Extremes of supOU processes Vicky Fasen August 16, 2005 - PowerPoint PPT Presentation

Extremes of supOU processes Vicky Fasen August 16, 2005 fasen@ma.tum.de Graduate Program ”Applied Algorithmic Mathematics” Munich University of Technology http://www-m4.ma.tum.de/pers/fasen/ Vicky Fasen – p. 1/22

Overview • Introduction ◮ i. d. i. s. r.m ◮ SupOU process ◮ Class of convolution equivalent tails ◮ Model assumptions of this talk • Extremal behavior • Conclusion Vicky Fasen – p. 2/22

i. d. i. s. r. m Definition A stochastic process Λ = { Λ( A ) : A ∈ B ( R + × R ) } is called an i. d. i. s. r. m. (infinitely divisible independently scattered random measure) on R + × R , if for disjoint sets ( A n ) n ∈ N in B ( R + × R ) , � ∞ � ∞ � � (r. m.) • Λ A n = Λ( A n ) a. s. n =1 n =1 (i. s.) • (Λ( A n )) n ∈ N is an independent sequence • Λ( A ) is infinitely divisible for every A ∈ B ( R + × R ) (i. d.) Vicky Fasen – p. 3/22

i. d. i. s. r. m We consider only i. d. i. s. r. m. with characteristic function E [exp ( iu Λ( A ))] = exp(Π( A ) ψ ( u )) for u ∈ R , A ∈ B ( R + × R + ) , where • ψ is the cumulant generating function of a Lévy process with generating triplet ( m, σ 2 , ν ) E [exp ( iuL ( t ))] = exp ( tψ ( u )) • Π( dω ) = π ( dr ) × λ ( dt ) for ω = ( r, t ) ∈ R + × R , where λ is the Lebesgue measure and π is a probability measure on R + ( m, σ 2 , ν, π ) are called the generating quadruple of Λ Vicky Fasen – p. 4/22

i. d. i. s. r. m We consider only i. d. i. s. r. m. with characteristic function E [exp ( iu Λ( A ))] = exp(Π( A ) ψ ( u )) for u ∈ R , A ∈ B ( R + × R + ) , where • ψ is the cumulant generating function of a Lévy process with generating triplet ( m, σ 2 , ν ) E [exp ( iuL ( t ))] = exp ( tψ ( u )) • Π( dω ) = π ( dr ) × λ ( dt ) for ω = ( r, t ) ∈ R + × R , where λ is the Lebesgue measure and π is a probability measure on R + ( m, σ 2 , ν, π ) are called the generating quadruple of Λ Λ is called Lévy random field Vicky Fasen – p. 4/22

Compound Poisson random measure Let ν be finite. Then ∞ � N = ε ( R k , Γ k ,Z k ) k =1 where • ( R k ) i. i. d. with d. f. π • (Γ k ) jump times of a Poisson process with intensity µ = ν ( R ) • ( Z k ) i. i. d. with d. f. ν/µ is a Poisson random measure with intensity π ( dr ) × dt × ν ( dx ) ∞ � � Λ( A ) = x dN ( A, x ) = Z k 1 { ( R k , Γ k ) ∈ A } R k =1 is a compound Poisson random measure Vicky Fasen – p. 5/22

Compound Poisson random measure Let ν be finite. Then ∞ � N = ε ( R k , Γ k ,Z k ) k =1 where • ( R k ) i. i. d. with d. f. π • (Γ k ) jump times of a Poisson process with intensity µ = ν ( R ) • ( Z k ) i. i. d. with d. f. ν/µ is a Poisson random measure with intensity π ( dr ) × dt × ν ( dx ) ∞ � � Λ( A ) = x dN ( A, x ) = Z k 1 { ( R k , Γ k ) ∈ A } R k =1 is a Lévy jump field Vicky Fasen – p. 5/22

Compound Poisson random measure Let ν be finite. Then ∞ � N = ε ( R k , Γ k ,Z k ) k =1 where • ( R k ) i. i. d. with d. f. π • (Γ k ) jump times of a Poisson process with intensity µ = ν ( R ) • ( Z k ) i. i. d. with d. f. ν/µ is a Poisson random measure with intensity π ( dr ) × dt × ν ( dx ) N ( t ) ∞ � � Λ( R + × [0 , t ]) = Z k 1 { ( R k , Γ k ) ∈ R + × [0 ,t ] } = Z k k =1 k =1 Vicky Fasen – p. 5/22

