FRACTIONAL POISSON FIELDS AND MARTINGALES By Giacomo Aletti∗ Nikolai Leonenko† and Ely Merzbach‡ ∗ADAMSS Center & Università degli Studi di Milano, Italy †Cardiff University, United Kingdom ‡Bar-Ilan University, Israel WepresentnewpropertiesfortheFractionalPoissonprocessand theFractionalPoissonfieldontheplane.Amartingalecharacteriza- tionforFractionalPoissonprocessesisgiven.Weextendthisresultto FractionalPoissonfields,obtainingsomeothercharacterizations.The fractionaldifferentialequationsarestudied.Weconsideramoregen- eral Mixed-Fractional Poisson process and show that this process is 6 thestochasticsolutionofasystemoffractionaldifferential-difference 1 equations.Finally,wegivesomesimulationsoftheFractionalPoisson 0 2 field on the plane. g u 1. Introduction. There are several different approaches to the fundamental concept of Frac- A tional Poisson process (FPP) on the real line. The “Renewal” approach consists of considering the 1 characterizationofthePoissonprocessasasumofindependentnon-negativerandomvariables,and instead of the assumption that these random variables have an exponential distribution, we assume ] R that they have the Mittag-Leffler distribution (see [31, 32, 41]). In [6], the renewal approach to P the Fractional Poisson process is developed and it is proved that its one-dimensional distributions . h coincidewiththesolutiontofractionalizedstateprobabilities.Anotherequivalentapproachisusing t a “inverse subordinator”, a kind of Fractional Poisson process can be constructed [33]. m In[26],followingthismethod,theFPPisgeneralizedanddefined,obtainingaFractionalPoisson [ random field (FPRF) parametrized by points of the Euclidean space R2, as it has been done for + 2 Fractional Brownian fields, see, e.g., [18, 23, 29, 21]. v The starting point of our extension will be the set-indexed Poisson process which is a well-known 6 concept, see, e.g., [43, 36, 37, 18, 23]. 3 1 Inthispaper,wefirstpresentamartingalecharacterizationoftheFractionalPoissonprocess.We 8 extend this characterization to FPRF using the concept of increasing path and strong martingales. 0 This characterization permits us to give a definition of a set-indexed Fractional Poisson process. . 1 We study the fractional differential equation for FPRF. Finally, we study Mixed-Fractional Poisson 0 6 processes. 1 The paper is organized as follows. In the next section, we collect some known results from the : v theory of subordinators and inverse subordinators, see [9, 45, 46, 35] among others. In Section 3, we i X proveamartingalecharacterizationoftheFPP,whichisageneralizationoftheWatanabeTheorem. r In Section 4, another generalization called “Mixed-Fractional Poisson process” is introduced and a some distributional properties are studied as well as Watanabe characterization is given. Section 5 is devoted to FPRF. We begin by computing covariance for this process, then we give some char- acterizations using increasing paths and intensities. We present a Gergely-Yeshow characterization and discuss random time changes. Fractional differential equations are discussed on Section 6. Finally, we present some simulations for the FPRF. Other different generalizations of FPP and fields can be found in [47, 48, 8, 38, 39]. 