Refracted Lvy processes Ronnie Loeffen 1 Johann Radon Institute for - PowerPoint PPT Presentation

Lévy processes and scale functions What is a refracted Lévy process and does it exist? Future work Refracted Lévy processes Ronnie Loeffen 1 Johann Radon Institute for Computational and Applied Mathematics Austrian Academy of Sciences December 4, 2008 1 based on joint work with Andreas Kyprianou, University of Bath

Lévy processes and scale functions What is a refracted Lévy process and does it exist? Future work Outline Lévy processes and scale functions What is a refracted Lévy process and does it exist? Future work

Lévy processes and scale functions What is a refracted Lévy process and does it exist? Future work Spectrally negative Lévy processes • We let X on (Ω , {F t : t ≥ 0 } , P ) be a Lévy process. We use P x ( · ) = P ( ·| X 0 = x ) . • In this talk we consider only the case when X is spectrally negative, i.e. X has only jumps downwards and X does not have monotone increasing or monotone decreasing paths. • The Laplace exponent of X , ψ ( λ ) := log E ( e λ X 1 ) is well defined for λ ≥ 0 and is given by � ψ ( λ ) = γλ + 1 2 σ 2 λ 2 − ( 1 − e − λ x − λ x 1 { x ≤ 1 } ) ν ( d x ) . ( 0 , ∞ ) Here ( γ, σ, ν ) is the Lévy triplet of X , where γ ∈ R , σ ≥ 0 and ν is a measure on ( 0 , ∞ ) satisfying � ( 1 ∧ x 2 ) ν ( d x ) < ∞ . ( 0 , ∞ )

Lévy processes and scale functions What is a refracted Lévy process and does it exist? Future work Spectrally negative Lévy processes • We let X on (Ω , {F t : t ≥ 0 } , P ) be a Lévy process. We use P x ( · ) = P ( ·| X 0 = x ) . • In this talk we consider only the case when X is spectrally negative, i.e. X has only jumps downwards and X does not have monotone increasing or monotone decreasing paths. • The Laplace exponent of X , ψ ( λ ) := log E ( e λ X 1 ) is well defined for λ ≥ 0 and is given by � ψ ( λ ) = γλ + 1 2 σ 2 λ 2 − ( 1 − e − λ x − λ x 1 { x ≤ 1 } ) ν ( d x ) . ( 0 , ∞ ) Here ( γ, σ, ν ) is the Lévy triplet of X , where γ ∈ R , σ ≥ 0 and ν is a measure on ( 0 , ∞ ) satisfying � ( 1 ∧ x 2 ) ν ( d x ) < ∞ . ( 0 , ∞ )

Lévy processes and scale functions What is a refracted Lévy process and does it exist? Future work Spectrally negative Lévy processes • We let X on (Ω , {F t : t ≥ 0 } , P ) be a Lévy process. We use P x ( · ) = P ( ·| X 0 = x ) . • In this talk we consider only the case when X is spectrally negative, i.e. X has only jumps downwards and X does not have monotone increasing or monotone decreasing paths. • The Laplace exponent of X , ψ ( λ ) := log E ( e λ X 1 ) is well defined for λ ≥ 0 and is given by � ψ ( λ ) = γλ + 1 2 σ 2 λ 2 − ( 1 − e − λ x − λ x 1 { x ≤ 1 } ) ν ( d x ) . ( 0 , ∞ ) Here ( γ, σ, ν ) is the Lévy triplet of X , where γ ∈ R , σ ≥ 0 and ν is a measure on ( 0 , ∞ ) satisfying � ( 1 ∧ x 2 ) ν ( d x ) < ∞ . ( 0 , ∞ )

Lévy processes and scale functions What is a refracted Lévy process and does it exist? Future work Spectrally negative Lévy processes (continued) • Lévy-Itô decomposition: � γλ + 1 ( 1 − e − λ x ) ν ( d x ) 2 σ 2 λ 2 ψ ( λ ) = − ( 1 , ∞ ) � �� � � �� � Brownian motion plus drift compound Poisson process � ( 1 − e − λ x − λ x ) ν ( d x ) − . ( 0 , 1 ] � �� � limit of compensated cpp • The process X has sample paths of unbounded variation if � ( 0 , 1 ] x ν ( d x ) = ∞ . and only if σ > 0 or • If X is of bounded variation, then we can write � X t = ct + ∆ X s , 0 < s ≤ t � where c = γ + ( 0 , 1 ] x ν ( d x ) > 0.

Lévy processes and scale functions What is a refracted Lévy process and does it exist? Future work Spectrally negative Lévy processes (continued) • Lévy-Itô decomposition: � γλ + 1 ( 1 − e − λ x ) ν ( d x ) 2 σ 2 λ 2 ψ ( λ ) = − ( 1 , ∞ ) � �� � � �� � Brownian motion plus drift compound Poisson process � ( 1 − e − λ x − λ x ) ν ( d x ) − . ( 0 , 1 ] � �� � limit of compensated cpp • The process X has sample paths of unbounded variation if � ( 0 , 1 ] x ν ( d x ) = ∞ . and only if σ > 0 or • If X is of bounded variation, then we can write � X t = ct + ∆ X s , 0 < s ≤ t � where c = γ + ( 0 , 1 ] x ν ( d x ) > 0.

