A New Fractional Process: A Fractional Non-homogeneous Poisson - PowerPoint PPT Presentation

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 Definitions 1 Limit theorems 2 Application to the CTRW 3 Summary and Outlook 4 A New Fractional Process: A Fractional Non-homogeneous Poisson Process Enrico Scalas

Definitions Limit theorems Application to the CTRW Summary and Outlook References Overview Definitions 1 Limit theorems 2 Application to the CTRW 3 Summary and Outlook 4 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 (i) ( N h (iii) ( N hf homogeneous λ ( t )) α ( t )) inhomogeneous (ii) ( N ( t )) (iv) ( N α ( 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) ( N h λ ( t )) with intensity parameter λ > 0: λ ( t ) = x ) = e − λ t ( λ t ) x p λ x ( t ) := P ( N h x = 0 , 1 , 2 , . . . , x ! (ii) The inhomogeneous Poisson process (NHPP) ( N ( t )) with intensity λ ( t ) : [0 , ∞ ) − → [0 , ∞ ) and rate function � t Λ( s , t ) = λ ( u )d u . s 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 )). 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 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 ∞ c j z j � E c a , b ( z ) = j !Γ( aj + b ) , with j =0 c j = 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 h α (10 , x ) h α (40 , x ) h α (1 , x ) 0.3 0.6 0.3 0.2 0.4 0.2 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 �→ 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. 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) ( N hf α ( t )) is defined as N hf α ( t ) := N h λ ( Y α ( t )) for t ≥ 0 , 0 < α < 1. Its marginal distribution is given by � ∞ e − λ u ( λ u ) x p α x ( t ) = P { N λ ( Y α ( t )) = x } = h α ( t , u )d u x ! 0 = ( λ t α ) x E x +1 α,α x +1 ( − λ t α ) , 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 ))) 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 { N h 1 (Λ( Y α ( t ) + v )) − N h 1 (Λ( v )) = x } , x = 0 , 1 , 2 , . . . � ∞ = p x ( u , v ) h α ( t , u )d u 0 Theorem (Leonenko et al. (2017)) Let I α ( t , v ) = N h 1 (Λ( Y α ( t ) + v )) − N h 1 (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 λ ( u + v )[ − p x ( u , v ) + p x − 1 ( u , v )] h α ( t , u )d u , x ( t , v ) = 0 with initial condition f α x (0 , v ) = δ 0 ( 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 Definitions 1 Limit theorems 2 Application to the CTRW 3 Summary and Outlook 4 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 N h λ ( t ) is λ t , i.e. N h λ ( t ) − λ t is a martingale. (Watanabe characterisation) One-dimensional central limit theorem N h λ ( t ) − λ t d √ − t →∞ N (0 , 1) − − → λ 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 ) − λ t � J 1 √ − λ →∞ B − − → λ t ≥ 0 Functional scaling limit: � N h λ ( ct ) � J 1 − c →∞ ( λ t ) t ≥ 0 − − → 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 Watanabe (1964): The compensator of N h λ ( t ) is λ t , i.e. N h λ ( t ) − λ t is a martingale. (Watanabe characterisation) One-dimensional central limit theorem N h λ ( t ) − λ t d √ − t →∞ N (0 , 1) − − → λ 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 ) − λ t � J 1 √ − − − → λ →∞ B λ t ≥ 0 Functional scaling limit: � N h λ ( ct ) � J 1 − c →∞ ( λ t ) t ≥ 0 − − → 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. J 1 -topology to a standard Brownian motion ( B ( t )) t ≥ 0 . � N h λ ( t ) − λ t � J 1 √ − λ →∞ B . − − → λ t ≥ 0 As B has continuous paths and Y α has non-decreasing paths, it follows that � N h λ ( Y α ( t )) − λ Y α ( t ) � J 1 √ − λ →∞ [ B ( Y α ( t ))] t ≥ 0 . − − → λ 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 N h λ ( t ) is λ t , i.e. N h λ ( t ) − λ t is a martingale. (Watanabe characterisation) One-dimensional central limit theorem N h λ ( t ) − λ t d √ − t →∞ N (0 , 1) − − → λ 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 ) − λ t � J 1 √ − − − → λ →∞ B λ t ≥ 0 Functional scaling limit: � N h λ ( ct ) � J 1 − c →∞ ( λ t ) t ≥ 0 − − → 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 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 d A ( t ). A New Fractional Process: A Fractional Non-homogeneous Poisson Process Enrico Scalas