2. Inverse Subordinators. This section collects some known resuts from the theory of subordinators and inverse subordinators [9, 45, 46, 35]. 1 imsart-aap ver. 2011/11/15 file: ALM-ARXIVv2.tex date: August 3, 2016 2 G. ALETTI, N. LEONENKO, E. MERZBACH 2.1. Subordinators and their inverse. Consider a non-decreasing Lévy process L = {L(t), t ≥ 0},startingfrom0,whichiscontinuousfromtherightwithleftlimits(cadlag),continuousinproba- bility,withindependentandstationaryincrements.SuchaprocessisknownasaLévysubordinator with Laplace exponent (cid:90) φ(s) = µs+ (1−e−sx)Π(dx), s ≥ 0, (0,∞) where µ ≥ 0 is the drift and the Lévy measure Π on R ∪{0} satisfies + (cid:90) ∞ min(1,x)Π(dx) < ∞. 0 This means that Ee−sL(t) = e−tφ(s), s ≥ 0. Consider the inverse subordinator Y(t), t ≥ 0, which is given by the first-passage time of L : Y(t) = inf{u ≥ 0 : L(u) > t},t ≥ 0. TheprocessY(t), t ≥ 0,isnon-decreasinganditssamplepathsarea.s.continuousifanditssample paths are a.s. right continuous if L is strictly increasing. We have {(u ,t ): L(u ) < t ,i = 1,...,n} = {(u ,t ): Y(t ) > u ,i = 1,...,n}, i i i i i i i i and for any z > 0 (cid:110) (cid:111) P{Y(t) > x} = P{L(x) ≤ t} = P e−zL(x) ≥ e−zt ≤ exp{tz−xφ(z)}, which implies that for any p > 0,EYp(t) < ∞, since (cid:90) ∞ (cid:90) ∞ EYp(t) = p xp−1(1−P{Y(t) ≤ x})dx ≤ petz xp−1e−xφ(z)dx < ∞. 0 0 Let U(t) = EY(t) be the renewal function. Since (cid:90) ∞ 1 U˜(s) = U(t)e−stdt = , sφ(s) 0 then U˜ characterizes the inverse process Y, since φ characterizes L. We get a covariance formula [45, 46] (cid:90) min(t,s) Cov(Y(t),Y(s)) = (U(t −τ)+U(t −τ))dU(τ)−U(t )U(t ). 1 2 1 2 0 Themostimportantexampleisconsideredinthenextsection,buttherearesomeotherexamples. imsart-aap ver. 2011/11/15 file: ALM-ARXIVv2.tex date: August 3, 2016 FRACTIONAL POISSON FIELDS AND MARTINGALES 3 2.2. Inverse stable subordinators. Let L = {L (t),t ≥ 0}, be an α−stable subordinator with α α φ(s) = sα,0 < α < 1. The density of L (1) is of the form [44] α ∞ 1 (cid:88) Γ(αk+1) 1 1 g (x) = (−1)k+1 sin(πkα) = W (−x−α). (2.1) α π k! xαk+1 x −α,0 k=1 Here we use the Wright’ s generalized Bessel function (see, e.g., [17]) (cid:88)∞ zk W (z) = , z ∈ C, (2.2) γ,β Γ(1+k)Γ(β +γk) k=0 where γ > −1, and β ∈ R. The set of jump times of L is a.s. dense. The Lévy subordinator is α strictly increasing, since the process L admits a density. α Then the inverse stable subordinator Y (t) = inf{u ≥ 0 : L (u) > t} α α has density [35, p.110] d t fα(t,x) = P{Yα(t) ≤ x} = x−1−α1gα(tx−α1), x > 0, t > 0. (2.3) dx α The Laplace transform of the density f (t,x) is a (cid:90) ∞ e−stf (t,x)dt = sα−1e−xsα, s ≥ 0, (2.4) a 0 Its paths are continuous and nondecreasing. For α = 1/2, the inverse stable subordinator is the running supremum process of Brownian motion, and for α ∈ (0,1/2] this process is the local time at zero of a strictly stable Lévy process of index α/(1−α). Let (cid:88)∞ zk E (z) = , α > 0, z ∈ C (2.5) α Γ(αk+1) k=0 be the Mittag-Leffler function [17], and recall the following: i) The Laplace transform of function E (−tα) is of the form α (cid:90) ∞ sα−1 e−stE (−λtα)dt = , 0 < α < 1, t ≥ 0,(cid:60)(s) > |λ|1/α. α λ+sα 0 (ii) The function E (λtα) is an eigenfunction at the the fractional Caputo-Djrbashian derivative α Dα with eigenvalue λ [35, p.36] t DαE (λtα) = λE (λtα), 0 < α < 1,λ ∈ R, t α α where Dα is defined as (see [35]) t 1 (cid:90) t du(τ) dτ Dαu(t) = , 0 < α < 1. (2.6) t Γ(1−α) dτ (t−τ)α 0 Proposition 2.1 (See [9, 45, 46]). The α-stable inverse subordinators satisfy the following properties: imsart-aap ver. 2011/11/15 file: ALM-ARXIVv2.tex date: August 3, 2016 4 G. ALETTI, N. LEONENKO, E. MERZBACH (i) (cid:88)∞ (−stα)n Ee−sYα(t) = = E (−stα), s > 0. α Γ(αn+1) n=0 (ii) Both processes L (t),t ≥ 0 and Y (t) are self-similar α α L (at) Y (at) α d α d = L (t), = Y (t), a > 0. a1/α α aα α (iii) For 0 < t < ··· < t , 1 k ∂kE(Y (t )···Y (t )) 1 1 α 1 α k = . ∂t1···∂tk Γk(α)[t1(t2−t1)···(tk −tk−1)]1−α In particular, (A) tα Γ(ν +1) EY (t) = ;E[Y (t)]ν = tαν, ν > 0; α α Γ(1+α) Γ(αν +1) (B) 1 (cid:90) min(t,s) (st)α Cov(Y (t),Y (s)) = ((t−τ)α+(s−τ)α)τα−1dτ − . (2.7) α α Γ(1+α)Γ(α) Γ2(1+α) 0 2.3. Mixture of inverse subordinators. This subsection collects some results from the theory of inverse subordinators, see [45, 46, 35, 5, 27]. Different kinds of inverse subordinators can be considered. Let L and L be two independent stable subordinators. The mixture of them L = α1 α2 α1,α2 {L (t),t ≥ 0} is defined by its Laplace transform: for s ≥ 0, C + C = 1, C ≥ 0, C ≥ α1,α2 1 2 1 2 0, α < α , 1 2 Ee−sLα1,α2(t) = exp{−t(C1sα1 +C2sα2)}. (2.8) It is clear that 1 1 Lα1,α2(t) = (C1)α1Lα1(t)+(C2)α2Lα2(t), t ≥ 0, 1 1 1 1 is not self-similar, unless α1 = α2 = α, since Lα1,α2(at) =d (C1)α1aα1Lα1(t)+(C2)α2aα2Lα2(t). 1 This expression is equal to aαLα1,α2(t) if and only if for any t > 0 α1 = α2 = α, in which case the process L can be reduced to the classical stable subordinator (up to constant). α1,α2 The inverse subordinator is defined by Y (t) = inf{u ≥ 0 : L (u) > t}, t ≥ 0. (2.9) α1,α2 α1,α2 We assume that C (cid:54)= 0 without loss of generality (the case C = 0 reduces to the previous case of 2 2 single inverse subordinator). It was proved in [27] that 1 1 U˜(t) = (C1sα1 +C2sα2)s,U(t) = C2tα2Eα2−α1,α2+1(−CC21tα2−α1), (2.10) where E (z) is the two-parametric Generalized Mittag-Leffler function ([15, 17]) α,β (cid:88)∞ zk E (z) = , α > 0,β > 0, z ∈ C. α,β Γ(αk+β) k=0 imsart-aap ver. 2011/11/15 file: ALM-ARXIVv2.tex date: August 3, 2016 FRACTIONAL POISSON FIELDS AND MARTINGALES 5 Also for the Laplace transform of the density f (t,u) = d P{Y (t) ≤ u}, u ≥ 0, of inverse α1,α2 du α1,α2 subordinator Y = {Y (t),t ≥ 0}, we have the following expression [34]: α1,α2 α1,α2 (cid:90) ∞ 1 f˜ (s,u) = e−stf (t,u)dt = [C sα1 +C sα2]e−u[C1sα1+C2sα2], s ≥ 0, (2.11) α1,α2 α1,α2 s 1 2 0 and the Laplace transform of f˜is given by (cid:90) ∞e−puf˜ (s,u)du = φ(s) = C1sα1−1+C2sα2−1, p ≥ 0. (2.12) 0 α1,α2 s(p+φ(s)) p+C1sα1 +C2sα2 From [5, Theorem 2.3] we have the following expression for u ≥ 0,t > 0: ∞ f (t,u) = C1 (cid:88) 1(−C2|u|)rW (−C1|u|)+ α1,α2 λtα1 r! λtα2 −α1,1−α2r−α1 λtα1 r=0 ∞ + C2 (cid:88) 1(−C1|u|)rW (−C2|u|). (2.13) λtα2 r! λtα1 −α2,1−α1r−α2 λtα2 r=0 One can also consider the tempered stable inverse subordinator, the inverse subordinator to the Poisson process, the compound Poisson process with positive jumps, the Gamma and the inverse Gaussian Lévy processes. For additional details see [45, 46, 27]. 3. Fractional Poisson Processes and Martingales. 3.1. Preliminaries. The first definition of FPP N = {N (t),t ≥ 0} was given in [31] (see also α α [32]) as a renewal process with Mittag-Leffler waiting times between the events ∞ (cid:88) N (t) = max{n : T +...+T ≤ t} = 1 , t ≥ 0, α 1 n {T1+...+Tj≤t} j=1 where {T }, j = 1,2,... are iid random variables with the strictly monotone Mittag-Leffler distri- j bution function F (t) = P(T ≤ t) = 1−E (−λtα), t ≥ 0,0 < α < 1, j = 1,2,... α j α The following stochastic representation for FPP is found in [33]: N (t) = N(Y (t)), t ≥ 0, α ∈ (0,1), α α whereN = {N(t),t ≥ 0},istheclassicalhomogeneousPoissonprocesswithparameterλ > 0,which is independent of the inverse stable subordinator Y . One can compute the following expression for α the one-dimensional distribution of FPP: (cid:90) ∞ e−λx(λt)k (α) P(N (t) = k) = p (t) = f (t,x)dx α k k! α 0 (λtα)k (cid:88)∞ (k+j)! (−λtα)j (λtα)k = = E(k)(−λtα) k! j! Γ(α(j +k)+1) k! α j=1 = (λtα)kEk+1 (−λtα), k = 0,1,2...,t ≥ 0, 0 < α < 1, α,αk+1 imsart-aap ver. 2011/11/15 file: ALM-ARXIVv2.tex date: August 3, 2016 6 G. ALETTI, N. LEONENKO, E. MERZBACH (k) where f is given by (2.3), E (z) is the Mittag-Leffler function (2.5), E (z) is the k−th derivative α α α of E (z), and Eγ (z) is the three-parametric Generalized Mittag-Leffler function defined as follows α α,β [17, 40]: Eγ (z) = (cid:88)∞ (γ)jzj ,α > 0,β > 0, γ > 0, z ∈ C, (3.1) α,β j!Γ(αj +β) j=0 where (cid:40) 1 if j = 0; (γ) = j γ(γ +1)···(γ +j −1) if j = 1,2,... is the Pochhammer symbol. Finally, in [6, 7] it is shown that the marginal distribution of FPP satisfies the following system of fractional differential-difference equations: Dαp(α)(t) = −λ(p(α)(t)−p(α) (t)), k = 0,1,2.. t k k k−1 with initial conditions: p(α)(0) = 1,p(α)(0) = 0,k ≥ 1, and p(α)(t) = 0, where Dα is the fractional 0 k −1 t Caputo-Djrbashian derivative (2.6). See also [13]. Remark. Note that (cid:90) ∞ EN (t) = E(cid:2)E[N(Y (t))|Y (t)](cid:3) = [EN(u)]f (t,u)du = λtα/Γ(1+α), α α α α 0 where f (t,u) is given by (2.3), and [27] showed that α λ(min(t,s))α Cov(N (t),N (s)) = +λ2Cov(Y (t),Y (s)), (3.2) α α α α Γ(1+α) where Cov(Y (t),Y (s)) is given in (2.7) while Cov(N(t),N(s)) = λmin(t,s). In particular, α α (cid:104) 2 1 (cid:105) λtα VarN (t) = λ2t2α − + α Γ(1+2α) Γ2(1+α) Γ(1+α) λ2t2α (cid:16)αΓ(α) (cid:17) λtα = −1 + , t ≥ 0. Γ2(1+α) Γ(2α) Γ(1+α) The definition of the Hurst index for renewal processes is discussed in [15]. In the same spirit, one can define the analogous of the Hurst index for the FPP as (cid:26) (cid:27) VarN (T) α H = inf β : lim sup < ∞ ∈ (0,2). T2β T→∞ To prove the formula (3.2), one can use the conditional covariance formula [42, Exercise 7.20.b]: (cid:0) (cid:1) (cid:0) (cid:1) Cov(Z ,Z ) = E Cov(Z ,Z |Y) +Cov E(Z |Y),E(Z |Y) , 1 2 1 2 1 2 where Z ,Z and Y are random variables, and 1 2 (cid:0) (cid:1) Cov(Z ,Z |Y) = E (Z −E(Z |Y))(Z −E(Z |Y)) . 