Lévy processes and scale functions What is a refracted Lévy process and does it exist? Future work Spectrally negative Lévy processes (continued) • Lévy-Itô decomposition: � γλ + 1 ( 1 − e − λ x ) ν ( d x ) 2 σ 2 λ 2 ψ ( λ ) = − ( 1 , ∞ ) � �� � � �� � Brownian motion plus drift compound Poisson process � ( 1 − e − λ x − λ x ) ν ( d x ) − . ( 0 , 1 ] � �� � limit of compensated cpp • The process X has sample paths of unbounded variation if � ( 0 , 1 ] x ν ( d x ) = ∞ . and only if σ > 0 or • If X is of bounded variation, then we can write � X t = ct + ∆ X s , 0 < s ≤ t � where c = γ + ( 0 , 1 ] x ν ( d x ) > 0.

Lévy processes and scale functions What is a refracted Lévy process and does it exist? Future work Scale functions • For each q ≥ 0 there exists a function W ( q ) : R → [ 0 , ∞ ) , called the ( q -)scale function of X , which satisfies W ( q ) ( x ) = 0 for x < 0 and is characterized on [ 0 , ∞ ) as a strictly increasing and continuous function whose Laplace transform is given by � ∞ 1 e − λ x W ( q ) ( x ) d x = for λ > Φ( q ) , ψ ( λ ) − q 0 where Φ( q ) = sup { λ ≥ 0 : ψ ( λ ) = q } is the right-inverse of ψ . ψ q 0 0 Φ (0) Φ (q) λ

Lévy processes and scale functions What is a refracted Lévy process and does it exist? Future work Scale functions (continued) • Scale functions appear in various fluctuation identities for spectrally negative Lévy processes, e.g. the two-sided exit problem: a = inf { t > 0 : X t > a } , τ − 0 = inf { t > 0 : X t < 0 } and Let τ + x ∈ [ 0 , a ] . Then = W ( q ) ( x ) � � a 1 { τ + e − q τ + W ( q ) ( a ) . E x a <τ − 0 } • Providing the scale function is smooth enough, we have Γ W ( q ) ( x ) = qW ( q ) ( x ) , where Γ is the infinitesimal generator of the process X . • Drawback: There is a limited number of examples where the scale function can be given in closed-form, see however Hubalek & Kyprianou (2007).

Lévy processes and scale functions What is a refracted Lévy process and does it exist? Future work Scale functions (continued) • Scale functions appear in various fluctuation identities for spectrally negative Lévy processes, e.g. the two-sided exit problem: a = inf { t > 0 : X t > a } , τ − 0 = inf { t > 0 : X t < 0 } and Let τ + x ∈ [ 0 , a ] . Then = W ( q ) ( x ) � � a 1 { τ + e − q τ + W ( q ) ( a ) . E x a <τ − 0 } • Providing the scale function is smooth enough, we have Γ W ( q ) ( x ) = qW ( q ) ( x ) , where Γ is the infinitesimal generator of the process X . • Drawback: There is a limited number of examples where the scale function can be given in closed-form, see however Hubalek & Kyprianou (2007).

Lévy processes and scale functions What is a refracted Lévy process and does it exist? Future work Scale functions (continued) • Scale functions appear in various fluctuation identities for spectrally negative Lévy processes, e.g. the two-sided exit problem: a = inf { t > 0 : X t > a } , τ − 0 = inf { t > 0 : X t < 0 } and Let τ + x ∈ [ 0 , a ] . Then = W ( q ) ( x ) � � a 1 { τ + e − q τ + W ( q ) ( a ) . E x a <τ − 0 } • Providing the scale function is smooth enough, we have Γ W ( q ) ( x ) = qW ( q ) ( x ) , where Γ is the infinitesimal generator of the process X . • Drawback: There is a limited number of examples where the scale function can be given in closed-form, see however Hubalek & Kyprianou (2007).

Lévy processes and scale functions What is a refracted Lévy process and does it exist? Future work What are scale functions used for these days? • They serve as building blocks for obtaining more sophisticated identities of (modified) spectrally negative Lévy processes. • They have been used as key tools for, amongst others, solving optimal stopping/control problems and deriving smooth pasting principles involving spectrally negative Lévy processes.

Lévy processes and scale functions What is a refracted Lévy process and does it exist? Future work What are scale functions used for these days? • They serve as building blocks for obtaining more sophisticated identities of (modified) spectrally negative Lévy processes. • They have been used as key tools for, amongst others, solving optimal stopping/control problems and deriving smooth pasting principles involving spectrally negative Lévy processes.

Lévy processes and scale functions What is a refracted Lévy process and does it exist? Future work Defining/constructing a refracted Lévy process • Let X be a spectrally negative Lévy process of bounded variation. Recall that we can write � X t = ct + ∆ X s , c>0 . 0 < s ≤ t • If we let X model the capital reserves of an insurance company, then c represents the premium rate and the jumps of X represent the claims. • We now want to modify the process X . Let 0 < δ < c and define Y by Y t = X t − δ t . Note that Y is still a spectrally negative Lévy process. • Let b be the threshold level. • The refracted Lévy process U is then constructed as follows.