Definitions Limit theorems Application to the CTRW Summary and Outlook References A New Fractional Process: A Fractional Non-homogeneous Poisson Process Enrico Scalas University of Sussex Joint work with Nikolai Leonenko (Cardiff University) and Mailan Trinh Fractional PDEs: Theory, Algorithms and Applications, ICERM, Jun 18 - 22, 2018 19 June, 2018 A New Fractional Process: A Fractional Non-homogeneous Poisson Process Enrico Scalas Definitions Limit theorems Application to the CTRW Summary and Outlook References Overview 1 Definitions 2 Limit theorems 3 Application to the CTRW 4 Summary and Outlook A New Fractional Process: A Fractional Non-homogeneous Poisson Process Enrico Scalas Definitions Limit theorems Application to the CTRW Summary and Outlook References Overview 1 Definitions 2 Limit theorems 3 Application to the CTRW 4 Summary and Outlook A New Fractional Process: A Fractional Non-homogeneous Poisson Process Enrico Scalas Definitions Limit theorems Application to the CTRW Summary and Outlook References Classification of Poisson processes standard fractional h hf homogeneous (i) (Nλ(t)) (iii) (Nα (t)) inhomogeneous (ii) (N(t)) (iv) (Nα(t)) A New Fractional Process: A Fractional Non-homogeneous Poisson Process Enrico Scalas Definitions Limit theorems Application to the CTRW Summary and Outlook References The standard (non-fractional) case h (i) The homogeneous Poisson process (HPP) (Nλ(t)) with intensity parameter λ > 0: (λt)x pλ(t) := (Nh(t) = x) = e−λt ; x = 0; 1; 2;::: x P λ x! (ii) The inhomogeneous Poisson process (NHPP) (N(t)) with intensity λ(t) : [0; 1) −! [0; 1) and rate function Z t Λ(s; t) = λ(u)du: s For x = 0; 1; 2;:::, the distribution of the increment is e−Λ(v;t+v)(Λ(v; t + v))x p (t; v) := fN(t + v) − N(v) = xg = : x P x! h Note that N(t) = N1 (Λ(t)). A New Fractional Process: A Fractional Non-homogeneous Poisson Process Enrico Scalas Definitions Limit theorems Application to the CTRW Summary and Outlook References The (inverse) α-stable subordinator Let Lα = fLα(t); t ≥ 0g, be an α-stable subordinator with Laplace transform α E [exp(−sLα(t))] = exp(−ts ); 0 < α < 1; s ≥ 0 and Yα = fYα(t); t ≥ 0g, be an inverse α-stable subordinator defined by Yα(t) = inffu ≥ 0 : Lα(u) > tg: Let hα(t; ·) denote the density of the distribution of Yα(t). Its Laplace transform can be expressed via the three-parameter Mittag-Leffler function (a.k.a Prabhakar function). 1 α E [exp(−sYα(t))] = Eα,1(−st ); where 1 X cj zj E c (z) = ; with a;b j!Γ(aj + b) j=0 j c = c(c + 1)(c + 2) ::: (c + j − 1); a > 0; b > 0; c > 0; z 2 C: A New Fractional Process: A Fractional Non-homogeneous Poisson Process Enrico Scalas Definitions Limit theorems Application to the CTRW Summary and Outlook References 1.2 0.6 0.5 α = 0.1 1 0.5 α = 0.6 0.4 α = 0.9 0.8 0.4 ) ) ) 0.3 , x , x , x 0.6 0.3 (1 (10 (40 α α α h 0.2 0.4 h 0.2 h 0.1 0.2 0.1 0 0 0 0 5 10 0 10 20 30 0 50 100 x x x Figure: Plots of the probability densities x 7! hα(t; x) of the distribution of the inverse α-stable subordinator Yα(t) for different parameter α = 0:1; 0:6; 0:9 and as a function of time: the plot on the left is generated for t = 1, the plot in the middle for t = 10 and the plot on the right for t = 40. The x scale is not kept constant. A New Fractional Process: A Fractional Non-homogeneous Poisson Process Enrico Scalas Definitions Limit theorems Application to the CTRW Summary and Outlook References The fractional case (iii) The fractional homogeneous Poisson process (FHPP) hf hf h (Nα (t)) is defined as Nα (t) := Nλ(Yα(t)) for t ≥ 0; 0 < α < 1. Its marginal distribution is given by Z 1 x α −λu (λu) px (t) = PfNλ(Yα(t)) = xg = e hα(t; u)du 0 x! α x x+1 α = (λt ) Eα,αx+1(−λt ); x = 0; 1; 2;::: (iv) The fractional non-homogenous Poisson process (FNPP) could be defined in the following way: Recall that the NPP can be expressed via the HPP: h N(t) = N1 (Λ(t)): h Analogously define Nα(t) := N(Yα(t)) = N1 (Λ(Yα(t))) A New Fractional Process: A Fractional Non-homogeneous Poisson Process Enrico Scalas Definitions Limit theorems Application to the CTRW Summary and Outlook References The governing equation for the FNPP We can define the marginals α h h fx (t; v) := PfN1 (Λ(Yα(t) + v)) − N1 (Λ(v)) = xg; x = 0; 1; 2;::: Z 1 = px (u; v)hα(t; u)du 0 Theorem (Leonenko et al. (2017)) h h Let Iα(t; v) = N1 (Λ(Yα(t) + v)) − N1 (Λ(v)) be the fractional increment process. Then, its