1 2 1 1 2 2 Really, if G (u,v) = P{Y (t) ≤ u,Y (s) ≤ v}, t,s α α imsart-aap ver. 2011/11/15 file: ALM-ARXIVv2.tex date: August 3, 2016 FRACTIONAL POISSON FIELDS AND MARTINGALES 7 then E(N(Y (t))|Y (t)) = E(N(1))·Y (t) = λY (t), and α α α α (cid:16) (cid:90) ∞(cid:90) ∞ (cid:17) (cid:0) (cid:1) Cov(Y (t),Y (s)) = Var N(1) min(u,v)G (du,dv) +Cov λY (t),λY (s) α α t,s α α 0 0 = λE(Y (min(t,s)))+λ2Cov(Y (t),Y (s)), α α α since, for example, if s ≤ t, then v = Y (s) ≤ Y (t) = u, and α α (cid:90) ∞(cid:90) ∞ (cid:90) ∞ (cid:90) ∞ (cid:90) ∞ vG (du,dv) = v G (du,dv) = vdP{Y (s) ≤ v} = E(Y (s)). t,s t,s α α 0 0 0 0 0 Remark. For more than one random variable in the condition, the conditional covariance formula becomes more complicated, it can be seen even for the conditional variance formula: (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) Var(Z) = E Var(Z|Y ,Y ) +E Var[E(Z|Y ,Y )]|Y +Var E(Z|Y ) . 1 2 1 2 1 1 The corresponding formulas can be found in [10]. That is why for random fields we develop another technique, see Appendix. 3.2. Watanabe characterization. Let (Ω,F,P) be a complete probability space. Recall that the F −adapted, P-integrable stochastic process M = {M(t),t ≥ 0} is an F −martingale (sub- t t martingale) if E(M(t)|F ) = (≥)M(s), 0 ≤ s ≤ t, a.s., where {F } is a non-decreasing family s t of sub-sigma fields of F. The following theorem is known as the Watanabe characterization for homogeneous Poisson processes (see, [49] and [12, p. 25]): Theorem 3.1. Let N = {N(t),t ≥ 0} be a F −adapted, simple locally finite point process. t Then N is a homogeneous Poisson process iff there is a constant λ > 0, such that the process M(t) = N(t)−λt is an F −martingale. t We extend the well-known Watanabe characterization for FPP. Theorem 3.2. Let X = {X(t), t ≥ 0} be a simple locally finite point process. Then X is a FPP iff there exist a constant λ > 0, and an α-stable subordinator L = {L (t), t ≥ 0}, 0 < α < 1, such α α that, denoted by Y (t) = inf{s : L (s) ≥ t} its inverse stable subordinator, the process α α M = {M(t), t ≥ 0} = {X(t)−λY (t), t ≥ 0} α is a martingale with respect to the induced filtration F = σ(X(s),Y (s),s ≤ t). t α Proof. IfX isaFPP,thenX(t) = N(Y (t)),whereY istheinverseofanα-stablesubordinator α α and N is a Poisson process with intensity λ > 0. Therefore N(Y (t)) − λY (t) is a martingale. Notice that Y (t) is continuous increasing and α α α adapted; therefore it is the predictable intensity of the sub-martingale X. Conversely, it is enough to prove that X(t) = N(Y (t)), where N is a Poisson process, indepen- α dent of Y . α Consider the inverse of Y (t) : α Z(t) = inf{s : Y (s) ≥ t}. α Notice that in general Z(t) does not equal to L (t), since