Fractional Skellam Processes with Applications to Finance

RESEARCH PAPER

FRACTIONAL SKELLAM PROCESSES WITH APPLICATIONS TO FINANCE

Alexander Kerss 1, Nikolai N. Leonenko 2, Alla Sikorskii 3

Abstract The recent literature on high frequency financial data includes models that use the difference of two Poisson processes, and incorporate a Skellam distribution for forward prices. The exponential distribution of inter-arrival times in these models is not always supported by data. Fractional generalization of Poisson process, or fractional Poisson process, overcomes this limitation and has Mittag-Leffler distribution of inter-arrival times. This paper defines fractional Skellam processes via the time changes in Poisson and Skellam processes by an inverse of a standard stable subordinator. An application to high frequency financial data set is provided to illustrate the advantages of models based on fractional Skellam processes. MSC 2010 : Primary 60E05, 60G22; Secondary 60G51, 26A33 Key Words and Phrases: fractional Poisson process, fractional Skellam process, Mittag-Leffler distribution, high frequency financial data

1. Introduction The advent of high frequency financial data has spurred new modeling techniques to describe characteristics of trade by trade data. Recent literature on the subject includes [1, 2, 3, 10] where models based on the difference of two point processes are proposed. Difference of Poisson processes is considered in [3, 10], and Hawkes processes are discussed in [1, 2]. This paper extends the models where the forward price of a risky asset c 2014 Diogenes Co., Sofia pp. 532–551 , DOI: 10.2478/s13540-014-0184-2 FRACTIONAL SKELLAM PROCESSES . . . 533 is modeled via the difference of two independent Poisson processes, also known as Skellam processes. A drawback of the existing models, which may be at odds with empirical facts, is exponential inter-arrival time, or time between trades. Mainardi et al. [21, 22] studied the fractional Poisson process, where the exponential waiting time distribution is replaced by a Mittag-Leffler distribution, see also [4, 17, 28, 32]. Meerschaert et al. [23] showed that the same fractional Poisson process can also be obtained via an inverse stable time change. Using a time change in models for financial data has been popularized in the last decade based on the idea that it is not the calendar time that drives the changes in price, but rather the information flow or activity time is what matters for modeling of the prices [15, 14, 18]. In this paper we define fractional Skellam processes via time changes in ordinary Skellam processes. The resulting fractional Skellam models incorporate Mittag-Leffler distribution of inter-arrival times and may provide better fit to high frequency financial data.

2. Preliminaries This section collects deﬁnitions and some results on the Skellam process, subordinators and the fractional Poisson process. These results will be used in the next section for the construction of the fractional Skellam processes. 2.1. Skellam processes Definition 2.1. A Skellam process is deﬁned as S(t)=N (1)(t) − N (2)(t),t≥ 0, where N (1)(t), t ≥ 0andN (2)(t), t ≥ 0 are two independent homogeneous Poisson processes with intensities λ(1) > 0andλ(2) > 0, respectively.

The Skellam distribution has been introduced in [30] and [16], and the Skellam processes are considered in [3]. The probability mass function of S(t)isoftheform (1) k/2 (1) (2) λ √ −t(λ +λ ) (1) (2) sk(t)=P(S(t)=k)=e I|k| 2t λ λ , λ(2) (2.1) k ∈ Z = {0, ±1, ±2,...}, where Ik is the modified Bessel function of the first kind [31, p. 114] ∞ z/ 2n+k I z ( 2) . k( )= n n k n=0 !( + )! The Skellam process is a Lévy process with

E[exp{−θS(t)}]=exp{−tψS(1)(θ)}, 534 A. Kerss, N. N. Leonenko, A. Sikorskii and the L´evy exponent is given by ∞ −θy ψS(1)(θ)= (1 − e )ν(dy), −∞ where L´evy measure ν is the linear combination of two Dirac measures: (1) (2) ν = λ δ{1} + λ δ{−1}. It is easy to see that the moment generating function of the Skellam process is

(1) (2) (1) θ (2) −θ M(θ,t):=E[eθS(t)]=e−t(λ +λ −λ e −λ e ),θ∈ R. (2.2) Themeanandthevarianceare ES(t)=(λ(1) − λ(2))t, VarS(t)=(λ(1) + λ(2))t, (2.3) and the covariance function Cov(S(t),S(s)) = (λ(1) + λ(2))min(t, s),t,s>0. The next result on the Skellam processes is straightforward, but to the best of our knowledge, it has not appeared in the literature.

