. Furthermore, the eigenvalues of the matrix B, η_1,⋯,η_q, are distinct and have strictly negative real parts and conditions (<ref>), (<ref>) and (<ref>) are satisfied.

To verify the condition (3.2), define the matrix P as follows

    P = [       1       ⋯       1;     η_1       ⋯     η_q;       ⋮       ⋯       ⋮; η_1^q-1       ⋯ η_q^q-1;         ].


  * After generating the arrival times Υ__n, n=1, ⋯, md, by the above algorithm, generate the state process (𝐘_Υ__n)_n∈ℕ in (<ref>) with an initial value 𝐘_Υ_0

    𝐘_Υ__n   =(I+Z_n^2𝐞𝐚^')e^B(Υ__n-Υ__n-1)𝐘_Υ__n-1+α_0Z_n^2𝐞
       =e^B(Υ__n-Υ__n-1)𝐘_Υ__n-1+𝐞(α_0+
    𝐚^'e^B(Υ__n-Υ__n-1)𝐘_Υ__n-1)Z^2_n.


  * Similar to (<ref>) 𝐘_t for t∈[Υ__n-1,Υ__n^-] can be written as 𝐘_t=e^B(t-Υ_n-1)𝐘_Υ_n-1. 
So, by (<ref>), generate the process (V_Υ_n)_n∈ℕ^0 with 

    V_Υ__n   =α_0+𝐚^'𝐘_Υ__n^-
       =α_0+𝐚^'e^B(Υ__n-Υ__n-1)𝐘_Υ__n-1.


  * Since the driving process (S_t)_t≥0 has one jump at time Υ_n over [Υ_n-1,Υ_n], it follows from (<ref>) that 

    G_Υ_n-G_Υ_n-1=∫_0^Υ_n√(V_u)dS_u-∫_0^Υ_n-1√(V_u)dS_u   =∫_Υ_n-1^Υ_n√(V_u)dS_u
       =√(V_Υ_n)Z_n.

Then generate the process (G_Υ_n)_n∈ℕ^0 by (<ref>) such that G_Υ_0=G_0=0.

  * Finally, using the values of V_Υ_n and G_Υ_n provided by the previous steps, evaluate the sampled processes (V_il)_i∈ℕ^0 and (G_il)_i∈ℕ^0 for l=τ/ϱ where ϱ is some integer by the following steps:


  * For il∈[Υ__n-1,Υ__n), n∈ℕ, it follows from (<ref>) that 𝐘_il=e^B(il-Υ__n-1)𝐘_Υ__n-1. So, using (<ref>), we have

    V_il   =α_0+𝐚^'𝐘_il^-
       =α_0+𝐚^'e^B(il-Υ_n-1)𝐘_Υ_n-1.

Note that if il=Υ__n-1, by (<ref>)

    V_il=V_Υ__n-1=α_0+𝐚^'e^B(Υ__n-1-Υ__n-2)𝐘_Υ__n-2.


  * Using the fact that the driving process (S_t)_t≥0 has no jump over [Υ__n-1,Υ__n), for n∈ℕ, it follows from (<ref>) for il∈[Υ__n-1,Υ__n)

    G_il-G_Υ__n-1=∫_Υ__n-1^il√(V_u)dS_u=0.

Thus, G_il=G_Υ_n-1. If il=Υ__n-1, then it follows from (<ref>) that

    G_il=G_Υ__n-1=G_Υ__n-2+√(V_Υ__n-1)Z_n-1.




To detect the PC structure of a process, it is shown that the proposed sample spectral coherence statistic can 
be used to test whether a discrete-time process is PC <cit.>.  
In this method, for n samples X_0, X_1, ⋯, X_n-1, and a fixed M, the following sample spectral coherence statistic is plotted

    |γ̂(P, Q ,M)|^2=|∑_m=0^M-1d_X(ζ__P+m)d_X
    (ζ__Q+m)|^2/∑_m=0^M-1|d_X(ζ__P+m)|^2∑_m=0^M-1|d_X(ζ__Q+m)|^2,

where P, Q=0, ⋯, n-1, d_X(ζ__P)=∑_k=0^n-1X_ke^ikζ__P and ζ__P=2π P/n for P=0,⋯, n-1.

The perception of this sample spectral coherence is aided by plotting the coherence values only at points where the following α-threshold is exceeded <cit.>

    x_α=1-e^log(α)/(M-1).

For a PC time series with period ϱ, it is expected that the sample spectral coherence statistic has a significant value on pairs (P,Q) for |P-Q|=kn/ϱ, k=0, ⋯, ϱ-1. The plot of the support of the sample spectral coherence statistic can be used to identify the type of model and analysis of the time series X_0, X_1, ⋯, X_n-1, see <cit.>:


  * If only the main diagonal appears, then the time series X_i is a stationary time series.

  * If there are some significant values of statistic and they seem to lie along the parallel equally spaced diagonal lines, then 
the time series X_i is PC with period ϱ=n/L, where L is the line spacing.

  * If there are some significant values of statistic but they occur in some non-regular places, then 
the time series X_i is a non-stationary time series.

In the following, we provide an example to investigate the process.