Underlying Lévy process Let Λ be an i. d. i. s. r. m. We denote by L = ( L ( t )) t ∈ R the underlying driving Lévy process with L ( t ) = Λ( R + × [0 , t ]) L has the characteristic triplet ( m, σ 2 , ν ) Vicky Fasen – p. 6/22

supOU process Definition The supOU process (superposition of Ornstein-Uhlenbeck processes) Y is defined by � e − r ( t − s ) 1 [0 , ∞ ) ( t − s ) d Λ( r, s ) Y ( t ) = R + × R where � log | x | ν ( dx ) < ∞ • | x |≥ 2 � • λ − 1 = r − 1 π ( dr ) < ∞ R + Vicky Fasen – p. 7/22

Special cases • Λ a compound Poisson random measure ( Λ( A ) = � ∞ k =1 Z k 1 { ( R k , Γ k ) ∈ A } ): N ( t ) � e − r ( t − s ) 1 [0 , ∞ ) ( t − s ) d Λ( r, s ) = � e − R k ( t − Γ k ) Z k Y ( t ) = R + × R k = −∞ � t e − λ ( t − s ) dL ( s ) • OU-process : π ( λ ) = 1 : Y ( t ) = −∞ • π discrete with π ( λ k ) = p k and � ∞ k =1 p k = 1 . Then � t ∞ e − λ k ( t − s ) dL k ( s ) � Y ( t ) = −∞ k =1 where ( L k ) are independent Lévy processes with characteristic triplet ( p k m, p k σ 2 , p k ν ) Vicky Fasen – p. 8/22

Properties of a supOU process • ( m, σ 2 , ν ) determines the marginal distribution : Y = σ 2 1 y � � � σ 2 m Y = m + | y | ν ( dy ) λ 2 λ | y | > 1 � ∞ 1 ν [ y, ∞ ) ν Y [ x, ∞ ) = dy λ y x • π determines the correlation function ρ : � ∞ r − 1 e − hr π ( dr ) ρ ( h ) = λ 0 e. g. π ( dr ) = Γ(2 H + 1) − 1 r 2 H e − r dr for r > 0 , H > 0 , then ρ ( h ) = ( h + 1) − 2 H for h ≥ 0 Vicky Fasen – p. 9/22

Properties of a supOU process • ( m, σ 2 , ν ) determines the marginal distribution : Y = σ 2 1 y � � � σ 2 m Y = m + | y | ν ( dy ) λ 2 λ | y | > 1 � ∞ 1 ν [ y, ∞ ) λ − 1 = R + r − 1 π ( dr ) � ν Y [ x, ∞ ) = dy λ y x • π determines the correlation function ρ : � ∞ r − 1 e − hr π ( dr ) ρ ( h ) = λ 0 e. g. π ( dr ) = Γ(2 H + 1) − 1 r 2 H e − r dr for r > 0 , H > 0 , then ρ ( h ) = ( h + 1) − 2 H for h ≥ 0 Vicky Fasen – p. 9/22

Properties of a supOU process • ( m, σ 2 , ν ) determines the marginal distribution : Y = σ 2 1 y � � � σ 2 m Y = m + | y | ν ( dy ) λ 2 λ | y | > 1 � ∞ 1 ν [ y, ∞ ) ν Y [ x, ∞ ) = dy λ y x • π determines the correlation function ρ : � ∞ r − 1 e − hr π ( dr ) ρ ( h ) = λ 0 e. g. π ( dr ) = Γ(2 H + 1) − 1 r 2 H e − r dr for r > 0 , H > 0 , then ρ ( h ) = ( h + 1) − 2 H for h ≥ 0 Reference: Barndorff-Nielsen (2001), Barndorff-Nielsen and Shephard (2001) Vicky Fasen – p. 9/22

Examples supOU process 40 30 20 10 0 200 400 600 800 1000 1200 1400 1600 1800 2000 OU process 40 30 20 10 0 200 400 600 800 1000 1200 1400 1600 1800 2000 Vicky Fasen – p. 10/22