Lemma 2.1. The Skellam process is a stochastic solution of the following system of diﬀerential equations: d s t λ(1) s t − s t − λ(2) s t − s t ,k∈ Z dt k( )= ( k−1( ) k( )) ( k( ) k+1( )) (2.4) with the initial conditions s0(0) = 1 and sk(0) = 0 for k =0 . The moment generating function of the Skellam process solves the differential equation dM θ,t ( ) M θ,t λ(1) eθ − λ(2) e−θ − ,θ∈ R dt = ( )( ( 1) + ( 1)) (2.5) with the initial condition M(θ,0) = 1.

P r o o f. Using the properties of the modiﬁed Bessel function [31, p. 115] Ik(x)=I−k(x) for any integer n and all x,and

dIν (z) 1 = (Iν−1(z)+Iν+1(z)) , dz 2 diﬀerentiate equation (2.1) to get d s t λ(1) s t − s t − λ(2) s t − s t ,k∈ Z. dt k( )= ( k−1( ) k( )) ( k( ) k+1( )) FRACTIONAL SKELLAM PROCESSES . . . 535

Thus (2.4) holds. Now multiply both sides of (2.4) by eθk and sum over k to get dM θ,t ( ) M θ,t λ(1) eθ − λ(2) e−θ − dt = ( )( ( 1) + ( 1)) with the initial condition M(θ,0) = 1, thus equation (2.5) holds. This equation clearly has solution (1) (2) (1) s (2) −s M(s, t)=e−t(λ −λ −λ e +λ e ),θ∈ R, which agrees with equation (2.2) above. 2 2.2. Subordinators and inverse subordinators ALévy subordinator L(t), t ≥ 0 is a homogeneous positive nondecreas- ing Lévy process with Laplace transform E[e−θL(t)]=e−tφL(1)(θ),θ≥ 0, where the Laplace exponent ψL(1)(·)isgivenby −θx φL(1)(θ)=bθ + (1 − e )ν(dx) (0, ∞) with drift b ≥ 0andtheLévy measure ν satisfying (0, ∞)(1∧x)ν(dx) < ∞. A standard α-stable subordinator D(t)isaLévy subordinator with α Laplace transform E[e−θD(t)]=e−tθ , θ>0, t ≥ 0withα ∈ (0, 1). The inverse α-stable subordinator E(t) is defined as the inverse or first passage time of a stable subordinator D(t), that is E(t)=inf{u ≥ 0:D(u) >t},t≥ 0, see, for example, [7], [25, p. 101]. Note that E(t), t ≥ 0 is non-Markovian with non-stationary and non-independent increments. From [8], the moments are tαkk! E[Ek(t)] = . (2.6) Γ(αk +1) The covariance function of this process is computed in [33], see also [19, equation (9)]: Cov[E(t),E(s)] = 1 min(t,s) (st)α ((t − τ)α +(s − τ)α) τ α−1dτ − ,t,s≥ 0. Γ(1 + α)Γ(α) 0 Γ(1 + α) (2.7) The Laplace transform of the inverse stable subordinator is −θE(t) α E[e ]=Eα(−θt ),θ>0,t≥ 0,α∈ (0, 1), (2.8) where 536 A. Kerss, N. N. Leonenko, A. Sikorskii

∞ zj E z ,z∈ C,α∈ , α( )= αj (0 1) j=0 Γ( +1) is the one-parameter Mittag-Leﬄer function, see for example [7, 20]. 2.3. Fractional Poisson process The fractional Poisson process can be obtained as a renewal process with Mittag-Leﬄer waiting times between events [21]:

