A New Fractional Process: A Fractional Non-homogeneous Poisson Process Enrico Scalas

Transcription

1 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, June, 2018

2 Overview 1 Definitions 2 Limit theorems 3 Application to the CTRW 4 Summary and Outlook

3 Overview 1 Definitions 2 Limit theorems 3 Application to the CTRW 4 Summary and Outlook

4 Classification of Poisson processes homogeneous inhomogeneous standard (i) (N h λ (t)) (ii) (N(t)) fractional (iii) (Nhf α (t)) (iv) (N α (t))

5 The standard (non-fractional) case (i) The homogeneous Poisson process (HPP) (Nλ h (t)) with intensity parameter λ > 0: p λ x (t) := P(N h λ (λt)x (t) = x) = e λt, x = 0, 1, 2,... x! (ii) The inhomogeneous Poisson process (NHPP) (N(t)) with intensity λ(t) : [0, ) [0, ) and rate function Λ(s, t) = t s λ(u)du. For x = 0, 1, 2,..., the distribution of the increment is p x (t, v) := P{N(t + v) N(v) = x} = e Λ(v,t+v) (Λ(v, t + v)) x. x! Note that N(t) = N h 1 (Λ(t)).

6 The (inverse) α-stable subordinator Let L α = {L α (t), t 0}, be an α-stable subordinator with Laplace transform E [exp( sl α (t))] = exp( ts α ), 0 < α < 1, s 0 and Y α = {Y α (t), t 0}, be an inverse α-stable subordinator defined by Y α (t) = inf{u 0 : L α (u) > t}. Let h α (t, ) denote the density of the distribution of Y α (t). Its Laplace transform can be expressed via the three-parameter Mittag-Leffler function (a.k.a Prabhakar function). E [exp( sy α (t))] = E 1 α,1( st α ), where E c a,b (z) = j=0 c j z j j!γ(aj + b), with c j = c(c + 1)(c + 2)... (c + j 1), a > 0, b > 0, c > 0, z C.

7 hα(1, x) hα(10, x) hα(40, x) α = 0.1 α = 0.6 α = x x x Figure: Plots of the probability densities x h α (t, x) of the distribution of the inverse α-stable subordinator Y α (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.

8 The fractional case (iii) The fractional homogeneous Poisson process (FHPP) (Nα hf (t)) is defined as Nα hf (t) := Nλ h(y α(t)) for t 0, 0 < α < 1. Its marginal distribution is given by px α λu (λu)x (t) = P{N λ (Y α (t)) = x} = e h α (t, u)du 0 x! = (λt α ) x Eα,αx+1 ( λtα ), 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: N(t) = N h 1 (Λ(t)). Analogously define N α (t) := N(Y α (t)) = N h 1 (Λ(Y α(t)))

9 The governing equation for the FNPP We can define the marginals f α x (t, v) := P{N h 1 (Λ(Y α (t) + v)) N h 1 (Λ(v)) = x}, x = 0, 1, 2,... = 0 p x (u, v)h α (t, u)du Theorem (Leonenko et al. (2017)) Let I α (t, v) = N1 h(λ(y α(t) + v)) N1 h (Λ(v)) be the fractional increment process. Then, its marginal distribution satisfies the following fractional differential-integral equations (x = 0, 1,...) D α t f α x (t, v) = 0 λ(u + v)[ p x (u, v) + p x 1 (u, v)]h α (t, u)du, with initial condition fx α (0, v) = δ 0 (x) and f 1 α (0, v) 0.

10 Overview 1 Definitions 2 Limit theorems 3 Application to the CTRW 4 Summary and Outlook

11 Limit theorems for the Poisson process Watanabe (1964): The compensator of Nλ h (t) is λt, i.e. (t) λt is a martingale. (Watanabe characterisation) N h λ One-dimensional central limit theorem Nλ h (t) λt λt d N (0, 1) t Functional central limit theorem: convergence in D([0, )) w.r.t. J 1 -topology to a standard Brownian motion (B(t)) t 0. ( N h ) λ (t) λt λ Functional scaling limit: ( N h ) λ (ct) c t 0 t 0 J 1 B λ J 1 (λt) t 0 c

12 Limit theorems for the Poisson process Watanabe (1964): The compensator of Nλ h (t) is λt, i.e. (t) λt is a martingale. (Watanabe characterisation) N h λ One-dimensional central limit theorem Nλ h (t) λt λt d N (0, 1) t Functional central limit theorem: convergence in D([0, )) w.r.t. J 1 -topology to a standard Brownian motion (B(t)) t 0. ( N h ) λ (t) λt λ Functional scaling limit: ( N h ) λ (ct) c t 0 t 0 J 1 B λ J 1 (λt) t 0 c

13 Random time change and continuous mapping theorem We have convergence in D([0, )) w.r.t. J 1 -topology to a standard Brownian motion (B(t)) t 0. ( N h ) λ (t) λt λ t 0 J 1 λ B. As B has continuous paths and Y α has non-decreasing paths, it follows that ( N h ) λ (Y α (t)) λy α (t) λ (Thm in Whitt (2002)) t 0 J 1 λ [B(Y α(t))] t 0.

