A Fractional Non-Homogeneous Poisson Process

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, ∞) −→ [0, ∞) 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) := {N(t + v) − N(v) = x} = . 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 = {Lα(t), t ≥ 0}, be an α-stable subordinator with Laplace transform α E [exp(−sLα(t))] = exp(−ts ), 0 < α < 1, s ≥ 0

and Yα = {Yα(t), t ≥ 0}, be an inverse α-stable subordinator deﬁned by Yα(t) = inf{u ≥ 0 : Lα(u) > t}.

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 ∞ 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 ∈ 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 ∞ x α −λu (λu) px (t) = P{Nλ(Yα(t)) = x} = 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) := P{N1 (Λ(Yα(t) + v)) − N1 (Λ(v)) = x}, x = 0, 1, 2,... Z ∞ = 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 ∞ 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 √ −−−→d N (0, 1) λt t→∞ Functional central limit theorem: convergence in D([0, ∞)) w.r.t. J1-topology to a standard Brownian motion (B(t))t≥0. h N (t) − λt J λ √ −−−→1 B λ t≥0 λ→∞ Functional scaling limit: h Nλ(ct) J1 −−−→ (λt)t≥0 c→∞ 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, ∞)) w.r.t. J1-topology to a standard Brownian motion (B(t))t≥0.

h N (t) − λt J λ √ −−−→1 B. λ t≥0 λ→∞

As B has continuous paths and Yα has non-decreasing paths, it follows that h Nλ(Yα(t)) − λYα(t) J1 √ −−−→ [B(Yα(t))]t≥0 . λ t≥0 λ→∞

(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

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 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({Nα(s), s ≤ t}) 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({Nα(s), s ≤ t}) 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. Then,

N(Yα(T )) − Λ(Yα(T )) d p −−−−→ N (0, 1). (1) Λ(Yα(T )) T →∞ Proof: apply Thm. 14.5.I. in Daley and Vere-Jones (2008).

A New Fractional Process: A Fractional Non-homogeneous Poisson Process Enrico Scalas Deﬁnitions Limit theorems Application to the CTRW Summary and Outlook References

0.6 0.45 0.4

0.4 0.35 0.5 0.35 0.3 0.4 0.3

) 0.25 ) ) , x , x , x 0.25 3 9 0.3 12 0.2 (10 (10 (10 1 1 1 . . 0.2 . 0 0 0 ϕ ϕ ϕ 0.15 0.2 0.15 0.1 0.1 0.1 0.05 0.05

0 0 0 -10 0 10 20 -5 0 5 10 -5 0 5 10 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 104 of the right hand side of (1) for diﬀerent times t = 10, 109, 1012 for α = 0.1 to illustrate convergence to the standard normal distribution.

A New Fractional Process: A Fractional Non-homogeneous Poisson Process Enrico Scalas Deﬁnitions Limit theorems Application to the CTRW Summary and Outlook References

0.6 0.4 0.45

0.35 0.4 0.5 0.35 0.3 0.4 0.3 0.25 ) ) )

, x , x 0.25 , x

(1 0.3 0.2 (10 (20 9 . 9 9 . . 0 0.2 0 0 ϕ ϕ 0.15 ϕ 0.2 0.15 0.1 0.1 0.1 0.05 0.05

0 0 0 -5 0 5 10 -5 0 5 -5 0 5 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 104 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.

A New Fractional Process: A Fractional Non-homogeneous Poisson Process Enrico Scalas Limit α → 1 Proposition

Let (Nα(t))t≥0 be the FNPP. Then, we have the limit

J1 Nα −−−→ 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

P Λ(Yα(t)) −−−→ Λ(t), t ∈ R+ α→1 By the continuous mapping theorem we need to show

d Yα(t) −−−→ t ∀t ∈ R+. α→1 This can be proven by convergence of the respective Laplace transforms:

α α→1 −ys L{hα(·, y)}(s, y) = Eα(−ys ) −−−→ e = L{δ0(· − y)}(s, y). Deﬁnitions Limit theorems Application to the CTRW Summary and Outlook References

A New Fractional Process: A Fractional Non-homogeneous Poisson Process Enrico Scalas Deﬁnitions Limit theorems Application to the CTRW Summary and Outlook References

A scaling limit (one-dimensional limit)

Assume F0 = {∅, Ω}. Theorem

Let (Nα(t))t≥0 be the FNPP. Suppose the function t 7→ Λ(t) is regularly varying with index β > 0, i.e. for x ∈ [0, ∞)