Nα(t)=max{n ≥ 0:T1 + ...+ Tn ≤ t},t≥ 0,α∈ (0, 1), (2.9) where {Tj}, j ≥ 1 are independent identically distributed random variables with Mittag-Leffler distribution function α Fα(x)=P[Tj ≤ x]=1−Eα(−λx ),x≥ 0,α∈ (0, 1), and Fα(x)=0forx<0. The density function of Mittag-Leffler distribution d f x F x λxα−1E −λxα ,x≥ , ( )=dx α( )= α,α( ) 0 where ∞ zj E z ,z∈ C,α> ,β> α,β( )= αj β 0 0 j=0 Γ( + ) is the two parameter Mittag-Leffler function, see [13]. The three parameter Mittag-Leffler function (also known as the Prabhakar function) ∞ γ zr Eγ z ( )r , α,β,γ ∈ C,Reα > ,Reβ > ,Reγ > , α,β( )= r αr β ( ) 0 ( ) 0 ( ) 0 r=0 !Γ( + ) where (γ)r =Γ(γ + r)/Γ(γ) whenever the Gamma function Γ is defined, and (γ)0 =1forγ =0,see[13]. From [21, equation (3.10)] (see also [4]), the probability mass function of the fractional Poisson process is ∞ (λtα)n (r + n)! (−λtα)r qk(t):=P(Nα(t)=n)= n! r! Γ(α(n + r)+1) r=0 (2.10) α n α n n+1 α (λt ) (n) α =(λt ) E (−λt )= Eα (−λt ) α,αn+1 n! (n) th where n ∈ N,andEα is the n derivative of the one-parameter Mittag- Leffler function. It was shown in [4] that the probability mass function of the fractional Poisson process satisfies the system of fractional differential equations α Dt q0(t)=−λq0(t) α Dt qn(t)=λ(qn−1(t) − qn(t)) FRACTIONAL SKELLAM PROCESSES . . . 537

α with the initial condition q0(0) = 1, qn(0) = 0, n ≥ 1. Here Dt is the Caputo fractional derivative t df τ Dαf t 1 ( ) 1 dτ, <α< . t ( )= α 0 1 Γ(1 − α) 0 dτ (t − τ) It is also proven in [23] that the definition of the fractional Poisson process as a renewal process with Mittag-Leffler distribution of inter-arrival times is equivalent to the time change definition Nα(t)=N1(E(t)), where N1(t), t ≥ 0 is a homogeneous Poisson process with parameter λ>0andE(t), t ≥ 0 is the inverse stable subordinator independent of N1(t). From [4], the mean and variance are λtα E[Nα(t)] = , Γ(α +1) tαλ t2αλ2 1 1 Var[Nα(t)] = + − . Γ(1 + α) α Γ(2α) αΓ(α)2 From [19], the covariance function of the fractional Poisson process is α λ(min(t, s)) 2 Cov[Nα(s),Nα(t)] = + λ Cov[E(s),E(t)], Γ(1 + α) where Cov[E(s),E(t)] is given by equation (2.7) so that for 0 ≤ s ≤ t λsα Cov[Nα(s),Nα(t)] = Γ(1 + α) αt2α αs2α tsα + λ2 B(α +1,α; s/t)+ B(α +1,α) − , Γ2(1 + α) Γ2(1 + α) Γ2(1 + α) (2.11) where B is the Beta function, and B(α, β; ·) is an incomplete Beta function.

3. Fractional Skellam processes We now introduce the main results of this paper which generalize the Skellam process to a setting where the inter-arrival times are no longer exponential but instead are of Mittag-Leﬄer type.

Definition 3.1. Let N (1)(t)andN (2)(t) be two independent homogeneous Poisson processes with intensities λ(1) > 0andλ(1) > 0. Let E(1)(t) and E(2)(t) be two independent inverse stable subordinators with indices α(1) ∈ (0, 1) and α(2) ∈ (0, 1) respectively, which are also independent of the two Poisson processes. The stochastic process X(t)=N (1)(E(1)(t)) − N (2)(E(2)(t)) is called a fractional Skellam process of type I. 538 A. Kerss, N. N. Leonenko, A. Sikorskii

A fractional Sekellam process of type I X(t) has marginal laws of fractional Skellam type I denoted by X(t) ∼ fSk(k, t; λ(1),α(1),λ(2),α(2)), which is a new four parameter distribution.