14 Limit theorems for the Poisson process Watanabe (1964): The compensator of Nλ h (t) is λt, i.e. (t) λt is a martingale. (Watanabe characterisation) N h λ One-dimensional central limit theorem Nλ h (t) λt λt d N (0, 1) t Functional central limit theorem: convergence in D([0, )) w.r.t. J 1 -topology to a standard Brownian motion (B(t)) t 0. ( N h ) λ (t) λt λ Functional scaling limit: ( N h ) λ (ct) c t 0 t 0 J 1 B λ J 1 (λt) t 0 c

15 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. Brémaud (1981)) Definition Let (Ω, F, P) be a probability space and (N(t)) t 0 be a point process adapted to (F N t ) 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 (F t ) t 0, where F t := F 0 F N t, F 0 = σ(a(t), t 0), (N(t)) t 0 is a Poisson process with intensity da(t).

16 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). Construction of a suitable filtration: N α (t) = N h 1 (Λ(Y α(t))). F Nα t := σ({n α (s), s t}) F 0 := σ(y α (t), t 0) F t := F 0 F Nα t

17 A central limit theorem F Nα t := σ({n α (s), s t}) F 0 := σ(y α (t), t 0) F t := F 0 F Nα t Proposition Let (N(Y α (t))) t 0 be the FNPP adapted to the filtration (F t ) t 0 as defined in previous slide. Then, N(Y α (T )) Λ(Y α (T )) Λ(Yα (T )) d N (0, 1). (1) T Proof: apply Thm I. in Daley and Vere-Jones (2008).

18 ϕ0.1(10 3,x) ϕ0.1(10 9,x) ϕ0.1(10 12,x) x x x Figure: The red line shows the probability density function of the standard normal distribution, the limit distribution according previous proposition. The blue histograms depict samples of size 10 4 of the right hand side of (1) for different times t = 10, 10 9, for α = 0.1 to illustrate convergence to the standard normal distribution.

19 ϕ0.9(1, x) ϕ0.9(10, x) ϕ0.9(20, x) x x x Figure: The red line shows the probability density function of the standard normal distribution, the limit distribution according to previous theorem. The blue histograms depict samples of size 10 4 of the right hand side of (1) for different times t = 1, 10, 20 for α = 0.9 to illustrate convergence to the standard normal distribution.

20 Proposition Limit α 1 Let (N α (t)) t 0 be the FNPP. Then, we have the limit J N 1 α N in D([0, )). α 1 Idea of the proof: According to Theorem VIII.3.36 on p. 479 in Jacod and Shiryaev (2003) it suffices to show Λ(Y α (t)) P α 1 Λ(t), t R + By the continuous mapping theorem we need to show Y α (t) d α 1 t t R +. This can be proven by convergence of the respective Laplace transforms: L{h α (, y)}(s, y) = E α ( ys α ) α 1 e ys = L{δ 0 ( y)}(s, y).

21 Limit theorems for the Poisson process Watanabe (1964): The compensator of Nλ h (t) is λt, i.e. (t) λt is a martingale. (Watanabe characterisation) N h λ One-dimensional central limit theorem Nλ h (t) λt λt d N (0, 1) t Functional central limit theorem: convergence in D([0, )) w.r.t. J 1 -topology to a standard Brownian motion (B(t)) t 0. ( N h ) λ (t) λt λ Functional scaling limit: ( N h ) λ (ct) c t 0 t 0 J 1 B λ J 1 (λt) t 0 c

22 A scaling limit (one-dimensional limit) Assume F 0 = {, Ω}. Theorem Let (N α (t)) t 0 be the FNPP. Suppose the function t Λ(t) is regularly varying with index β > 0, i.e. for x [0, ) Λ(xt) Λ(t) t x β. Then the following limit holds for the FNPP: N α (t) Λ(t α ) d t (Y α(1)) β.