Let (S_t)_t≥0 be a SLCP process with period τ=6.5. Furthermore, the lengths of the 
successive subintervals of each period interval are 0.5, 2.5, 3, 0.5 where corresponding occurrence rates 
of the semi-Lévy Poisson process on these subintervals are assumed as 4, 10, 5, 30. 
Moreover, the distribution of jump sizes on these subintervals are N(2, 4), N(1.5, 2.5), N(2.5, 1.5) and N(1.75, 3), 
where N(μ, σ^2) denotes a Normal distribution with mean μ and variance σ^2.
In this example, we consider SLD-COGARCH(1,3) process with parameters α_0=10^-6, α_1=0.005, β_1=2.1, 
β_2=6 and β_3=0.6. Thus, the matrix B is

    B = [    0    1    0;    0    0    1; -0.6   -6 -2.1;      ]
 
and conditions (<ref>), (<ref>) and (<ref>) are satisfied. For such SLD-COGARCH process, we simulate 
G_Υ_n 
for the duration of m=30 period intervals by the specified parameters, 𝐘_0=( 0.37×10^-3, 0.05×10^-3, 
0.19×10^-3)^' and G_0=0 by the suggested simulation algorithm. 
Then, by the step 6 of the last algorithm, we get equally space samples with length l=0.25. So, we have 780 discretized 
samples of this 30 period intervals.


In Figure 1, we see the differenced series (G_il^(l))_i=0, 1, ⋯, 779 (top) and its sample autocorrelation (bottom left). The sample spectral coherence statistic of this series for a specified collection of pairs (P,Q) and M=240 that exceed the threshold corresponding to α=0.05 which is presented at the bottom right. If X_i:=G_il^(l), then the parallel lines for the sample spectral coherence confirm the series (X_i)_i=0, 1, ⋯, 779 is PC with period ϱ=780/30=26. This implies that, for i, j=0, 1, ⋯, 779, 

    E(G_il^(l))=E(X_i)=E(X_i+ϱ)=E(G_il+ϱ l^(l))

and

    cov(G_il^(l),G_jl^(l))=cov(X_i,X_j)=cov(X_i+ϱ,X_j+ϱ)=cov(G_il+ϱ l^(l),G_jl+ϱ 
    l^(l)).


So, it follows from (<ref>) and (<ref>) that the series (G_t^(l))_t=0, l, ⋯, 779l has a PC structure with period τ=ϱ l=6.5. 



§ REAL DATA ANALYSIS


One of the striking features of financial time series is that the log returns have negligible correlation while its square log returns are significantly correlated <cit.>. Many of these time series are collected at a high-frequency and typically show a PC structure in their squared log returns <cit.>. In this section, we evaluate the results of Corollary 3.4 and show that the SLD-COGARCH process can capture the periodic structure of high-frequency financial data. For this, we use the 15-minute log returns of the Dow Jones Industrial Average (DJIA) index.

This data set is recorded between 9:35 to 16:00 from September 4th, 2015 to August 14th, 2018. There was a total of 735 trading days not including the weekends and holidays with 26 15-minute observations per day, resulting in the total of n = 19110 15-minute observations. Figure 2, shows the DJIA index for the specified times.

 
In Figure 3, we show the 15-minute log returns of DJIA (top), their squares (middle) and sample autocorrelation function (ACF) of the squared log returns (bottom left). It is clear from ACF that 15-minute squared log returns have a seasonal structure with period 26. To show more precisely the PC structure of these data, we sample last 100 trading days. These data were recorded from March 22th, 2018 to August 14th, 2018 which contain 2600 observations. Then, we present the sample spectral coherence statistic of their squared log returns for a specified collection of pairs (P,Q) and M=100. It is shown that it exceeds the threshold corresponding to α=0.05 in Figure 3 (bottom right). Therefore, sample spectral coherence statistic clearly shows that the squared log returns have a PC structure with period ϱ=2600/100=26.

Currently we show that how well SLD-COGARCH process fits the 15-minute log returns of DJIA index. For this, we first simulate the driving process (S_t)_t≥0 which is specified in Example 4.1 in the case that δ=0 and the distribution of jump sizes are assumed to be N(0,4), N(0,2.5), N(0,1.5) and N(0,3). Second, we simulate the SLD-COGARCH(1,3) process for the duration of m=735 period intervals with the initial value 𝐘_0=( 0.37×10^-3, 0.05×10^-3, 0.19×10^-3)^' and the parameters α_0=0.8×10^-6, α_1=0.0275, β_1=2.1, β_2=6 and β_3=0.6 by the suggested simulation algorithm. Then, by the step 6 of the last algorithm, we get 19110 equally space samples by length l=0.25. 

Figure 4 shows the increments of the sampled process (G_il)_i=0,⋯,19109, (top) and their squares (middle) and ACF of the squared increments (bottom left). We select the last 2600 simulated data as sample and present the sample spectral coherence statistic of their squared increments for a specified collection of pairs (P,Q) and M=550. It is shown that it exceeds the threshold corresponding to α=0.05 in Figure 4 (bottom right). The ACF and sample spectral coherence statistic demonstrate that the squared increments have a PC structure with period ϱ=2600/100=26. Therefore, these graphs show that SLD-COGARCH process can be consistent with 15-minute log returns of DJIA index.