Theorem 3.1. Let X(t) be a fractional Skellam process of type I. Its probability mass function is given by ∞ (1) k (1) (2) n P(X(t)=k)= λ(1)tα λ(1)λ(2)tα +α n=0 ×En+k+1 − λ(1)tα(1) En+1 − λ(2)tα(2) α(1),α(1)(n+k)+1 α(2),α(2)n+1 for k ∈ Z, k ≥ 0 and when k<0 ∞ (2) |k| (1) (2) n P(X(t)=k)= λ(2)tα λ(1)λ(2)tα +α n=0 ×En+|k|+1 − λ(2)tα(2) En+1 − λ(1)tα(1) . α(2),α(2)(n+|k|)+1 α(1),α(1)n+1 The moment generating function is E eθX(t) E λ(1)tα(1) eθ − E λ(2)tα(2) e−θ − ,θ∈ R. = α(1) ( 1) α(2) ( 1) (3.1)

Proof.SinceN (1)(E(1)(t)) and N (2)(E(2)(t)) are independent, ∞ (1) (1) (2) (2) P(X(t)=k)= P(N (E (t)) = n + k)P(N (E (t)) = n)Ik≥0 n=0 ∞ (1) (1) (2) (2) + P(N (E (t)) = n)P(N (E (t)) = n + |k|)Ik<0. n=0 Now use the expression for the probability mass function of the fractional Poisson process given in equation (2.10) to complete the calculation. When k>0, ∞ (1) k (1) (2) n P(X(t)=k)= λ(1)tα λ(1)λ(2)tα +α n=0 ×En+k+1 − λ(1)tα(1) En+1 − λ(2)tα(2) . α(1),α(1)(n+k)+1 α(2),α(2)n+1 The case k<0 is treated similarly. The moment generating function is computed using that of the fractional Poisson process. Denote by h(·,t) the density of E(t), then FRACTIONAL SKELLAM PROCESSES . . . 539

∞ E eθNα(t) = E eθN(u)h(u, t)du 0 ∞ θ θ = eλu(e −1)h(u, t)du = E eλ(e −1)E(t) = E(λ(eθ − 1)tα), 0 using formula (2.8) for the Laplace transform of the inverse stable subordinator. Note that formula (2.8) remains true for all θ ∈ R.Thiscanbe seen from the proof of [8, Theorem 4.3] and (2.6): ∞ θkEk t ∞ θtα k E eθE(t) E ( ) ( ) E θtα . = k = αk = α( ) k=0 ! k=0 Γ( +1) Therefore for the fractional Skellam process of type I, (1) (1) (2) (2) E eθX(t) = E eθN (E (t)) E e−θN (E (t)) E λ(1)tα(1) eθ − E λ(2)tα(2) e−θ − . = α(1) ( 1) α(2) ( 1) 2

Remark 3.1. The moments of all orders can be obtained either from the moment generating function (3.1) or using the moments of the fractional Poisson processes. For example, the ﬁrst moment of X(t) ∼ fSk is λ(1)tα(1) λ(2)tα(2) E[X(t)] = − . (3.2) Γ(α(1) +1) Γ(α(2) +1) The variance is (1) (1) tα λ(1) t2α (λ(1))2 1 1 Var[X(t)] = + − Γ(1 + α(1)) α(1) Γ(2α(1)) α(1)Γ(α(1))2 (2) (2) tα λ(2) t2α (λ(2))2 1 1 + + − . Γ(1 + α(2)) α(2) Γ(2α(2)) α(2)Γ(α(2))2 (3.3) A random variable X is called over dispersed if Var[X] − E[X] > 0. From inspection of equations (3.2) and (3.3) it is clear that the fractional Skel- lam law of type I has the property of over dispersion. Figure 1 displays the probability mass function for the fractional Skellam distribution with selected parameter values.

The covariance function for the fractional Skellam process of type I can be computed by substituting the expression for the covariance function of the fractional Poisson process (2.11) into the equation below: 540 A. Kerss, N. N. Leonenko, A. Sikorskii

Figure 1. Probability mass function for the fractional Skel- lam distribution at times t =1, ··· , 5

Cov[X(t),X(s)] = Cov[N (1)(E(1)(t)),N(1)(E(1)(s))] + Cov[N (2)(E(2)(t)),N(2)(E(2)(s))].

Definition 3.2. Let S(t)=N (1)(t) − N (2)(t), t ≥ 0 be a Skellam process. Let E(t), t ≥ 0 be an inverse stable subordinator of exponent α ∈ (0, 1) independent of N (1)(t)andN (2)(t). The stochastic process Y (t)=S(E(t)) is called a fractional Skellam process of type II.

Fractional Skellam process of type II Y (t) has marginal laws of fractional Skellam type II, for which we shall write Y (t) ∼ fSk(k, t; λ(1),λ(2),α).