Λ(xt) −−−→ xβ. Λ(t) t→∞

Then the following limit holds for the FNPP:

Nα(t) d β −−−→ (Yα(1)) . Λ(tα) t→∞

A New Fractional Process: A Fractional Non-homogeneous Poisson Process Enrico Scalas Deﬁnitions Limit theorems Application to the CTRW Summary and Outlook References

A functional scaling limit

Assume F0 = {∅, Ω}. Theorem

Let (Nα(t))t≥0 be the FNPP. Suppose the function t 7→ Λ(t) is regularly varying with index β > 0, i.e. for x ∈ [0, ∞)

Λ(xt) −−−→ xβ. Λ(t) t→∞

Then the following limit holds for the FNPP: Nα(tτ) J1 β −−−→ Yα(τ) . (2) α t→∞ Λ(t ) τ≥0 τ≥0

A New Fractional Process: A Fractional Non-homogeneous Poisson Process Enrico Scalas Deﬁnitions Limit theorems Application to the CTRW Summary and Outlook References

Proof

Using Thm. 2 on p. 81 in Grandell (1976), it suﬃces to show that Λ(Yα(tτ)) J1 β −−−→ Yα(τ) α t→∞ Λ(t ) τ≥0 τ≥0

1 Convergence of ﬁnite-dimensional distributions: By self-similarity of Yα and L´evy’scontinuity theorem. (Details in the next slides)

2 Tightness: As τ 7→ Λ(Yα(tτ)) and τ 7→ Yα(τ) are continuous and increasing. Thm VI.3.37(a) in Jacod and Shiryaev (2003) ensures tightness.

A New Fractional Process: A Fractional Non-homogeneous Poisson Process Enrico Scalas Deﬁnitions Limit theorems Application to the CTRW Summary and Outlook References

Proof (II)

n Let t > 0 be ﬁxed at ﬁrst, τ = (τ1, τ2, . . . , τn) ∈ R+ and h·, ·i denote the scalar product in Rn. Then, Λ(tαY (τ)) Λ(tαY (τ )) Λ(tαY (τ )) Λ(tαY (τ )) α = α 1 , α 2 ,..., α n ∈ n Λ(tα) Λ(tα) Λ(tα) Λ(tα) R+

Its characteristic function is given by

Λ(Y (tτ)) Λ(tαY (τ)) ϕ (u) := exp i u, α = exp i u, α t E Λ(tα) E Λ(tα) Z Λ(tαx) = exp i u, α hα(τ, x)dx n Λ(t ) R+ Z " n α # Y Λ(t xk ) = exp iuk α hα(τ1, . . . , τn; x1,..., xn)dx1 ... dxn n Λ(t ) R+ k=1

A New Fractional Process: A Fractional Non-homogeneous Poisson Process Enrico Scalas Deﬁnitions Limit theorems Application to the CTRW Summary and Outlook References

Proof (III)

We may estimate

α Λ(t x) exp i u, hα(τ, x) ≤ hα(τ, x). Λ(tα) By dominated convergence

Z Λ(tαx) lim ϕt (u) = lim exp i u, α hα(τ, x)dx t→∞ t→∞ n Λ(t ) R+ Z Λ(tαx) = lim exp i u, α hα(τ, x)dx n t→∞ Λ(t ) R+ Z D βE β = exp i u, x hα(τ, x)dx = E[exp(ihu, (Yα(τ)) i)]. n R+

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 1.5 1.2

1 1

0.8 ) 1 0.8 ) ) , x , x , x 3 0.6 0.6 (10 (10 (100 9 . 9 9 . . 0 0 0 φ φ 0.4 φ 0.5 0.4

0.2 0.2

0 0 0 0 2 4 0 2 4 0 2 4 x x x

Figure: Red line: probability density function φ of the distribution of the 0.7 random variable (Y0.9(1)) , the limit distribution according to previous Theorem. The blue histogram is based on 104 samples of the random variables on the right hand side of (2) for time points t = 10, 100, 103 to illustrate the convergence result.