§ APPENDIX


P1: Proof of Lemma 2.1

For any t≥0 there exist r=1, ⋯, d such that t∈ A_r+md=(s__r-1+md, s__r+md], where m=[t/τ]. 
So, it follows from Definition 2.1 and Definition 2.2 that

    E(e^iuS_t)   =E(e^iu(δ t+∑_n=1^N(t)Z_n))
    =e^iuδ tE(e^iu∑_n=1^N(t)Z_n)
       =e^iuδ tE(e^iu∑_n=1^N(s__1)Z_n^(1)× 
    e^iu∑_n=N(s__1)+1^N(s__2)Z_n^(2)×⋯× 
    e^iu∑_n=N(s__r-1+md)+1^N(t)Z_n^(r))
       =e^iuδ tE(∏_j=1^d∏_k=0^m-1
    e^iu∑_n=N(s__j-1+kd)+1^N(s__j+kd)Z_n^(j)∏_j=1^r-1e^iu∑_n=N(s__j-1+md)+1^N(s__j+md)Z_n^(j)
               × e^iu∑_n=N(s__r-1+md)+1^N(t)Z_n^(r)).

Since, by Definition 2.2, the jump sizes Z_n^(j), n∈ℕ, are independent for j=1,⋯,d, we have that

    E(e^iuS_t)   =e^iuδ t∏_j=1^d∏_k=0^m-1
    E(e^iu∑_n=N(s__j-1+kd)+1^N(s__j+kd)Z_n^(j))
    ∏_j=1^r-1E(e^iu∑_n=N(s__j-1+md)+1^N(s__j+md)Z_n^(j))
         × E(e^iu∑_n=N(s__r-1+md)+1^N(t)Z_n^(r)).

By the fact that the jump sizes Z_n^(j), n∈ℕ, in each partition A_j+kd=(s__j-1+kd,s__j+kd], 
k∈ℕ^0, 
are i.i.d. random variables with distribution F_j, it follows from Definition 2.1 and conditional expected value that

    E(e^iu∑_n=N(s__j-1+kd)+1^N(s__j+kd)Z_n^(j))=E(E(e^iu∑_n=1^N(s__j+kd)-
    N(s__j-1+kd)Z_n^(j)| N(s__j+kd)-N(s__j-1+kd)=N))
               =∑_N=0^∞E(e^iu∑_n=1^NZ_n^(j))P(N(s__j+kd)-N(s__j-1+kd)=N)
               =∑_N=0^∞(E(e^iuZ_n^(j)))^N(Λ(s__j+kd)-Λ(s__j-1+kd))^N
    e^-(Λ(s__j+kd)-Λ(s__j-1+kd))/N!
               =e^-(Λ(s__j+kd)-Λ(s__j-1+kd))∑_N=0^∞(∫_ℝe^iuz(Λ(s__j+kd)-Λ(s__j-1+kd))F_j(dz)
    )^N/N!
               =e^-(Λ(s__j+kd)-Λ(s__j-1+kd))e^∫_ℝe^iuz
    (Λ(s__j+kd)-Λ(s__j-1+kd))F_j(dz)
               =e^∫_ℝ(e^iuz-1)λ_j|A_j|F_j(dz),

where |A_j|=|A_j+kd|=s__j-s__j-1. The last equality follows from (<ref>) and the fact that the semi-Lèvy Poisson 
process 
N(t) has the occurrence rate λ_j 
on A_j+kd for k∈ℕ^0. An argument similar to that used in (<ref>) shows that 

    E(e^iu∑_n=N(s__r-1+md)+1^N(t)Z_n^(r))=e^∫_ℝ(e^iuz-1)λ_r(t-s__r-1+md)F_r
    (dz)

By replacing from (<ref>) and (<ref>) in (<ref>), it reads as

    E(e^iuS_t)   =e^iuδ t∏_j=1^de^∫_ℝ(e^iuz-1)mλ_j|A_j|F_j(dz)∏_j=1^r-1e^∫_ℝ(e^iuz-1)λ_j|A_j|F_j(dz)
         × e^∫_ℝ(e^iuz-1)λ_r(t-s__r-1+md)F_r(dz)
       =e^(iu(δ t+∫_|z|≤1zν_t(dz))+∫_ℝ(e^iuz-1-iuzI_|z|≤1)ν_t(dz)),

where

    ν_t(dz)=∑_j=1^r-1(m+1)λ_j|A_j|F_j(dz)+∑_j=r^dmλ_j|A_j|F_j(dz)+
    λ_r(t-s__r-1+md)F_r(dz).



P2: Proof of Theorem 3.1



Proof (a): Let (S_t)_t≥0 be the driving process introduced by Definition 2.2, Υ__n be the time of n^th jump and Υ__0=0. Following the method of Brockwell et al. <cit.>, and as (S_t)_t≥0  has no jump over [Υ__n,Υ__n+1), it follows by (<ref>) that [S,S]_t=0 for t∈[Υ__n,Υ__n+1). So (<ref>) implies that 𝐘_t satisfies d𝐘_t=B𝐘_tdt for t∈[Υ__n,Υ__n+1).  Hence

    e^-Btd𝐘_t=Be^-Bt𝐘_tdt

and

    d(e^-Bt𝐘_t)=0.

Integrating from Υ__n to t∈[Υ__n,Υ__n+1), we have that

    𝐘_t=e^B(t-Υ__n)𝐘_Υ__n.

By (<ref>) and (<ref>), we have that 𝐘_t has a 
jump of size 𝐞(α_0+𝐚^'𝐘_Υ__n^-)Z__n^2 
at Υ__n. So, 𝐘_t at Υ__n can be written as

    𝐘_Υ__n   =𝐘_Υ__n^-+𝐞
    (α_0+𝐚^'𝐘_Υ__n^-)Z__n^2
       =(I+Z__n^2𝐞𝐚^')𝐘_Υ__n^-+
    α_0Z__n^2𝐞.