Theorem 3.2. Let Y (t)=S(E(t)) be fractional Skellam process of type II, and let rk(t)=P (Y (t)=k), k ∈ Z. The marginal distribution is given by k/2 λ(1) ∞ u r t 1 e−u(λ(1)+λ(2))I u λ(1)λ(2) du, k( )= α (2) |k| 2 Φα α t λ 0 t (3.4) FRACTIONAL SKELLAM PROCESSES . . . 541 where ∞ −z n z ( ) , <α< Φα( )= n − nα − α 0 1 n=0 !Γ(1 ) is the Wright function, also known as the Mainardi function [11].The marginal distribution satisfies the following system of fractional differential equations: α (1) (2) Dt rk(t)=λ (rk−1(t) − rk(t)) − λ (rk(t) − rk+1(t)) (3.5) with the initial conditions r0(0) = 1 and rk(0) = 0 for k =0 . The moment generating function L(θ,t)=EeθX(t) is (1) (2) (1) θ (2) −θ α L(θ,t)=Eα(−(λ + λ − λ e − λ e )t ), (3.6) and for every θ ∈ R it satisfies the fractional differential equation α (1) θ (2) −θ Dt L(θ,t)=(λ (e − 1) + λ (e − 1))L(θ,t) (3.7) with the initial condition L(θ,0) = 1.

Proof.Withsk(t)=P (S(t)=k) as before in (2.1), use conditioning argument to write ∞ rk(t)= sk(u)h(u, t)du, (3.8) 0 where h(·,t) is the density of E(t). Using the expression for the probability mass function of the Skellam process (2.1) and the fact that u h u, t 1 , ( )=tα Φα tβ see [27, equation (3.7)], equation (3.4) follows. To derive the governing fractional diﬀerential equation, note that from [26, Theorem 4.1], for t>0, u>0, h(u, t)satisﬁes ∂ Dαh u, t − h u, t , t ( )= ∂u ( ) where the Riemann-Liouville fractional derivative for 0 <α<1is t α 1 d −α Dt f(t)= f(t − s)s ds. Γ(1 − α) dt 0 Then integration by parts yields ∞ ∞ ∂ Dαr t s u Dαh u, t du − s u h u, t du t k( )= k( ) t ( ) = k( )∂u ( ) 0 0 ∞ ∂ = h(u, t) sk(u)du − sk(0)h(0+,t), 0 ∂u 542 A. Kerss, N. N. Leonenko, A. Sikorskii

−α and h(0+,t)=t /Γ(1 − α), see [12, Lemma 2.1]. Since sk(0) = 0 for k = 0, the boundary term disappears except when k = 0. Also, from (3.8), rk(0) = sk(0) = 1. Since for 0 <α<1 the Caputo and Riemann-Liouville derivatives are related by −α α α t Dt rk(t)=Dt rk(t) − rk(0) , Γ(1 − α) for both cases, k =0andk =0,wehave ∞ α ∂ Dt rk(t)= h(u, t) sk(u)du. 0 ∂u Now apply(2.4) to get ∞ α (1) (2) Dt rk(t)= h(u, t) λ (sk−1(u) − sk(u)) − λ (sk(u) − sk+1(u)) du 0 and arrive at (3.5) using (3.8). Through the use of conditioning and equation (2.8), the moment generating function ∞ E[eθX(t)]=E[eθS(E(t))]= E[eθS(u)]h(u, t)du 0 ∞ (1) (2) (1) θ (2) −θ = e−u(λ +λ −λ e −λ e )h(u, t)du 0 (1) (2) (1) θ (2) −θ α = Eα(−(λ + λ − λ e − λ e )t ). Since the one-parameter Mittag-Leﬀeler function is the eigenfunction for α α α the Caputo derivative [20, 24], Dt Eα(−λt )=−λEα(−λt ), and equation (3.7) follows. Note that equation (3.7) can also be obtained by multiplying both sides of equation (3.5) by e−θk and summing over k ∈ Z to get α (1) θ (2) −θ Dt L(θ,t)=(λ (e − 1) + λ (e − 1))L(θ,t), which has the solution (3.6). 2

Remark 3.2. The mean, variance and covariance functions for the fractional Skellam process of type II are obtained from [19, Theorem 2.1 ], moments of the Skellam process (2.3) and the time-change process: tα(λ(1) − λ(2)) E[Y (t)] = , Γ(1 + α) α (1) (2) t (λ + λ ) 2 2 1 Var[Y (t)] = + λ(1) − λ(2) t2α − , Γ(1 + α) Γ(2α +1) Γ(1 + α)2 and for 0 ≤ s ≤ t FRACTIONAL SKELLAM PROCESSES . . . 543

α (1) (2) s (λ + λ ) 2 Cov[Y (t),Y(s)] = + λ(1) − λ(2) Cov[E(t),E(s)], Γ(1 + α) where the covariance function for the inverse stable subordinator is given by (2.11) and [19, Equation (9)]. Fractional Skellam law of type II also has the property of overdispersion, as does fractional Skellam law of type I.

