Lecture 2: The Wiener-Hopf factorisation A. E. Kyprianou Department - PowerPoint PPT Presentation

Lecture 2: The Wiener-Hopf factorisation Lecture 2: The Wiener-Hopf factorisation A. E. Kyprianou Department of Mathematical Sciences, University of Bath 1/ 23

Lecture 2: The Wiener-Hopf factorisation Random walks 2/ 23

Lecture 2: The Wiener-Hopf factorisation Random walks Random walk: Consider the discrete time process S = { S n : n ≥ 0 } where n � S 0 = 0 and S n = ξ i , i =1 with { ξ i : i ≥ 1 } an i.i.d. sequence with common distribution F . 2/ 23

Lecture 2: The Wiener-Hopf factorisation Random walks Random walk: Consider the discrete time process S = { S n : n ≥ 0 } where n � S 0 = 0 and S n = ξ i , i =1 with { ξ i : i ≥ 1 } an i.i.d. sequence with common distribution F . Duality: Feller’s classic Duality Lemma for random walks says that for any n = 0 , 1 , 2 ... the independence and common distribution of increments implies that { S n − k − S n : k = 0 , 1 , ..., n } has the same law as {− S k : k = 0 , 1 , ..., n } . 2/ 23

Lecture 2: The Wiener-Hopf factorisation Random walks Random walk: Consider the discrete time process S = { S n : n ≥ 0 } where n � S 0 = 0 and S n = ξ i , i =1 with { ξ i : i ≥ 1 } an i.i.d. sequence with common distribution F . Duality: Feller’s classic Duality Lemma for random walks says that for any n = 0 , 1 , 2 ... the independence and common distribution of increments implies that { S n − k − S n : k = 0 , 1 , ..., n } has the same law as {− S k : k = 0 , 1 , ..., n } . Infinite divisibiltiy: Let Γ p be a geometrically distributed random variable with parameter p which is independent of the random walk S . The random variable Γ p � S Γ p = ξ i i =1 is infinitely divisible. 2/ 23

Lecture 2: The Wiener-Hopf factorisation Wiener-Hopf for random walks 3/ 23

Lecture 2: The Wiener-Hopf factorisation Wiener-Hopf for random walks Assume min { F (0 , ∞ ) , F ( −∞ , 0) } > 0 , and F has no atoms. 3/ 23

Lecture 2: The Wiener-Hopf factorisation Wiener-Hopf for random walks Assume min { F (0 , ∞ ) , F ( −∞ , 0) } > 0 , and F has no atoms. Fix 0 < p < 1 and define G = inf { k = 0 , 1 , ..., Γ p : S k = j =0 , 1 ,..., Γ p S j } max and D := inf { k = 0 , 1 , ..., Γ p : S k = j =0 , 1 ,..., Γ p S j } . min where Γ p is a geometrically distributed random variable with parameter p which is independent of the random walk S . 3/ 23

Lecture 2: The Wiener-Hopf factorisation Wiener-Hopf for random walks Assume min { F (0 , ∞ ) , F ( −∞ , 0) } > 0 , and F has no atoms. Fix 0 < p < 1 and define G = inf { k = 0 , 1 , ..., Γ p : S k = j =0 , 1 ,..., Γ p S j } max and D := inf { k = 0 , 1 , ..., Γ p : S k = j =0 , 1 ,..., Γ p S j } . min where Γ p is a geometrically distributed random variable with parameter p which is independent of the random walk S . Set. N = inf { n > 0 : S n > 0 } . In words, the first visit of S to (0 , ∞ ) after time 0 . 3/ 23

Lecture 2: The Wiener-Hopf factorisation D G N 4/ 23

Lecture 2: The Wiener-Hopf factorisation The Wiener-Hopf factorisation Theorem: S G is independent of S Γ p − S G and both are infinitely divisible with the latter equal in distribution to S D . S -S G S G D G N 5/ 23