Integrating of (<ref>) this time from Υ__n-1 to Υ__n^-, we have that 

    𝐘_Υ__n^-=e^B(Υ__n-Υ__n-1)𝐘_Υ__n-1.

Therefore 

    𝐘_Υ__n=(I+Z__n^2𝐞𝐚^')
    e^B(Υ__n-Υ__n-1)𝐘_Υ__n-1+α_0Z__n^2𝐞.

By assuming t∈[Υ__N(t),Υ__N(t)+1), (<ref>) reads as

    𝐘_t=e^B(t-Υ__N(t))𝐘_Υ__N(t)

and (<ref>) as

    𝐘_Υ__N(t)=(I+Z__N(t)^2𝐞𝐚^')
    e^B(Υ__N(t)-Υ__N(t)-1)𝐘_Υ__N(t)-1
    +α_0Z__N(t)^2𝐞.

By successive replacement from (<ref>) in (<ref>) we have that

    𝐘_t   =e^B(t-Υ__N(t))[(I+Z__N(t)^2𝐞𝐚^')
    e^B(Υ__N(t)-Υ__N(t)-1)×
                   ⋯×(I+Z__N(s)+2^2𝐞𝐚^')
    e^B(Υ__N(s)+2-Υ__N(s)+1)]𝐘_Υ__N(s)+1
       +e^B(t-Υ__N(t))[(α_0Z__N(t)^2𝐞)+∑_i=0^N(t)-N(s)-3(I+Z__N(t)^2𝐞𝐚^')
    e^B(Υ__N(t)-Υ__N(t)-1)×
                   ⋯×(I+Z__N(t)-i^2𝐞𝐚^')
    e^B(Υ__N(t)-i-Υ__N(t)-i-1)(α_0Z__N(t)-i-1^2𝐞)].

Similar to (<ref>) 𝐘_t at Υ__N(s)+1 can be written as

    𝐘_Υ__N(s)+1=(I+Z__N(s)+1^2𝐞𝐚^')
    𝐘_Υ__N(s)+1^^-+α_0Z__N(s)+1^2𝐞.

By integrating of (<ref>) from s to Υ__N(s)+1, we have that

    𝐘_Υ__N(s)+1^^-=e^B(Υ__N(s)+1-s)𝐘_s.

Replacing this in (<ref>) implies that

    𝐘_Υ__N(s)+1=(I+Z__N(s)+1^2𝐞𝐚^')
    e^B(Υ__N(s)+1-s)𝐘_s+α_0Z__N(s)+1^2𝐞.

By replacing from (<ref>) in (<ref>), it reads as

    𝐘_t   =e^B(t-Υ__N(t))[(I+Z__N(t)^2𝐞𝐚^')
    e^B(Υ__N(t)-Υ__N(t)-1)×
                 ⋯×(I+Z__N(s)+2^2𝐞𝐚^')
    e^B(Υ__N(s)+2-Υ__N(s)+1)(I+Z__N(s)+1^2𝐞𝐚^')
    e^B(Υ__N(s)+1-s)]𝐘_s
       +e^B(t-Υ__N(t))[(α_0Z__N(t)^2𝐞)+∑_i=0^N(t)-N(s)-2(I+Z__N(t)^2𝐞𝐚^')
    e^B(Υ__N(t)-Υ__N(t)-1)×
                 ⋯×(I+Z__N(t)-i^2𝐞𝐚^')
    e^B(Υ__N(t)-i-Υ__N(t)-i-1)(α_0Z__N(t)-i-1^2𝐞)]
       =:J_s,t𝐘_s+𝐊_s,t.


Proof (b): We show that the distribution of J_s,t and 𝐊_s,t are periodic in both indices s and t, with the same period τ. Following the fact that J_s,t and 𝐊_s,t which are defined by (<ref>) depend on s and t through


  * the jump sizes Z_·'s which by Definition 2.1 are independent and periodically identically distributed random variables with distribution F_j, j=1, ⋯, d,

  * Υ__N(t)-Υ__N(t)-1, ⋯, Υ__N(s)+2-Υ__N(s)+1 which are inter-arrival times of N(t) which by Definition 2.1 has periodically stationary increment with period τ, So are periodic with period τ. Also N(t)-N(s) are periodic by the same reason,

  * t-Υ__N(t) is the current life at time t which is periodic with period τ. Also J_s,t is a function of Υ__N(s)+1-s which excess life and is periodic with period τ, 

it follows that 

    J_s,td=J_s+τ,t+τ    and    𝐊_s,td=𝐊_s+τ,t+τ.

So, (J_s,t,𝐊_s,t) are periodic in both indices s and t, with the same period τ.



Proof (c): By (<ref>) it is clear that 
(J_s,t,𝐊_s,t) and (J_r,u,𝐊_r,u) are constructed from segments of the driving process (S_t)_t≥0 in the intervals (s,t] and (r,u], respectively. Thus (J_s,t,𝐊_s,t) and (J_r,u,𝐊_r,u) are independent for 0≤ s≤ t≤ r≤ u.

Now we show that 𝐘_s is independent of (J_s,t,𝐊_s,t). By (<ref>) we have that

    𝐘_s=J_0,s𝐘_0+𝐊_0,s.

As (J_0,s,𝐊_0,s) and (J_s,t,𝐊_s,t) are independent and that 𝐘_0 is independent of the driving process (S_t)_t≥0, assumed by (<ref>), it follows that 𝐘_s and (J_s,t,𝐊_s,t) are independent.