4. An empirical investigation of waiting times in high frequency financial data We consider transaction records for the September 2011 Eurofx over a three month horizon from the 22nd June until expiration on the 22nd Sep- tember 2011. The Eurofx is a type of forward asset known as a future, and the data set was obtained directly from the Chicago mercantile exchange. The market is open from 12pm Sunday evening until Friday at 5pm with a one hour close each day between 4pm and 5pm. The price of the forward asset at time t is denoted by F (t), t = 1, 2, ..., N. For this period there are N =5, 465, 779 timestamped transactions recorded over market opening hours. Of these records, 71% of transactions get completed at the previous trade price. No tick change from one trade to the next, and single tick price changes account for 98% of all transactions. Close symmetry between negative and positive tick jumps of the same magnitude is seen. The count for jumps of three ticks up or down is 1,411 and 1,419 respectively a difference of only eight counts, with a similarly finding for jumps of a four ticks. The frequency of both positive and negative jumps in general decreases as the jump size increases but does not hold true for an absolute jump size of eight ticks, which has a higher frequency than both six and seven tick jumps. The data contain the transacted price along with timestamps binned to the nearest second, when multiple trades occur during the same second interval, the trades are recorded in the order they are filled but with iden- tical time stamps. Since we are interested in the inter arrival time between trades this rounding off in timestamps will cause a data loss. A second issue is market micro structure noise in the form of the bid-ask bounce. The futures contract is very liquid and it not uncommon to see strings of transactions occurring in rapid succession bouncing from the bid to the ask, a difference of a single tick. We note though, that our data set does not implicitly state the bid and ask prices we have only interpreted the price bounce to be such a spread. The bid price and the ask price have not changed but the transaction record details a series of positive and negative returns of a single tick. 544 A. Kerss, N. N. Leonenko, A. Sikorskii

We ﬁlter the series by only recording the transactions if they go outside the bid ask spread. The spread is ﬁxed to a single tick of 0.0001 by setting F (0)bid = F (0) and F (t)ask = F (t)bid +0.0001 and computing F (t)bid as ⎧ ⎨⎪F (t − 1)bid if F (t − 1)bid ≤ F (t) ≤ F (t − 1)ask F (t)bid = F (t)ifF (t) F(t − 1)ask

The resulting ﬁltered transaction chain still contains 5, 465, 779 records but we now deleted all entries where the bid price has not changed from previous bid price, that is no up or down jump has occurred, leaving 682, 550 records.

Figure 2. Price path plot of original data and ﬁltered data

Next we consider the up and down jump processes in two models for the spot prices. First is the model from [3], where the price is modeled by the Skellam process. The second model is proposed by us and it uses fractional Skellam process of type I to model the price movements. In the empirical analysis of these models, we separate up and down jumps seen in Figure 2. In the case of Skellam processes, which is the diﬀerence of two independent Poisson processes, the absence of simultaneous jumps FRACTIONAL SKELLAM PROCESSES . . . 545 for the two processes follows from a general result: two independent L´evy processes have no common points of discontinuity almost surely [25, p.106]. As follows from the lemma below, absence of simultaneous jumps also holds for two components in fractional Skellam process of type I.

Lemma 4.1. Let X(t)=N (1)(E(1)(t)) − N (2)(E(2)(t)) be the fractional Skellam process of type I. The processes N (1)(E(1)(t)) and N (2)(E(2)(t)) have no common points of discontinuity almost surely.