Lecture 2: The Wiener-Hopf factorisation Sketch proof (1) (2) H H + H (1) G (1) (2) N N 6/ 23

Lecture 2: The Wiener-Hopf factorisation Sketch proof D 7/ 23

Lecture 2: The Wiener-Hopf factorisation Sketch proof Let ν be geometrically distributed with parameter P ( N > Γ p ) . We have ν � H ( i ) S G = i =1 where { H ( i ) : i = 1 , 2 , ... } are independent having the same distribution as S N conditioned on { N ≤ Γ p } . The variable S Γ p − S G is equal in distribution to S Γ p conditional on { Γ p < N } . Infinite divisibility follows as a consequence of the fact that S G is a geometric sum of i.i.d. random variables. (This also implies infinite divisibility of S D ). Duality implies that the S Γ p − S G is equal in distribution to S D , and hence, also infinitely divisible. Said another way: Let S Γ p = max n ≤ Γ p S n and S Γ p = min n ≤ Γ p S n , then S Γ p = d S Γ p ⊕ S Γ p . 8/ 23

Lecture 2: The Wiener-Hopf factorisation The Wiener-Hopf factorisation for Lévy process 9/ 23

Lecture 2: The Wiener-Hopf factorisation The Wiener-Hopf factorisation for Lévy process Let X be any Lévy process and e q be an exponentially distributed random variable which is independent of X . Define X e q = sup X s and X e q = inf s ≤ e q X s . s ≤ e q 9/ 23

Lecture 2: The Wiener-Hopf factorisation The Wiener-Hopf factorisation for Lévy process Let X be any Lévy process and e q be an exponentially distributed random variable which is independent of X . Define X e q = sup X s and X e q = inf s ≤ e q X s . s ≤ e q Theorem: The random variables X e q and X e q − X e q are independent and infinitely divisible. Moreover X e q − X e q = d X e q . In particular X e q = d X e q ⊕ X e q 9/ 23

Lecture 2: The Wiener-Hopf factorisation The Wiener-Hopf factorisation for Lévy process Let X be any Lévy process and e q be an exponentially distributed random variable which is independent of X . Define X e q = sup X s and X e q = inf s ≤ e q X s . s ≤ e q Theorem: The random variables X e q and X e q − X e q are independent and infinitely divisible. Moreover X e q − X e q = d X e q . In particular X e q = d X e q ⊕ X e q Analytically speaking: q E (e i θX e q ) = q + Ψ( θ ) = E (e i θX e q ) E (e i θX e q ) =: Ψ + q ( θ )Ψ − q ( θ ) . 9/ 23

Lecture 2: The Wiener-Hopf factorisation The Wiener-Hopf factorisation for Lévy process q q + Ψ( θ ) = Ψ + q ( θ )Ψ − q ( θ ) 10/ 23

Lecture 2: The Wiener-Hopf factorisation The Wiener-Hopf factorisation for Lévy process q q + Ψ( θ ) = Ψ + q ( θ )Ψ − q ( θ ) Dividing through by q and taking limits as q ↓ 0 , it turns out that we have a further factorisation Ψ( θ ) = κ + ( − i θ ) κ − (i θ ) , where κ ± are so-called Bernstein functions and necessarily take the form � κ ± ( λ ) = η ± + δ ± λ + (1 − e − λx ) ν ± (d x ) , (0 , ∞ ) with η ± ≥ 0 such that η + η − = 0 , δ ± ≥ 0 and ν ± satisfy � (1 ∧ x ) ν ± (d x ) < ∞ . (0 , ∞ ) 10/ 23

Lecture 2: The Wiener-Hopf factorisation Wiener-Hopf factorisation, financial and insurance mathematics 11/ 23

Lecture 2: The Wiener-Hopf factorisation Wiener-Hopf for spectrally negative Lévy processes 12/ 23

Lecture 2: The Wiener-Hopf factorisation Wiener-Hopf for spectrally negative Lévy processes Work with Laplace exponent instead of characteristic exponent, θ ≥ 0 E (e θX t ) = e ψ ( θ ) t where ψ ( θ ) = − Ψ( − i θ ) . 12/ 23

