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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

homogeneous inhomogeneous standard fractional (i) (Nh

λ(t))

(iii) (Nhf

α (t))

(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

(i) The homogeneous Poisson process (HPP) (Nh

λ(t)) with

intensity parameter λ > 0: pλ

x (t) := P(Nh λ(t) = x) = e−λt (λt)x

x! , x = 0, 1, 2, . . . (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 px(t, v) := P{N(t + v) − N(v) = x} = e−Λ(v,t+v)(Λ(v, t + v))x x! . Note that N(t) = Nh

1 (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 E [exp(−sLα(t))] = exp(−tsα), 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). E [exp(−sYα(t))] = E 1

α,1(−stα), where

E c

a,b(z) = ∞

• j=0

cjzj j!Γ(aj + b), with cj = 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

x

5 10

hα(1, x)

0.2 0.4 0.6 0.8 1 1.2

x

10 20 30

hα(10, x)

0.1 0.2 0.3 0.4 0.5 0.6

x

50 100

hα(40, x)

0.1 0.2 0.3 0.4 0.5

α = 0.1 α = 0.6 α = 0.9

Figure: Plots of the probability densities x → hα(t, x) of the distribution

• f 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) (Nhf

α (t)) is defined as Nhf α (t) := Nh λ(Yα(t)) for

t ≥ 0, 0 < α < 1. Its marginal distribution is given by pα

x (t) = P{Nλ(Yα(t)) = x} =

∞ e−λu (λu)x x! hα(t, u)du = (λtα)xE x+1

α,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) = Nh

1 (Λ(t)).

Analogously define Nα(t) := N(Yα(t)) = Nh

1 (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 f α

x (t, v) := P{Nh 1 (Λ(Yα(t) + v)) − Nh 1 (Λ(v)) = x},

x = 0, 1, 2, . . . = ∞ px(u, v)hα(t, u)du

Theorem (Leonenko et al. (2017))

Let Iα(t, v) = Nh

1 (Λ(Yα(t) + v)) − Nh 1 (Λ(v)) be the fractional

increment process. Then, its marginal distribution satisfies the following fractional differential-integral equations (x = 0, 1, . . .) Dα

t f α x (t, v) =

∞ λ(u + v)[−px(u, v) + px−1(u, v)]hα(t, u)du, with initial condition f α

x (0, v) = δ0(x) and f α −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

Watanabe (1964): The compensator of Nh

λ(t) is λt, i.e.

Nh

λ(t) − λt is a martingale. (Watanabe characterisation)

One-dimensional central limit theorem Nh

λ(t) − λt

√ λt

d

− − − →

t→∞ N(0, 1)

Functional central limit theorem: convergence in D([0, ∞)) w.r.t. J1-topology to a standard Brownian motion (B(t))t≥0. Nh

λ(t) − λt

√ λ

• t≥0

J1

− − − →

λ→∞ B

Functional scaling limit: Nh

λ(ct)

c

• t≥0

J1

− − − →

c→∞ (λt)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

Watanabe (1964): The compensator of Nh

λ(t) is λt, i.e.

Nh

λ(t) − λt is a martingale. (Watanabe characterisation)

One-dimensional central limit theorem Nh

λ(t) − λt

√ λt

d

− − − →

t→∞ N(0, 1)

Functional central limit theorem: convergence in D([0, ∞)) w.r.t. J1-topology to a standard Brownian motion (B(t))t≥0. Nh

λ(t) − λt

√ λ

• t≥0

J1

− − − →

λ→∞ B

Functional scaling limit: Nh

λ(ct)

c

• t≥0

J1

− − − →

c→∞ (λt)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. Nh

λ(t) − λt

√ λ

• t≥0

J1

− − − →

λ→∞ B.

As B has continuous paths and Yα has non-decreasing paths, it follows that Nh

λ(Yα(t)) − λYα(t)

√ λ

• t≥0

J1

− − − →

λ→∞ [B(Yα(t))]t≥0 .

(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

Watanabe (1964): The compensator of Nh

λ(t) is λt, i.e.

Nh

λ(t) − λt is a martingale. (Watanabe characterisation)

One-dimensional central limit theorem Nh

λ(t) − λt

√ λt

d

− − − →

t→∞ N(0, 1)

Functional central limit theorem: convergence in D([0, ∞)) w.r.t. J1-topology to a standard Brownian motion (B(t))t≥0. Nh

λ(t) − λt

√ λ

• t≥0

J1

− − − →

λ→∞ B

Functional scaling limit: Nh

λ(ct)

c

• t≥0

J1

− − − →

c→∞ (λt)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 process adapted to (FN

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 (Ft)t≥0, where Ft := F0 ∨ FN

t ,

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). Construction of a suitable filtration: Nα(t) = Nh

1 (Λ(Yα(t))).

FNα

t

:= σ({Nα(s), s ≤ t}) F0 := σ(Yα(t), t ≥ 0) Ft := F0 ∨ FNα

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 central limit theorem

FNα

t

:= σ({Nα(s), s ≤ t}) F0 := σ(Yα(t), t ≥ 0) Ft := F0 ∨ FNα

t

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))

• Λ(Yα(T))

d

− − − − →

T→∞ N(0, 1).