P r o o f. We use the deﬁnition of Mainardi, Gorenﬂo and Scalas [21] of fractional Poisson process as a renewal process. Since the sample paths of E(1)(t)andE(2)(t) are continuous almost surely, the discontinuities of the fractional Poisson process come from jumps of the outer Poisson process. Therefore, P N (1)(E(1)(t+)) >N(1)(E(1)(t)) and N (2)(E(2)(t+)) >N(2)(E(2)(t)) n m (1) (2) for some t > 0 = P Ti = Tj for some m, n ∈ N i=1 j=1 n m (1) (2) ≤ P Ti = Tj , m,n∈N i=1 j=1

(1) (2) where the independent random variables Ti and Tj are waiting times between events from (2.9). Since these random variables follow Mittag- Leffler distribution, the distribution of their sum has a density, and the probabilities of the events summed above all have probability zero. 2 We now proceed with the data analyzes by separating the up and down jump processes. Up jump process: To construct the up jump time series we remove all trades with negative jumps leaving 317, 212 observations, all duplicate time stamps are removed leaving only the last recorded entry for each second. A time series of 253, 092 entries remain representing the positive jump process. Figure 3 clearly shows that the exponential distribution provides a poor fit to the data which can be quantified with the 95% confidence interval (0.9512, 0.9554) for α(1) and so α(1) = 1. The Mittag-Leffler provides a closer fit to the data and supports our generalization to a fractional process in this setting. Down jump process: As with the up jump process to build the down jump series we remove the trades with negative jumps leaving 365, 338 observations, all duplicate time stamps are removed leaving only the last 546 A. Kerss, N. N. Leonenko, A. Sikorskii

Figure 3. Survival function for the up jump process recorded entry for each second. A time series of 281, 833 observations is left representing the down jump process. Similarly to the up jump process Figure 4, the exponential distribution does not provide a realistic match the empirically observed survival probabilities. This can be quantified through the 95% confidence interval for α(2) computed as (0.9557, 0.9597), concluding that α(2) =1aswouldbe the case if inter-arrival times where exponential in law. In summary, we have shown that the inter-arrival times between the jumps in both the positive and negative jump processes are clearly not exponential. The Mittag-Leffler law provides a closer fit to the data, however the fit is not perfect and even with the added flexibility of an additional parameter, the Mittag-Leffler does not seem to provide tails that are as heavy as the market suggests. This is true for our data set and more empirical work would be needed to see if this is a common feature amongst different FRACTIONAL SKELLAM PROCESSES . . . 547

Figure 4. Survival function for the down jump process asset classes. Further, although the magnitude of ninety eight percent of jumps is a single tick, there is the case to extend the models further to allow for jumps greater than one tick. It would then seem sensible to model the random component not as the diﬀerence between two fractional Poisson processes but instead as the diﬀerence of two compound fractional Poisson processes. Fractional compound Poisson processes have been studied in [6, 29]. 548 A. Kerss, N. N. Leonenko, A. Sikorskii

Appendix: Statistical analysis of the Mittag-Leﬄer distribution

Let T1,...,Tn be independent identically distributed random variables with Mittag-Leﬄer distribution. The moment estimators for the parame- ters from [9] are 2π αˆ = , and λˆ =exp{−αˆ(E[log(T )] + γ)}, 2(6Var[log(T )] + π2) where γ is Euler’s constant and n n E T 1 T , V T 1 T − E T 2. [log( )] := n log i ar[log( )] := n (log i [log( )]) i=1 i=1 For the above estimators to be of use we must have data where Var[log(T )] >π2/6=1.6449 so that the standard deviation of log(T ) is greater than 1.2825. The estimator for α is asymptotically normal as n →∞: √ α2(32 − 20α2 − α4) n(ˆα − α) −→ N 0, , 40 and we obtained an asymptotic (1 − )100% conﬁdence interval for α.

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1 Cardiff School of Mathematics Cardiff University, Senghennydd Road Cardiff CF24 4 YH, UK e-mail: KerssAD@cardiff.ac.uk

2 Cardiﬀ School of Mathematics Cardiﬀ University, Senghennydd Road FRACTIONAL SKELLAM PROCESSES . . . 551

Cardiﬀ CF24 4 YH, UK e-mail: nleonenko@cardiﬀ.ac.uk

3 Department of Statistics and Probability Michigan State University, 619 Red Cedar Road East Lansing, MI 48824, USA e-mail: [email protected] Received: February 6, 2014

Please cite to this paper as published in: Fract. Calc. Appl. Anal.,Vol.17, No 2 (2014), pp. 532–551; DOI: 10.2478/s13540-014-0184-2