Lamperti transform for multi-type CBI Nicoletta Gabrielli (based on - PowerPoint PPT Presentation

Lamperti transform for multi-type CBI Nicoletta Gabrielli (based on a joint work with J. Teichmann) University of Z¨ urich Department of Banking and Finance nicoletta.gabrielli@bf.uzh.ch October 23, 2014 N. Gabrielli (UZH) October 23, 2014 1 / 38

Structure of the talk Introduction 1 Real valued CB and Lamperti transform Extension to real valued CBI What is a multi–type CBI 2 Definitions Some additional results Lamperti transform for multi–type CBI 3 Multi–type CB and their time–change representation Extension to multi–type CBI Sketch of the proof N. Gabrielli (UZH) October 23, 2014 2 / 38

Outline Introduction 1 Real valued CB and Lamperti transform Extension to real valued CBI What is a multi–type CBI 2 Definitions Some additional results Lamperti transform for multi–type CBI 3 Multi–type CB and their time–change representation Extension to multi–type CBI Sketch of the proof

Branching processes Let (Ω , ( X t ) t ≥ 0 , ( F ♮ t ) t ≥ 0 , ( P x ) x ∈ R ≥ 0 ) be a time homogeneous Markov process. The process X is said to be an branching process if it satisfies the following property: Branching property For any t ≥ 0 and x 1 , x 2 ∈ R ≥ 0 , the law of X t under P x 1 + x 2 is the same , where each X ( i ) has the same distribution as as the law of X ( 1 ) + X ( 2 ) t t X under P x i , for i = 1 , 2. N. Gabrielli (UZH) October 23, 2014 4 / 38

Fourier–Laplace transform characterization There exists a function Ψ : R ≥ 0 × C ≤ 0 → C such that E x � � = e x Ψ( t , u ) , e uX t for all x ∈ R ≥ 0 and ( t , u ) ∈ R ≥ 0 × C ≤ 0 . On the set Q = R ≥ 0 × C ≤ 0 , the function Ψ satisfies the equation ∂ t Ψ( t , u ) = R (Ψ( t , u )) , Ψ( 0 , u ) = u . The branching mechanism The function R has the following L´ evy-Khintchine form � ∞ � � R ( u ) = β u + 1 e u ξ − 1 − u ξ ✶ {| ξ |≤ 1 } 2 u 2 α + M ( d ξ ) , 0 where β ∈ R , α ≥ 0 and M is a L´ evy measure with support in R ≥ 0 . N. Gabrielli (UZH) October 23, 2014 5 / 38

L´ evy processes A time homogeneous Markov process Z is a L´ evy process if the following three conditions are satisfied: L1) Z 0 = 0 P -a.s. L2) Z has independent and stationary increments, i.e. for all n ∈ N and 0 ≤ t 0 < t 1 < . . . < t n + 1 < ∞ (independence) the random variables { Z t j + 1 − Z t j } j = 0 ,..., n are independent, (stationarity) the distribution of Z t j + 1 − Z t j coincides with the distribution of Z ( t j + 1 − t j ) , L3) (stochastic continuity) for each a > 0 and s ≥ 0 , lim t → s P ( | Z t − Z s | > a ) = 0 . N. Gabrielli (UZH) October 23, 2014 6 / 38

Relation with infinitely divisible distributions If Z is a L´ evy process, then, for any t ≥ 0, the random variable Z t is infinitely divisible. The Fourier transform of a L´ evy process takes the form: E 0 � e � u , Z t � � = e t η ( u ) , u ∈ i R � � � η ( u ) = β u + 1 e u ξ − 1 − u ξ ✶ {| ξ |≤ 1 } 2 u 2 α + M ( d ξ ) , where β ∈ R , α ≥ 0 and M is a L´ evy measure in R . The Fourier transform can be extended in the complex domain and the resulting Fourier–Laplace transform is well defined in U := { u ∈ C | η ( R e ( u )) < ∞} . N. Gabrielli (UZH) October 23, 2014 7 / 38

Lamperti transform Theorem [Lamperti, 1967] Let Z be a L´ evy process with no negative jumps with L´ evy exponent R , i.e. E 0 � � e uZ t = e tR ( u ) , u ∈ U . Define, for t ≥ 0 � � � s dr s > 0 | X t = x + Z θ t ∧ τ − θ t := inf > t . Z r 0 0 Then X is a CB process with branching mechanism R . N. Gabrielli (UZH) October 23, 2014 8 / 38

Lamperti transform Theorem 2 in [Caballero et al., 2013] Let Z be a L´ evy process with no negative jumps with L´ evy exponent R , i.e. E 0 � � = e tR ( u ) , e uZ t u ∈ U . The time–change equation X t = x + Z � t 0 X r dr admits a unique solution, which is a CB process with branching mechanism R . N. Gabrielli (UZH) October 23, 2014 9 / 38

Outline Introduction 1 Real valued CB and Lamperti transform Extension to real valued CBI What is a multi–type CBI 2 Definitions Some additional results Lamperti transform for multi–type CBI 3 Multi–type CB and their time–change representation Extension to multi–type CBI Sketch of the proof

Add immigration A CBI-process with branching mechanism R and immigration mechanism F is a Markov process Z taking values in R ≥ 0 satisfying there exist functions φ : R ≥ 0 × U → C and Ψ : R ≥ 0 × U → C such that E x � � e uX t = e φ ( t , u )+ x Ψ( t , u ) , for all x ∈ R ≥ 0 and ( t , u ) ∈ R ≥ 0 × U . On the set Q = R ≥ 0 × U , the functions φ and Ψ satisfy the following system : ∂ t φ ( t , u ) = F (Ψ( t , u )) , φ ( 0 , u ) = 0 , ∂ t Ψ( t , u ) = R (Ψ( t , u )) , Ψ( 0 , u ) = u . N. Gabrielli (UZH) October 23, 2014 11 / 38