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 Proposition (The fractional compound Poisson process)

Let (Nα(t))t≥0 be the FNPP and suppose the function t 7→ Λ(t) is regularly varying with index β ∈ R. Moreover let X1, X2,... be i.i.d. random variables independent of Nα. Assume that the law of X1 is in the domain of attraction of a stable law, i.e. there exist sequences (an)n∈N and (bn)n∈N and a stable L´evyprocess (S(t))t≥0 such that for

bntc X J1 S¯n(t) := an Xk − bn it holds that S¯n −−−→ S. n→∞ k=1 Then the fractional compound Poisson process PNα(t) Z(t) := SNα(t) = k=1 Xk has the following limit:

M1 β (cnZ(nt))t≥0 −−−→ S [Yα(t)] , n→∞ t≥0 where cn = abΛ(n)c. Deﬁnitions Limit theorems Application to the CTRW Summary and Outlook References

One-dimensional limit

The previous proposition implies for ﬁxed t > 0

Nα(nt) X d β cn Xk −−−→ S((Yα(t)) ) n→∞ k=1 In the one-dimensional case we can do better:

We do not need independence between N(t) and X1, X2,... (Anscombe (1952))

Additionally, X1, X2,... can be mixing (Mogyor´odi(1967), Cs¨org˝oand Fischler (1973))

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

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: N1(Yα(Λ(t))) We derived limit theorems for the FNPP ⇒ Parameter estimation Other related stochastic processes: Skellam processes, integrated processes

A New Fractional Process: A Fractional Non-homogeneous Poisson Process Enrico Scalas Deﬁnitions Limit theorems Application to the CTRW Summary and Outlook References

Thank you for your attention!

A New Fractional Process: A Fractional Non-homogeneous Poisson Process Enrico Scalas Deﬁnitions Limit theorems Application to the CTRW Summary and Outlook References

Anscombe, F. J. (1952). Large-sample theory of sequential estimation. Proc. Cambridge Philos. Soc. 48, 600–607. Bielecki, T. R. and M. Rutkowski (2002). Credit risk: modelling, valuation and hedging. Springer Finance. Springer-Verlag, Berlin. Brémaud,P. (1981). Point processes and queues. Springer-Verlag, New York-Berlin. Martingale dynamics, Springer Series in Statistics. Cox, D. R. (1955). Some statistical methods connected with series of events. J. Roy. Statist. Soc. Ser. B. 17, 129–157; discussion, 157–164. Csörg˝o,M. and R. Fischler (1973). Some examples and results in the theory of mixing and random-sum central limit theorems. Period. Math. Hungar. 3, 41–57. Collection of articles dedicated to the memory of AlfrédRényi,II.

A New Fractional Process: A Fractional Non-homogeneous Poisson Process Enrico Scalas Deﬁnitions Limit theorems Application to the CTRW Summary and Outlook References

Daley, D. J. and D. Vere-Jones (2008). An introduction to the theory of point processes. Vol. II (Second ed.). Probability and its Applications (New York). Springer, New York. General theory and structure. Grandell, J. (1976). Doubly stochastic Poisson processes. Lecture Notes in Mathematics, Vol. 529. Springer-Verlag, Berlin-New York. Grandell, J. (1991). Aspects of risk theory. Springer Series in Statistics: Probability and its Applications. Springer-Verlag, New York. Jacod, J. and A. N. Shiryaev (2003). Limit theorems for stochastic processes (Second ed.), Volume 288 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin.

A New Fractional Process: A Fractional Non-homogeneous Poisson Process Enrico Scalas Deﬁnitions Limit theorems Application to the CTRW Summary and Outlook References

Kingman, J. (1964). On doubly stochastic poisson processes. In Mathematical Proceedings of the Cambridge Philosophical Society, Volume 60, pp. 923–930. Cambridge Univ Press. Leonenko, N., E. Scalas, and M. Trinh (2017). The fractional non-homogeneous Poisson process. Statist. Probab. Lett. 120, 147–156. Mogyor´odi,J. (1967). Limit distributions for sequences of random variables with random indices. In Trans. Fourth Prague Conf. on Information Theory, Statistical Decision Functions. Random Processes (Prague, 1965), pp. 463–470. Academia, Prague. Watanabe, S. (1964). On discontinuous additive functionals and L´evymeasures of a Markov process. Japan. J. Math. 34, 53–70. Whitt, W. (2002). Stochastic-process limits. Springer Series in Operations Research. Springer-Verlag, New York. An introduction to stochastic-process limits and their application to queues.

A New Fractional Process: A Fractional Non-homogeneous Poisson Process Enrico Scalas Deﬁnitions Limit theorems Application to the CTRW Summary and Outlook References

Yannaros, N. (1994). Weibull renewal processes. Ann. Inst. Statist. Math. 46(4), 641–648.

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