23 A functional scaling limit Assume F 0 = {, Ω}. Theorem Let (N α (t)) t 0 be the FNPP. Suppose the function t Λ(t) is regularly varying with index β > 0, i.e. for x [0, ) Λ(xt) Λ(t) t x β. Then the following limit holds for the FNPP: ( ) Nα (tτ) ( J 1 Λ(t α Y α (τ) β) ). (2) t τ 0 τ 0

24 Proof Using Thm. 2 on p. 81 in Grandell (1976), it suffices to show that ( ) Λ(Yα (tτ)) ( J 1 Λ(t α Y α (τ) β) ) t τ 0 τ 0 1 Convergence of finite-dimensional distributions: By self-similarity of Y α and Lévy s continuity theorem. (Details in the next slides) 2 Tightness: As τ Λ(Y α (tτ)) and τ Y α (τ) are continuous and increasing. Thm VI.3.37(a) in Jacod and Shiryaev (2003) ensures tightness.

25 Proof (II) Let t > 0 be fixed at first, τ = (τ 1, τ 2,..., τ n ) R n + and, denote the scalar product in R n. Then, Λ(t α ( Y α (τ)) Λ(t α ) Y α (τ 1 )) Λ(t α = ) Λ(t α, Λ(tα Y α (τ 2 )) ) Λ(t α,..., Λ(tα Y α (τ n )) ) Λ(t α R n + ) Its characteristic function is given by [ ( ϕ t (u) := E exp i u, Λ(Y )] [ ( )] α(tτ)) Λ(t α = E exp i u, Λ(tα Y α (τ)) ) Λ(t α ) ( ) = exp i u, Λ(tα x) R n Λ(t α h α (τ, x)dx ) + [ n ( Λ(t α ) ] x k ) = exp iu k Λ(t α h α (τ 1,..., τ n ; x 1,..., x n )dx 1... dx n ) R n + k=1

26 Proof (III) We may estimate ) (i exp u, Λ(tα x) Λ(t α h α (τ, x) ) h α(τ, x). By dominated convergence lim ϕ t(u) = lim exp t t R n + = = R n + R n + lim exp t exp ( i u, Λ(tα x) Λ(t α ) ( i u, Λ(tα x) Λ(t α ) ) h α (τ, x)dx ) h α (τ, x)dx ( i u, x β ) h α (τ, x)dx = E[exp(i u, (Y α (τ)) β )].

27 φ0.9(10, x) φ0.9(100, x) φ0.9(10 3,x) x x x Figure: Red line: probability density function φ of the distribution of the random variable (Y 0.9 (1)) 0.7, the limit distribution according to previous Theorem. The blue histogram is based on 10 4 samples of the random variables on the right hand side of (2) for time points t = 10, 100, 10 3 to illustrate the convergence result.

28 Overview 1 Definitions 2 Limit theorems 3 Application to the CTRW 4 Summary and Outlook

29 Proposition (The fractional compound Poisson process) Let (N α (t)) t 0 be the FNPP and suppose the function t Λ(t) is regularly varying with index β R. Moreover let X 1, X 2,... be i.i.d. random variables independent of N α. Assume that the law of X 1 is in the domain of attraction of a stable law, i.e. there exist sequences (a n ) n N and (b n ) n N and a stable Lévy process (S(t)) t 0 such that for nt J S n (t) := a n X k b n it holds that S 1 n S. n k=1 Then the fractional compound Poisson process Z(t) := S Nα(t) = N α(t) k=1 X k has the following limit: where c n = a Λ(n). M (c n Z(nt)) 1 t 0 n ( S ( [Y α (t)] β)) t 0,

30 One-dimensional limit The previous proposition implies for fixed t > 0 N α(nt) c n k=1 X k d n S((Y α(t)) β ) In the one-dimensional case we can do better: We do not need independence between N(t) and X 1, X 2,... (Anscombe (1952)) Additionally, X 1, X 2,... can be mixing (Mogyoródi (1967), Csörgő and Fischler (1973))

31 Overview 1 Definitions 2 Limit theorems 3 Application to the CTRW 4 Summary and Outlook

32 Summary and Outlook We gave a reasonable definition of a fractional non-homogeneous Poisson process that fits into pre-existing theory and results. Other possible definitions of FNPP: N 1 (Y α (Λ(t))) We derived limit theorems for the FNPP Parameter estimation Other related stochastic processes: Skellam processes, integrated processes

33 Thank you for your attention!

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37 Yannaros, N. (1994). Weibull renewal processes. Ann. Inst. Statist. Math. 46(4),