Add immigration The immigration mechanism The function F has the following L´ evy-Khintchine form � ∞ � � e u ξ − 1 F ( u ) = bu + m ( d ξ ) , 0 with b ≥ 0 and m is a L´ evy measure on R ≥ 0 such that � ( 1 ∧ ξ ) m ( d ξ ) < ∞ . N. Gabrielli (UZH) October 23, 2014 12 / 38

Lamperti transform for CBI Theorem 2 in [Caballero et al., 2013] Let Z ( 1 ) be a L´ evy process with no negative jumps and Z ( 0 ) an independent subordinator such that E 0 � � = e tR ( u ) and E 0 � � e uZ ( 1 ) e uZ ( 0 ) = e tF ( u ) , u ∈ U . t t The time–change equation X t = x + Z ( 0 ) + Z ( 1 ) � t t 0 X r dr admits a unique solution, which is a CBI process with branching mechanism R and immigration mechanism F . N. Gabrielli (UZH) October 23, 2014 13 / 38

Outline Introduction 1 Real valued CB and Lamperti transform Extension to real valued CBI What is a multi–type CBI 2 Definitions Some additional results Lamperti transform for multi–type CBI 3 Multi–type CB and their time–change representation Extension to multi–type CBI Sketch of the proof

Definition Let (Ω , ( X t ) t ≥ 0 , ( F ♮ t ) t ≥ 0 , ( P x ) x ∈ R m ≥ 0 ) be a time homogeneous Markov process. The process X is said to be a multi–type CBI if it satisfies the following property: See [Duffie et al., 2003, Barczy et al., 2014] There exist functions φ : R ≥ 0 × U → C and Ψ : R ≥ 0 × U → C m such that E x � e � u , X t � � = e φ ( t , u )+ � x , Ψ( t , u ) � , for all x ∈ R m ≥ 0 and ( t , u ) ∈ R ≥ 0 × U , with U = C m ≤ 0 . N. Gabrielli (UZH) October 23, 2014 15 / 38

Generalized Riccati equations On the set Q = R ≥ 0 × U , the functions φ and Ψ satisfy the following system of generalized Riccati equations: ∂ t φ ( t , u ) = F (Ψ( t , u )) , φ ( 0 , u ) = 0 , ∂ t Ψ( t , u ) = R (Ψ( t , u )) , Ψ( 0 , u ) = u . L´ evy–Khintchine form for the vector fields The functions F and R k , for each k = 1 , . . . , m , have the following L´ evy-Khintchine form � � � e � u ,ξ � − 1 F ( u ) = � b , u � + m ( d ξ ) , R m ≥ 0 \{ 0 } � � � R k ( u ) = � β k , u � + 1 e � u ,ξ � − 1 − u k ξ k ✶ {| ξ |≤ 1 } 2 u 2 k α k + M k ( d ξ ) . R m ≥ 0 \{ 0 } N. Gabrielli (UZH) October 23, 2014 16 / 38

Admissible parameters The set of parameters satisfies the following restrictions b , β i ∈ R m , i = 1 , . . . , m , α i ≥ 0, m , M i , i = 1 , . . . , m , L´ evy measures. drift b ∈ R m ≥ 0 , ( β i ) k ≥ 0 , for all i = 1 , . . . , m and k � = i , jumps � supp m ⊆ R m ≥ 0 , and ( | ξ | ∧ 1 ) m ( d ξ ) < ∞ , supp M i ⊆ R m ≥ 0 , for all i = 1 , . . . , m and � � � ( | ( ξ ) − i | + | ξ i | 2 ) ∧ 1 M i ( d ξ ) < ∞ . N. Gabrielli (UZH) October 23, 2014 17 / 38

Outline Introduction 1 Real valued CB and Lamperti transform Extension to real valued CBI What is a multi–type CBI 2 Definitions Some additional results Lamperti transform for multi–type CBI 3 Multi–type CB and their time–change representation Extension to multi–type CBI Sketch of the proof

Remarks N. Gabrielli (UZH) October 23, 2014 19 / 38

Remarks It is possible to define m + 1 independent L´ evy processes Z ( 0 ) , Z ( 1 ) , . . . , Z ( m ) taking values in R m with L´ evy exponents F , R 1 , . . . , R m . N. Gabrielli (UZH) October 23, 2014 19 / 38

Remarks It is possible to define m + 1 independent L´ evy processes Z ( 0 ) , Z ( 1 ) , . . . , Z ( m ) taking values in R m with L´ evy exponents F , R 1 , . . . , R m . In [Kallsen, 2006] it has been proved that the time change equation m � X t = x + Z ( 0 ) Z ( k ) + t ≥ 0 , (*) ds , � t t 0 X ( k ) s k = 1 admits a weak solution, i.e. there exists a probability space containing two processes ( X , Z ) such that (*) holds in distribution. N. Gabrielli (UZH) October 23, 2014 19 / 38

Remarks It is possible to define m + 1 independent L´ evy processes Z ( 0 ) , Z ( 1 ) , . . . , Z ( m ) taking values in R m with L´ evy exponents F , R 1 , . . . , R m . In [Kallsen, 2006] it has been proved that the time change equation m � X t = x + Z ( 0 ) Z ( k ) + t ≥ 0 , (*) ds , � t t 0 X ( k ) s k = 1 admits a weak solution, i.e. there exists a probability space containing two processes ( X , Z ) such that (*) holds in distribution. Moreover X has the distribution of a multi–type CBI with immigration mechanism F and branching mechanism ( R 1 , . . . , R m ) . N. Gabrielli (UZH) October 23, 2014 19 / 38