Class of convolution equivalent tails S ( γ ) Let F be a d. f. on R with F ( x ) < 1 for every x ∈ R . F belongs to the class S ( γ ) , γ ≥ 0 , if (i) F belongs to the class L ( γ ) , γ ≥ 0 , i. e. for all y ∈ R locally uniformly x →∞ F ( x + y ) /F ( x ) = e − γy lim � e γx dF ( x ) < ∞ x →∞ F 2 ∗ ( x ) /F ( x ) = 2 (ii) lim R The class γ = 0 is called subexponential d. f. s denoted by S Examples: • γ = 0 : stable-, Weibull-, loggamma-, Pareto distribution • γ > 0 : generalized inverse Gaussian distribution Vicky Fasen – p. 11/22

Properties of S ( γ ) Let F be infinitely divisible with Lévy measure ν and γ ≥ 0 . Then ν (1 , · ] F ∈ S ( γ ) ⇔ ν (1 , ∞ ) ∈ S ( γ ) F ( x ) � e γx F ( dx ) ⇔ lim ν ( x, ∞ ) = x →∞ R Vicky Fasen – p. 12/22

Model In this talk we restrict our attention to a supOU process driven by a positive compound Poisson random measure , i. e. Z k is positive and ν ( R ) < ∞ . N ( t ) � e − R k ( t − Γ k ) Z k Y ( t ) = k = −∞ a) L (1) ∈ S ( γ ) ∩ MDA(Λ) : n →∞ n P ( L (1) > a n x + b n ) = e − x lim b) L (1) ∈ S ( γ ) ∩ MDA(Φ α ) = R α : n →∞ n P ( L (1) > a n x ) = x − α lim Vicky Fasen – p. 13/22

Overview • Introduction • Extremal behavior ◮ Tail behavior of Y ◮ Tail behavior of M ( h ) ◮ Point process behavior ◮ Running maxima • Conclusion Vicky Fasen – p. 14/22

Representation If L (1) ∈ L ( γ ) then � x 1 � � P ( L (1) > x ) = c ( x ) exp − a ( y ) dy , x > 0 , 0 where a, c : R + → R + x →∞ c ( x ) = c > 0 lim a is absolutely continuous x →∞ a ( x ) = 1 lim γ x →∞ a ′ ( x ) = 0 lim Vicky Fasen – p. 15/22

Tail behavior of Y ( t ) x →∞ a ( x ) /x = 0 lim x →∞ a ( x ) = 1 /γ lim a) L (1) ∈ S ( γ ) ∩ MDA(Λ) : E e γY ( t ) P ( Y ( t ) > x ) ∼ 1 a ( x ) E e γL (1) P ( L (1) > x ) for x → ∞ λ x b) L (1) ∈ S ( γ ) ∩ MDA(Φ α ) : 1 P ( Y ( t ) > x ) ∼ λα P ( L (1) > x ) for x → ∞ Vicky Fasen – p. 16/22

Tail behavior of M ( h ) Let h > 0 and M ( h ) = sup 0 ≤ t ≤ h Y ( t ) a) L (1) ∈ S ( γ ) ∩ MDA(Λ) : P ( M ( h ) > x ) ∼ h E e γY ( t ) E e γL (1) P ( L (1) > x ) for x → ∞ b) L (1) ∈ S ( γ ) ∩ MDA(Φ α ) : P ( M ( h ) > x ) ∼ [ h + ( λα ) − 1 ] P ( L (1) > x ) for x → ∞ Vicky Fasen – p. 17/22

Example: OU-Weibull Process L ( t ) = � N ( t ) Y ( t ) = � N ( t ) k = −∞ e − λ ( t − Γ k ) Z k k =1 Z k , Path of a OU−Weibull−process, p=0.5 30 25 Y( Γ k )=Z k +Y( Γ k ) −Z K 20 15 10 5 0 20 40 60 80 100 120 140 160 180 200 Vicky Fasen – p. 18/22

Example: OU-Weibull Process a − 1 Y (Γ k ) > a T x + b T T ( Y (Γ k ) − b T ) > x ⇐ ⇒ Path of a OU−Weibull−process, p=0.5 30 25 Y( Γ k )=Z k +Y( Γ k ) −Z K 20 u T (x) =a T x+b T 15 10 5 0 20 40 60 80 100 120 140 160 180 200 tT sT Vicky Fasen – p. 18/22