P3: Proof of Theorem 3.2

For fixed t∈[0,τ) and for m∈ℕ, it follows from (<ref>) that 

    𝐘_t+mτ=J_t+(m-1)τ,t+mτ𝐘_t+(m-1)τ+𝐊_t+(m-1)τ,t+mτ.

By successive replacement from (<ref>) we obtain

    𝐘_t+mτ   =(J_t+(m-1)τ,t+mτ⋯ J_t,t+τ)𝐘_t
    +[𝐊_t+(m-1)τ,t+mτ
          +J_t+(m-1)τ,t+mτ𝐊_t+(m-2)τ,t+(m-1)τ+⋯+ J_t+(m-1)τ,t+mτ⋯ J_t+τ,t+2τ𝐊_t,t+τ].

Since, by the part (b) of Theorem 3.1, (J_t,t+τ, 𝐊_t,t+τ), (J_t+τ,t+2τ, 𝐊_t+τ,t+2τ),⋯ are i.i.d., it follows from (<ref>) that

    𝐘_t+mτ   d=(J_t,t+τ⋯ J_t+(m-1)τ,t+mτ)𝐘_t
    +[𝐊_t,t+τ+J_t,t+τ𝐊_t+τ,t+2τ
       +⋯+ J_t,t+τ⋯ J_t+(m-2)τ,t+(m-1)τ𝐊_t+(m-1)τ,t+mτ]
       =(∏_i=1^mJ_t+(i-1)τ, t+iτ)𝐘_t
    +[𝐊_t,t+τ+∑_i=1^m-1J_t,t+τ⋯ J_t+(i-1)τ,t+iτ𝐊_t+iτ,t+(i+1)τ]
       =:H_m^(t)𝐘_t+𝐔_m^(t).

The following method of Brockwell et al. <cit.>, for the proof of this theorem we use the general theory of multivariate random recurrence equations, see Bougerol and Picard <cit.>. For this, we first show that there exist 
some r∈[1,∞] and (q× q)-matrix P such that

    E(log||J_t,t+τ||_P,r)<0     and     E(log^+||𝐊_t,t+τ||
    _P,r)<∞,

where log^+(x)=log(max{1,x}). Then, it follows from the strong law of large numbers that the Lyapunov exponent of the i.i.d. sequence (J_t+(n-1)τ,t+nτ)_n∈ℕ is strictly negative

    γ   =lim_n→∞1/nlog||J_t,t+τ⋯ J_t+(n-1)τ,t+nτ||_P,r
       ≤lim_n→∞1/n∑_i=1^nlog||J_t+(i-1)τ,t+iτ||_P,r<0

Therefore, as shown by Bougerol and Picard <cit.>, the sum 𝐔_m^(t) for all t∈[0,τ) is converges almost surely (a.s.).

Since the eigenvalues of the matrix B, η_1, ⋯, η_q, are distinct, we define the matrix P as follows

    P:= [       1       ⋯       1;     η_1       ⋯     η_q;       ⋮       ⋯       ⋮; η_1^q-1       ⋯ η_q^q-1 ].

So, the matrix C:=P^-1BP is the diagonal matrix with entries η_1, ⋯, η_q on the diagonal. Using the fact that P is a invertible matrix, it follows from the exponential matrix definition that, for t≥0, 

    e^Ct=e^P^-1BtP   =I+P^-1BtP+(P^-1BtP)^2/2!+⋯
       =P^-1[I+Bt+(Bt)^2/2!+⋯]P=P^-1e^BtP.

Hence, P^-1e^BtP=e^Ct is the diagonal matrix with entries e^η_1t, ⋯, e^η_qt on the diagonal. Since the eigenvalues of the matrix B have strictly negative real parts, it follows from the Definition 2.5 that, for r∈[1,∞], 

    ||e^Bt||_P,r=||Pe^CtP^-1||_P,r=||P^-1Pe^CtP^-1P||_r=||e^Ct||_r≤ e^η t.

By denoting

    C_i   :=(I+Z_i^2𝐞𝐚^')e^B(Υ__i-Υ__i-1),
    𝐃_i   :=α_0Z_i^2𝐞,     i∈ℕ,

in (<ref>) we have that

    J_t,t+τ   =e^B(t+τ-Υ__N(t+τ))C__N(t+τ)⋯ C__N(t)+2
    (I+Z__N(t)+1^2𝐞𝐚^')e^B(Υ__N(t)+1-t)
       =e^B(t+τ-Υ__N(t+τ))C__N(t+τ)⋯ C__N(t)+2C__N(t)+1e^B(Υ__N(t)-t)
       =e^B(t+τ-Υ__N(t+τ))[∏_i=0^N(t+τ)-N(t)-1C__N(t+τ)-i]e^B(Υ__N(t)-t).

So, by (<ref>) we have that

    ||J_t,t+τ||_P,r≤ e^η(t+τ-Υ__N(t+τ))[∏_i=0^N(t+τ)-N(t)-1
    ||C__N(t+τ)-i||_P,r]e^η(Υ__N(t)-t).