for example Z(t) is continuous, but L (t) α α is not. imsart-aap ver. 2011/11/15 file: ALM-ARXIVv2.tex date: August 3, 2016 8 G. ALETTI, N. LEONENKO, E. MERZBACH {Z(t), t ≥ 0} can be seen as a family of stopping times. Then, by the Doob optional sampling theorem [24, Theorem 7.29], M(Z(t)) = X(Z(t))−λY (Z(t)) α isstillamartingale.ThenX(Z(t))−λtisamartingale.SinceZ(t)isincreasing,X(Z(t))isasimple point process. Following the classical Watanabe characterization, X(Z(t)) is a classical Poisson process with parameter λ > 0. Call this process N(t) = X(Z(t)). Then X(t) = N(Y (t)) is a FPP. α 4. Mixed-Fractional Poisson Processes. 4.1. Definition. In this section, we consider a more general Mixed-Fractional Poisson process (MFPP) Nα1,α2 = {Nα1,α2(t),t ≥ 0} = {N(Y (t)),t ≥ 0}, (4.1) α1,α2 where the homogeneous Poisson process N with intensity λ > 0, and the inverse subordinator Yα1,α2 given by (2.9) are independent. We will show that Nα1,α2 is the stochastic solution of the system of fractional differential-difference equations: for k = 0,1,2,..., C Dα1p(α1,α2)(t)+C Dα2p(α1,α2)(t) = −λ(p(α1,α2)(t)−p(α1,α2)(t)), (4.2) 1 t k 2 t k k k−1 with initial conditions: p(α1,α2)(0) = 1,p(α1,α2)(0) = 0,p(α1,α2)(t) = 0, k ≥ 1, (4.3) 0 k −1 where Dα is the fractional Caputo-Djrbashian derivative (2.6), and for C ≥ 0,C > 0,C +C = 1, t 1 2 1 2 α ,α ∈ (0,1), 1 2 p(α1,α2)(t) = P{Nα1,α2(t) = k}, k = 0,1,2... k 4.2. Distribution Properties. UsingtheformulaeforLaplacetransformofthefractionalCaputo- Djrbashian derivative (see, [35, p.39]): (cid:90) ∞ e−stDαu(t)dt = sαu˜(s)−sα−1u(0),0 < α < 1, t 0 one can obtain from (4.2) with k = 0 the following equation C sα1p˜ (s)−C sα1−1+C sα2p˜ (s)−C sα2−1 = −λp˜ (s),p˜ (0) = 1, 1 0 1 2 0 2 0 0 for the Laplace transform (cid:90) ∞ p˜(α1,α2)(s) = p˜ (s) = e−stp(α1,α2)(t)dt, s ≥ 0. 0 0 0 0 Thus C1sα1−1+C2sα2−1 p˜ (s) = , s ≥ 0, 0 λ+C1sα1 +C2sα2 and using the formula for an inverse Laplace transform (see, [17]), for (cid:60)α > 0,(cid:60)β > 0,(cid:60)s > 0,(cid:60)(α−ρ) > 0,(cid:60)(α−β) > 0, and |asβ/(sα+b)| < 1: (cid:16) sρ−1 (cid:17) (cid:88)∞ L−1 ;t = tα−ρ (−a)rt(α−β)rEr+1 (−btα), (4.4) sα+asβ +b α,α+(α−β)r−ρ+1 r=0 imsart-aap ver. 2011/11/15 file: ALM-ARXIVv2.tex date: August 3, 2016 FRACTIONAL POISSON FIELDS AND MARTINGALES 9 one can find an exact form of the p(α1,α2)(t) in terms of generalized Mittag-Leffler functions (3.1): 0 p(α1,α2)(t) = (cid:88)∞ (cid:18)−C1tα2−α1(cid:19)rEr+1 (cid:18)− λ tα2(cid:19) (4.5) 0 C2 α2,(α2−α1)r+1 C2 r=0 −(cid:88)∞ (cid:18)−C1tα2−α1(cid:19)r+1Er+1 (cid:18)− λ tα2(cid:19). C2 α2,(α2−α1)(r+1)+1 C2 r=0 For k ≥ 1,we obtain from (4.2): p˜ (s)(λ+C sα1 +C sα2) = λp˜ (s), k 1 2 k−1 where (cid:90) ∞ p˜(α1,α2)(s) = p˜ (s) = e−stp(α1,α2)(t)dt, s ≥ 0. k k k 0 Thus from (4.2) we obtain the following expression for the Laplace transform of p(α1,α2)(t), k ≥ 0 : k (cid:18) λ (cid:19) (cid:18) λ (cid:19)k p˜ (s) = p˜ (s) = p˜ (s) (4.6) k λ+C1sα1 +C2sα2 k−1 λ+C1sα1 +C2sα2 0 λk(C1sα1−1+C2sα2−1) λk(C1sα1 +C2sα2) = = , k = 0,1,2... (λ+C1sα1 +C2sα2)k+1 s(λ+C1sα1 +C2sα2)k+1 On the other