marginal distribution satisfies the following fractional differential-integral equations (x = 0; 1;:::) Z 1 α α Dt fx (t; v) = λ(u + v)[−px (u; v) + px−1(u; v)]hα(t; u)du; 0 α α with initial condition fx (0; v) = δ0(x) and f−1(0; v) ≡ 0. A New Fractional Process: A Fractional Non-homogeneous Poisson Process Enrico Scalas Definitions Limit theorems Application to the CTRW Summary and Outlook References Overview 1 Definitions 2 Limit theorems 3 Application to the CTRW 4 Summary and Outlook A New Fractional Process: A Fractional Non-homogeneous Poisson Process Enrico Scalas Definitions Limit theorems Application to the CTRW Summary and Outlook References Limit theorems for the Poisson process h Watanabe (1964): The compensator of Nλ(t) is λt, i.e. h Nλ(t) − λt is a martingale. (Watanabe characterisation) One-dimensional central limit theorem Nh(t) − λt λ p −−−!d N (0; 1) λt t!1 Functional central limit theorem: convergence in D([0; 1)) w.r.t. J1-topology to a standard Brownian motion (B(t))t≥0. h N (t) − λt J λ p −−−!1 B λ t≥0 λ!1 Functional scaling limit: h Nλ(ct) J1 −−−! (λt)t≥0 c!1 c t≥0 A New Fractional Process: A Fractional Non-homogeneous Poisson Process Enrico Scalas Definitions Limit theorems Application to the CTRW Summary and Outlook References Limit theorems for the Poisson process h Watanabe (1964): The compensator of Nλ(t) is λt, i.e. h Nλ(t) − λt is a martingale. (Watanabe characterisation) One-dimensional central limit theorem Nh(t) − λt λ p −−−!d N (0; 1) λt t!1 Functional central limit theorem: convergence in D([0; 1)) w.r.t. J1-topology to a standard Brownian motion (B(t))t≥0. h N (t) − λt J λ p −−−!1 B λ t≥0 λ!1 Functional scaling limit: h Nλ(ct) J1 −−−! (λt)t≥0 c!1 c t≥0 A New Fractional Process: A Fractional Non-homogeneous Poisson Process Enrico Scalas Definitions Limit theorems Application to the CTRW Summary and Outlook References Random time change and continuous mapping theorem We have convergence in D([0; 1)) w.r.t. J1-topology to a standard Brownian motion (B(t))t≥0. h N (t) − λt J λ p −−−!1 B: λ t≥0 λ!1 As B has continuous paths and Yα has non-decreasing paths, it follows that h Nλ(Yα(t)) − λYα(t) J1 p −−−! [B(Yα(t))]t≥0 : λ t≥0 λ!1 (Thm. 13.2.2 in Whitt (2002)) A New Fractional Process: A Fractional Non-homogeneous Poisson Process Enrico Scalas Definitions Limit theorems Application to the CTRW Summary and Outlook References Limit theorems for the Poisson process h Watanabe (1964): The compensator of Nλ(t) is λt, i.e. h Nλ(t) − λt is a martingale. (Watanabe characterisation) One-dimensional central limit theorem Nh(t) − λt λ p −−−!d N (0; 1) λt t!1 Functional central limit theorem: convergence in D([0; 1)) w.r.t. J1-topology to a standard Brownian motion (B(t))t≥0. h N (t) − λt J λ p −−−!1 B λ t≥0 λ!1 Functional scaling limit: h Nλ(ct) J1 −−−! (λt)t≥0 c!1 c t≥0 A New Fractional Process: A Fractional Non-homogeneous Poisson Process Enrico Scalas Definitions Limit theorems Application to the CTRW Summary and Outlook References Cox processes: definition Idea: Poisson process with stochastic intensity. (Cox (1955)) ! actuarial risk models (e.g. Grandell (1991)) ! credit risk models (e.g. Bielecki and Rutkowski (2002)) ! filtering theory (e.g. Br´emaud(1981)) Definition Let (Ω; F; P) be a probability space and (N(t))t≥0 be a point N process adapted to (Ft )t≥0.(N(t))t≥0 is a Cox process if there exist a right-continuous, increasing process (A(t))t≥0 such that, conditional on the filtration (Ft )t≥0, where N Ft := F0 _Ft ; F0 = σ(A(t); t ≥ 0); (N(t))t≥0 is a Poisson process with intensity dA(t). A New Fractional Process: A Fractional Non-homogeneous Poisson Process Enrico Scalas Definitions Limit theorems Application to the CTRW Summary and Outlook References Cox processes and the FNPP Is the FNPP a Cox process? The FHPP is also a renewal process: handy criteria in Yannaros (1994), Grandell (1976), Kingman (1964). h Construction of a suitable filtration: Nα(t) = N1 (Λ(Yα(t))). Nα Ft := σ(fNα(s); s ≤ tg) F0 := σ(Yα(t); t ≥ 0) Nα Ft := F0 _Ft A New Fractional Process: A Fractional Non-homogeneous Poisson Process Enrico Scalas Definitions Limit theorems Application to the CTRW Summary and Outlook References A central limit theorem Nα Ft := σ(fNα(s); s ≤ tg) F0 := σ(Yα(t); t ≥ 0) Nα Ft := F0 _Ft Proposition Let (N(Yα(t)))t≥0 be the FNPP adapted to the filtration (Ft )t≥0 as defined in previous slide.