As (<ref>) and (<ref>) leads to

    ||C_i||_P,r   ≤ (1+Z_i^2||𝐞𝐚^'||_P,r)e^η(Υ_i-Υ_i-1)
    
    ||𝐃_i||_P,r   =α_0Z_i^2||𝐞||_P,r,

so, (<ref>) reads as

    ||J_t,t+τ||_P,r   ≤ e^η(t+τ-Υ__N(t+τ))[∏_i=0^N(t+τ)-N(t)-1
    (1+Z__N(t+τ)-i^2
    ||𝐞𝐚^'||_P,r)e^η(Υ__N(t+τ)-i-Υ__N(t+τ)-i-1)]
         × e^η(Υ__N(t)-t).

Therefore

    log||J_t,t+τ||_P,r≤ητ+∑_i=0^N(t+τ)-N(t)-1log(1+Z__N(t+τ)-i^2||𝐞𝐚^'||_P,r).

So, it follows from (<ref>) and (<ref>) that

    E(log||J_t,t+τ||_P,r)≤ητ+[∑_n,m=0^∞∑_i=0^n-1E(log(1+Z_m-i^2
    ||𝐞𝐚^'||_P,r))
       × P(N(t+τ)-N(t)=n, N(t+τ)=m)]
           <ητ+∑_n,m=0^∞n×-ητ/Λ(t+τ)-Λ(t)P(N(t+τ)-N(t)=n, N(t+τ)=m)
           =ητ+(Λ(t+τ)-Λ(t))-ητ/Λ(t+τ)-Λ(t)=0

To show that E(log^+||𝐊_t,t+τ||_P,r)<∞, observe from (<ref>) and (<ref>) that

    𝐊_t,t+τ=e^B(t+τ-Υ__N(t+τ))[𝐃__N(t+τ)+∑_i=0^N(t+τ)-N(t)-2
    C__N(t+τ)⋯ C__N(t+τ)-i𝐃__N(t+τ)-i-1].

So, by (<ref>) we have that

    ||𝐊_t,t+τ||_P,r≤ e^η(t+τ-Υ__N(t+τ))   [||𝐃__N(t+τ)||_P,r+
    (∑_i=0^N(t+τ)-N(t)-2||C__N(t+τ)||_P,r⋯||C__N(t+τ)-i||_p,r
         ×||𝐃__N(t+τ)-i-1||_P,r)].

The eigenvalues of the matrix B have strictly negative real parts, say ℛe(η_i)<0. So, it follows that η:=max_i=1,⋯,qℛe(η_i)<0 and for t≥0 

    e^η t≤1.

From this and (<ref>) we have that

    ||C_i||_P,r≤(1+Z_i^2||𝐞𝐚^'||_P,r).

Thus, (<ref>) reads as

    ||𝐊_t,t+τ||_P,r≤α_0Z__N(t+τ)^2||𝐞||_P,r
    +   (∑_i=0^N(t+τ)- N(t)-2(1+Z__N(t+τ)^2||𝐞𝐚^'||_P,r)×
         ⋯×(1+Z__N(t+τ)-i^2||𝐞𝐚^'||_P,r)
    ×α_0Z__N(t+τ)-i-1^2||𝐞||_P,r).

Since E(Z_i^2)<∞ and E(N(t+τ)-N(t))=Λ(t+τ)-Λ(t)<∞, it follows that E(||𝐊_t,t+τ||_P,r)<∞. So, by Jensen's inequality, we have that 

    E(log^+||𝐊_t,t+τ||_P,r)=E(log(max{1,||𝐊_t,t+τ||_P,r}))<∞.

Now, since E(log||J_t,t+τ||_P,r)<0 and E(log^+||𝐊_t,t+τ||_P,r)<∞, it follows from the general theory of random recurrence equations (see <cit.>) that H_m^(t), in (<ref>), converges a.s. to 0 as m→∞ and that 𝐔_m^(t) converges a.s. absolutely to some random vector 𝐔^(t) as m→∞, for fixed t∈[0,τ). Using the fact that the process (𝐘_t)_t≥0 has càdlàg paths, it follows that ||𝐘_t||_P,r is a.s. finite for fixed t∈[0,τ). So

    lim_m→∞||H_m^(t)𝐘_t||_P,r=0    a.s.,
 
and it follows the 𝐘_t+mτ converges in distribution to 𝐔^(t) as m→∞, for fixed t∈[0,τ).

It remains to show that 𝐔^(t) satisfies (<ref>) for fixed t∈[0,τ). It follows from (<ref>) and the part (c) of Theorem 3.1 that 𝐘_t+(m-1)τ is independent of (J_t+(m-1)τ,t+mτ, 𝐊_t+(m-1)τ,t+mτ). So, by the fact that (J_t+(m-1)τ,t+mτ,𝐊_t+(m-1)τ,t+mτ) and (J_t,t+τ,𝐊_t,t+τ) are i.i.d., for any m≥2, and that 𝐘_t+mτ converges in distribution to 𝐔^(t) as m→∞, we have that

    (J_t+(m-1)τ,t+mτ,𝐊_t+(m-1)τ,t+mτ,𝐘_t+(m-1)τ)d⟶(J_t,t+τ,𝐊_t,t+τ,𝐔^(t))  as   m→∞,

where 𝐔^(t) is independent of (J_t,t+τ,𝐊_t,t+τ) and d⟶ denotes the convergence in distribution. Hence, by (<ref>) and (<ref>), it follows (<ref>).