hand, one can compute the Laplace transform from the stochastic representation (4.1). If (cid:90) ∞ e−λx p(α1,α2)(t) = P{N(Y (t) = k} = (λx)kf (t,x)dx, (4.7) k α1,α2 k! α1,α2 0 where f (t,x) is given by (2.13), then using (2.11),(2.12) we have for k ≥ 0,s > 0 α1,α2 (cid:90) ∞ (cid:90) ∞ e−λx (cid:90) ∞ p˜ (s) = e−stp(α1,α2)(t)dt = (λx)k[ e−stf (t,x)dt]dx k k k! α1,α2 0 0 0 λk φ(s) (cid:90) ∞ = e−λxxke−xφ(s)dx k! s 0 Note that ∂k (cid:90) ∞ (cid:90) ∞ e−λxe−xφ(s)dx = (−1)k e−λxxke−xφ(s)dx ∂λk 0 0 ∂k 1 k! = = (−1)k ; ∂λk λ+φ(s) (λ+φ(s))k+1 thus p˜ (s) = λk φ(s) = λk(C1sα1 +C2sα2) , k s(λ+φ(s))k+1 s(λ+C1sα1 +C2sα2)k+1 the same expression as (4.6). We can formulate the result in the following form: Theorem 4.1. The MFPP Nα1,α2 defined in (4.1) is the stochastic solution of the system of fractional differential-difference equations (4.2) with initial conditions (4.3). imsart-aap ver. 2011/11/15 file: ALM-ARXIVv2.tex date: August 3, 2016 10 G. ALETTI, N. LEONENKO, E. MERZBACH Notethatin[5]onecanfindsomeotherstochasticrepresentationsoftheMFPP(4.1).Also,some analyticalexpressionforp(α1,α2)(t)isgivenby(4.5),whiletheanalyticalexpressionforp(α1,α2)(t),for 0 k k ≥ 1, are given by (4.7). Moreover, p(α1,α2)(t),for k ≥ 1, can be obtained by the following recurrent relation: k t (cid:90) p(α1,α2)(t) = p(α1,α2)(t−z)g(z)dz, k k−1 0 where (cid:90) ∞ λ g˜(s) = e−stg(z)dz = , 0 λ+C1sα1 +C2sα2 and from (4.4): ∞ g(z) = λ zα2−1(cid:88)(cid:16)− C1tα2−α1(cid:17)rEr+1 (cid:16)− λ tα2(cid:17). C2 C2 α2,α2+(α2−α1)r C2 r=0 4.3. Dependence. From [27, Theorem 2.1] and (2.10), we have the following expressions for moments in form of the function 1 U(t) = C tα2Eα2−α1,α2+1(−C1tα2−α1/C2), 2 ENα1,α2(t) = λU(t), 1 VarNα1,α2(t) = λ2C2t2α2[2Eα2−α1,α1+α2+1(−C1tα2−α1/C2) 2 −(Eα2−α1,α2+1(−C1tα2−α1/C2))2] 1 +λC tα2Eα2−α1,α2+1(−C1tα2−α1/C2), 2 (cid:110)(cid:90) min(t,s)(cid:16) Cov(Nα1,α2(t),Nα1,α2(s)) = λU(min(t,s))+λ2 U(t −τ) 1 0 (cid:17) (cid:111) +U(t −τ) dU(τ)−U(t )U(t ) . 2 1 2 We extend the Watanabe characterization for MFPP. Let Λ(t) : R → R be a non-negative + + right-continuous non-decreasing deterministic function such that Λ(0) = 0, Λ(∞) = ∞, and Λ(t)−Λ(t−) ≤ 1 for any t. Such a function will be called consistent. The Mixed-Fractional Non- homogeneous Poisson process (MFNPP) is defined as Nα1,α2 = {Nα1,α2(t),t ≥ 0} = {N(Λ(Y (t))),t ≥ 0}, Λ Λ α1,α2 where the homogeneous Poisson process N with intensity λ = 1, and the inverse subordinator Y given by (2.9) are independent. α1,α2 Theorem 4.2. Let X = {X(t), t ≥ 0} be a simple locally finite point process. X is a MFNPP iff there exist a consistent function Λ(t), and a mixed stable subordinator {L (t), t ≥ 0}, 0 < α1,α2 α < 1, 0 < α < 1, defined in (2.8), such that 1 2 M = {M(t), t ≥ 0} = {X(t)−Λ(Y (t)), t ≥ 0} α1,α2 isamartingalewithrespecttotheinducedfiltrationF = σ(X(s),Y (s),s ≤ t),whereY (t) = t α1,α2 α1,α2 inf{s : L (t) ≥ t} is the inverse mixed stable subordinator. α1,α2 imsart-aap ver. 2011/11/15 file: ALM-ARXIVv2.tex date: August 3, 2016

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