Lecture 2: The Wiener-Hopf factorisation Wiener-Hopf for spectrally negative Lévy processes Work with Laplace exponent instead of characteristic exponent, θ ≥ 0 E (e θX t ) = e ψ ( θ ) t where ψ ( θ ) = − Ψ( − i θ ) . Wiener-Hopf factorisation reads ψ ( λ ) = ( λ − ϕ ) κ − ( λ ) , λ ≥ 0 , where ϕ ≥ 0 is the largest root of ψ on [0 , ∞ ) (there are at most two). 12/ 23

Lecture 2: The Wiener-Hopf factorisation Wiener-Hopf for spectrally negative Lévy processes Work with Laplace exponent instead of characteristic exponent, θ ≥ 0 E (e θX t ) = e ψ ( θ ) t where ψ ( θ ) = − Ψ( − i θ ) . Wiener-Hopf factorisation reads ψ ( λ ) = ( λ − ϕ ) κ − ( λ ) , λ ≥ 0 , where ϕ ≥ 0 is the largest root of ψ on [0 , ∞ ) (there are at most two). ϕ = 0 ⇔ lim sup t ↑∞ X t = ∞ (i.e. X does not drift to −∞ ). 12/ 23

Lecture 2: The Wiener-Hopf factorisation Wiener-Hopf for spectrally negative Lévy processes Work with Laplace exponent instead of characteristic exponent, θ ≥ 0 E (e θX t ) = e ψ ( θ ) t where ψ ( θ ) = − Ψ( − i θ ) . Wiener-Hopf factorisation reads ψ ( λ ) = ( λ − ϕ ) κ − ( λ ) , λ ≥ 0 , where ϕ ≥ 0 is the largest root of ψ on [0 , ∞ ) (there are at most two). ϕ = 0 ⇔ lim sup t ↑∞ X t = ∞ (i.e. X does not drift to −∞ ). Theorem [scale functions]: There exists a continuous, non-decreasing function W : [0 , ∞ ) → [0 , ∞ ) satisfying � ∞ 1 e − λx W ( x )d x = for λ > ϕ. ψ ( λ ) 0 12/ 23

Lecture 2: The Wiener-Hopf factorisation Wiener-Hopf for spectrally negative Lévy processes Work with Laplace exponent instead of characteristic exponent, θ ≥ 0 E (e θX t ) = e ψ ( θ ) t where ψ ( θ ) = − Ψ( − i θ ) . Wiener-Hopf factorisation reads ψ ( λ ) = ( λ − ϕ ) κ − ( λ ) , λ ≥ 0 , where ϕ ≥ 0 is the largest root of ψ on [0 , ∞ ) (there are at most two). ϕ = 0 ⇔ lim sup t ↑∞ X t = ∞ (i.e. X does not drift to −∞ ). Theorem [scale functions]: There exists a continuous, non-decreasing function W : [0 , ∞ ) → [0 , ∞ ) satisfying � ∞ 1 e − λx W ( x )d x = for λ > ϕ. ψ ( λ ) 0 When ϕ = 0 integration by parts shows � 1 e − λx W (d x ) = κ − ( λ ) , λ > 0 . [0 , ∞ ) 12/ 23

Lecture 2: The Wiener-Hopf factorisation Wiener-Hopf and the ruin problem x v u P x ( − X τ − 0 ∈ d u, X τ − 0 − ∈ d v ) = { W ( x ) − W ( x − v ) } Π( − d u − v )d v 13/ 23

Lecture 2: The Wiener-Hopf factorisation Wiener-Hopf and the ruin problem x v u When Π is absolutely continuous with density π : P x ( − X τ − 0 ∈ d u, X τ − 0 − ∈ d v ) = { W ( x ) − W ( x − v ) } π ( − u − v )d u d v 14/ 23