P4: Proof of Corollary 3.3



Proof (a): By Theorem 3.2, 𝐔^(t) is the long run behavior of 𝐘_t+mτ for fixed t∈[0,τ) and large m, so that 𝐔^(t)d=J_t,t+τ𝐔^(t)+𝐊_t,t+τ and 𝐔^(t) is independent of (J_t,t+τ,𝐊_t,t+τ). Since, by the part (b) of Theorem 3.1, (J_t,t+τ, 𝐊_t,t+τ) and (J_t+(k-1)τ,t+kτ,𝐊_t+(k-1)τ,t+kτ), for k≥2, are i.i.d., we have that 

    𝐔^(t)   d=J_t,t+τ𝐔^(t)+𝐊_t,t+τ
       d=J_t+(k-1)τ,t+kτ𝐔^(t)+𝐊_t+(k-1)τ,t+kτ.

It also follows from the parts (a) and (c) of Theorem 3.1 that, for fixed t∈[0,τ) and k∈ℕ,

    𝐘_t+kτ=J_t+(k-1)τ,t+kτ𝐘_t+(k-1)τ+𝐊_t+(k-1)τ,t+kτ

and that 𝐘_t+(k-1)τ is independent of (J_t+(k-1)τ,t+kτ,𝐊_t+(k-1)τ,t+kτ). Thus if 𝐘_td=𝐔^(t) for t∈[0,τ), then (<ref>) and (<ref>) imply that 
𝐘_t+τd=𝐔^(t). Further, it follows by induction on k∈ℕ that

    𝐘_t+kτd=𝐔^(t).

We show that the process (𝐘_t)_t≥0 is strictly periodically stationary with period τ. If n=1, then 
for any t__1≥0 there exist t∈[0,τ) and k∈ℕ^0 such that t__1=t+kτ. So, by (<ref>) 
we have that

    𝐘_t__1d=𝐘_t__1+τ.

Now, let n≥2 and 0≤ t__1≤ t__2≤⋯≤ t__n. Since, by the part (a) of Theorem 3.1,

    𝐘_t__n=J_t__n-1,t__n𝐘_t__n-1+𝐊_t__n-1,t__n,

it follows that (𝐘_t)_t≥0 is a Markov process. So, by the fact that (J_t__n-1,t__n,𝐊_t__n-1,t__n)d=(J_t__n-1+τ,t__n+τ, 𝐊_t__n-1+τ,t__n+τ), we have that

    (𝐘_t__n|𝐘_t__n-1=𝐲_n-1,⋯, 𝐘_t__1=𝐲_1)
       =J_t__n-1,t__n𝐲_n-1+𝐊_t__n-1,t__n
       d=J_t__n-1+τ,t__n+τ𝐲_n-1+𝐊_t__n-1+τ,t__n+τ
       =(𝐘_t__n+τ|𝐘_t__n-1+τ=𝐲_n-1,⋯, 
    𝐘_t__1+τ=𝐲_1).

Therefore, an induction argument for n=2, 3, ⋯ and (<ref>) imply

    (𝐘_t__1, 𝐘_t__2, ⋯, 𝐘_t__n)d=(𝐘_t__1+τ, 𝐘_t__2+τ, ⋯, 𝐘_t__n+τ).

Hence, by (<ref>) it follows that

    (V_t__1, V_t__2, ⋯, V_t__n)d=(V_t__1+τ, V_t__2+τ, ⋯, V_t__n+τ).


Proof (b): By (<ref>) and (<ref>) we have that

    V_u   =α_0+𝐚^'𝐘_u^-
       =α_0+𝐚^'J_0,u^-𝐘_0+𝐚^'𝐊_0,u^-  ∀ u≥0.

As (J_0,u^-,𝐊_0,u^-) are constructed from segments of the driving process S_t in the interval (0,u^-], see Theorem 3.1 (c), and that 𝐘_0 is independent of (S_t)_t≥0, assumed by (<ref>), it follows that V_u and dS_u=S_u+du-S_u are independent. So, (<ref>) and semi-Lévy structure of S_t imply that V_u and dS_u=S_u+du-S_u both are periodic with the same period τ, say 

    √(V_u)dS_ud=√(V_u+τ)dS_u+τ.

Thus it follows from (<ref>) that

    E(G_t^(l))   =E(∫_t^t+l√(V_u)dS_u)
       =E(∫_t^t+l√(V_u+τ)dS_u+τ)=E(G_t+τ^(l))

and

    E(G_t^(l)G_t+h^(l))   =
    E(∫_t^t+l√(V_u)dS_u∫_t+h^t+h+l√(V_u)dS_u)
       =E(∫_t^t+l√(V_u+τ)dS_u+τ∫_t+h^t+h+l√(V_u+τ)dS_u+τ)
    =E(G_t+τ^(l)G_t+h+τ^(l)).

Hence, from (<ref>) and (<ref>) we have that

    cov(G_t^(l),G_t+h^(l))=cov(G_t+τ^(l),G_t+h+τ^(l)).



P5: Proof of Corollary 3.4



Proof (a): By the part (b) of Corollary 3.3, it follows that V_u and dS_u=S_u+du-S_u are independent for any u≥0. So, E(dS_u)=E(S_u+du)-E(S_u)=0 and 

    E(G_t^(l))   =E(∫_t^t+l√(V_u)dS_u)
       =∫_t^t+lE(√(V_u))E(dS_u)=0.

Since, by Definition 2.2, the jump sizes Z_n have finite second order moment it is easy to see that E(S_t)^2<∞. So, it follows from the proposition 3.17 of Cont and Tankov <cit.> that S_t is a martingale. Thus, from Itô isometry for square integrable martingales as integrators (e.g. Rogers and Williams <cit.>, IV 27) follows

    E(G_t^(l)G_t+h^(l))=E(∫_0^t+h+lV_uI_(t,t+l](u)I_(t+h,t+h+l](u)d[S,S]_u)=0

for h≥ l. Hence, cov(G_t^(l),G_t+h^(l))=0.