(1) 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

x

• 10

10 20

ϕ0.1(103, x)

0.1 0.2 0.3 0.4 0.5 0.6

x

• 5

5 10

ϕ0.1(109, x)

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45

x

• 5

5 10

ϕ0.1(1012, x)

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

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

x

• 5

5 10

ϕ0.9(1, x)

0.1 0.2 0.3 0.4 0.5 0.6

x

• 5

5

ϕ0.9(10, x)

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

x

• 5

5

ϕ0.9(20, x)

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45

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 Nα

J1

− − − →

α→1 N

in D([0, ∞)). Idea of the proof: According to Theorem VIII.3.36 on p. 479 in Jacod and Shiryaev (2003) it suffices to show Λ(Yα(t))

P

− − − →

α→1 Λ(t),

t ∈ R+ By the continuous mapping theorem we need to show Yα(t)

d

− − − →

α→1 t

∀t ∈ R+. This can be proven by convergence of the respective Laplace transforms: L{hα(·, y)}(s, y) = Eα(−ysα) α→1 − − − → e−ys = L{δ0(· − y)}(s, y).

Definitions Limit theorems Application to the CTRW Summary and Outlook References

Limit theorems for the Poisson process

Watanabe (1964): The compensator of Nh

λ(t) is λt, i.e.

Nh

λ(t) − λt is a martingale. (Watanabe characterisation)

One-dimensional central limit theorem Nh

λ(t) − λt

√ λt

d

− − − →

t→∞ N(0, 1)

Functional central limit theorem: convergence in D([0, ∞)) w.r.t. J1-topology to a standard Brownian motion (B(t))t≥0. Nh

λ(t) − λt

√ λ

• t≥0

J1

− − − →

λ→∞ B

Functional scaling limit: Nh

λ(ct)

c

• t≥0

J1

− − − →

c→∞ (λt)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

A scaling limit (one-dimensional limit)

Assume F0 = {∅, Ω}.

Theorem

Let (Nα(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→∞ xβ.

Then the following limit holds for the FNPP: Nα(t) Λ(tα)

d

− − − →

t→∞ (Yα(1))β.

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 → Λ(t) is regularly varying with index β > 0, i.e. for x ∈ [0, ∞) Λ(xt) Λ(t) − − − →

t→∞ xβ.

Then the following limit holds for the FNPP: Nα(tτ) Λ(tα)

• τ≥0

J1

− − − →

t→∞

• Yα(τ)β

τ≥0 .

(2)

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)) Λ(tα)

• τ≥0

J1

− − − →

t→∞

• Yα(τ)β

τ≥0 1 Convergence of finite-dimensional distributions: By

self-similarity of Yα and L´ evy’s continuity theorem. (Details in the next slides)

2 Tightness: As τ → Λ(Yα(tτ)) and τ → 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)

Let t > 0 be fixed at first, τ = (τ1, τ2, . . . , τn) ∈ Rn

+ and ·, · denote the

scalar product in Rn. Then, Λ(tαYα(τ)) Λ(tα) = Λ(tαYα(τ1)) Λ(tα) , Λ(tαYα(τ2)) Λ(tα) , . . . , Λ(tαYα(τn)) Λ(tα)

• ∈ Rn

+

Its characteristic function is given by ϕt(u) := E

• exp
• i
• u, Λ(Yα(tτ))

Λ(tα)

• = E
• exp
• i
• u, Λ(tαYα(τ))

Λ(tα)

• =
• Rn

+

exp

• i
• u, Λ(tαx)

Λ(tα)

• hα(τ, x)dx

=

• Rn

+

n

• k=1

exp

• iuk

Λ(tαxk) Λ(tα)

• hα(τ1, . . . , τn; x1, . . . , xn)dx1 . . . dxn

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

• exp
• i
• u, Λ(tαx)

Λ(tα)

• hα(τ, x)
• ≤ hα(τ, x).

By dominated convergence lim

t→∞ ϕt(u) = lim t→∞

• Rn

+

exp

• i
• u, Λ(tαx)

Λ(tα)

• hα(τ, x)dx

=

• Rn

+

lim

t→∞ exp

• i
• u, Λ(tαx)

Λ(tα)

• hα(τ, x)dx

=

• Rn

+

exp

• i
• u, xβ

hα(τ, x)dx = E[exp(iu, (Yα(τ))β)].

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

Definitions Limit theorems Application to the CTRW Summary and Outlook References

x

2 4

φ0.9(10, x)

0.2 0.4 0.6 0.8 1 1.2

x

2 4

φ0.9(100, x)

0.5 1 1.5

x

2 4

φ0.9(103, x)

0.2 0.4 0.6 0.8 1 1.2

Figure: Red line: probability density function φ of the distribution of the random variable (Y0.9(1))0.7, 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 → 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´ evy process (S(t))t≥0 such that for ¯ Sn(t) := an

⌊nt⌋

• k=1

Xk − bn it holds that ¯ Sn

J1

− − − →

n→∞ S.

Then the fractional compound Poisson process Z(t) := SNα(t) = Nα(t)

k=1 Xk has the following limit:

(cnZ(nt))t≥0

M1

− − − →

n→∞

• S
• [Yα(t)]β

t≥0 ,

where cn = a⌊Λ(n)⌋.

Definitions Limit theorems Application to the CTRW Summary and Outlook References

One-dimensional limit

The previous proposition implies for fixed t > 0 cn

Nα(nt)

• k=1

Xk

d

− − − →

n→∞ S((Yα(t))β)

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´

• di (1967),

Cs¨

• rg˝
• and 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. Br´ emaud, 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

• f events. J. Roy. Statist. Soc. Ser. B. 17, 129–157; discussion,

157–164. Cs¨

• rg˝
• , 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 Alfr´ ed R´ enyi, 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´

• di, 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´ evy measures 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