Proof (b): By (<ref>) we have that

    E((G_t^(l))^2)   =E(∫_t^t+l√(V_u)dS_u)^2
       =E(∫_t^t+l√(V_u+τ)dS_u+τ)^2=E((G_t+τ^(l))^2)

and

    E((G_t^(l))^2(G_t+h^(l))^2)   =
    E((∫_t^t+l√(V_u)dS_u)^2(∫_t+h^t+h+l√(V_u)dS_u)^2)
       =E((∫_t^t+l√(V_u+τ)dS_u+τ)^2(∫_t+h^t+h+l√(V_u+τ)dS_u+τ)^2)
       =E((G_t+τ^(l))^2(G_t+h+τ^(l))^2).

Hence, from (<ref>) and (<ref>) it follows that

    cov((G_t^(l))^2,(G_t+h^(l))^2)=cov((G_t+τ^(l))^2,(G_t+h+τ^(l))^2).



P6: Proof of Theorem 3.5

Suppose that (<ref>) and (<ref>) both hold. The following proof of Theorem 3.1, for any t≥0 it follows that t∈[Υ__N(t),Υ__N(t)+1). So, we have by (<ref>) that 

    𝐘_t=e^B(t-Υ__N(t))𝐘_Υ__N(t).

By (<ref>) and replacing from N(t) in n, (<ref>) reads as 

    𝐘_Υ__N(t)=𝐘_Υ^-__N(t)+𝐞V_Υ__N(t)Z__N(t)^2

and (<ref>) as

    𝐘_Υ^-__N(t)=e^B(Υ__N(t)-Υ__N(t)-1)𝐘_Υ__N(t)-1.

Replacing (<ref>) in (<ref>) implies that

    𝐘_Υ__N(t)=e^B(Υ__N(t)-Υ__N(t)-1)𝐘_Υ__N(t)-1
    +𝐞V_Υ__N(t)Z__N(t)^2.

So, it follows from (<ref>) and (<ref>) that

    𝐘_t   =e^B(t-Υ__N(t))[e^B(Υ__N(t)-Υ__N(t)-1)𝐘_Υ__N(t)-1
    +𝐞V_Υ__N(t)Z__N(t)^2]
       =e^B(t-Υ__N(t)-1)𝐘_Υ__N(t)-1
    +e^B(t-Υ__N(t))𝐞V_Υ__N(t)Z__N(t)^2.

By successive replacement from (<ref>) in (<ref>) we have that

    𝐘_t=e^Bt𝐘_0+∑_i=1^N(t)e^B(t-Υ__i)𝐞V_Υ__iZ__i^2,
         t≥0.

So, from this and (<ref>) follows that 

    𝐚^'𝐘_t   =𝐚^' e^Bt𝐘_0
    +∑_i=1^N(t)𝐚^'e^B(t-Υ__i)𝐞V_Υ__iZ__i^2
       ≥γ+∑_i=1^N(t)𝐚^'e^B(t-Υ__i)𝐞V_Υ__iZ__i^2.

By a similar method in Brockwell et al. <cit.>, we find by (<ref>) and (<ref>) that V_t=α_0+𝐚^'𝐘_t^-≥α_0+γ for t∈[0,Υ__1]. It also follows from (<ref>), (<ref>) and (<ref>) that

    V_Υ^+__1=α_0+𝐚^'𝐘_Υ__1   =α_0+𝐚^'e^BΥ__1𝐘_0+
    𝐚^'𝐞V_Υ__1Z__1^2
       ≥α_0+γ.

So, by induction one can easily verify V_t≥α_0+γ≥0 for any t≥0.



To prove the converse, suppose first that (<ref>) fails. Then, it follows from (<ref>) that, for  t∈[0,Υ__1],

    V_t=α_0+𝐚^'𝐘_t^-<α_0+γ.

Since P(Υ__1>0)>0, it follows from (<ref>) and γ=-α_0 that P(V_t<0)>0 for t∈[0,Υ__1]. Now assume that (<ref>) holds with γ>-α_0, but (<ref>) fails. Suppose that the support of the jumps distribution Z_i is unbounded. Let (t_1,t_2)⊂(0,∞) be an interval such that 𝐚^'e^Bt𝐞<c for any t∈(t_1,t_2) and for some c<0. By (<ref>), (<ref>) and (<ref>) we have that

    V_Υ__1=α_0+𝐚^'𝐘_Υ__1^-   =α_0
    +𝐚^'e^BΥ__1𝐘_0
       ≥α_0+γ.

It follows from (<ref>) and assumption γ>-α_0, that P(V_Υ__1≥α_0+γ>0)=1. So, one can easily verify that the set 

    A:={Υ__1<t_3<Υ__2, t_3-Υ__1∈(t_1, t_2), V_Υ__1>0}

has positive probability for t_3>t_2. On A, by (<ref>) we have that

    V_t_3=α_0+𝐚^'e^Bt_3𝐘_0+𝐚^'e^B(t_3-Υ__1)𝐞V_Υ__1Z_1^2.

Since 𝐚^'e^B(t_3-Υ__1)𝐞<c<0, by choosing Z_1 large enough it follows from (<ref>) that P(V_t_3<0)>0.

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