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

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Lecture 2: The Wiener-Hopf factorisation

Lecture 2: The Wiener-Hopf factorisation

• A. E. Kyprianou

Department of Mathematical Sciences, University of Bath

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Lecture 2: The Wiener-Hopf factorisation

Random walks

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Lecture 2: The Wiener-Hopf factorisation

Random walks

Random walk: Consider the discrete time process S = {Sn : n ≥ 0} where S0 = 0 and Sn =

n

• i=1

ξi, with {ξi : i ≥ 1} an i.i.d. sequence with common distribution F.

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Lecture 2: The Wiener-Hopf factorisation

Random walks

Random walk: Consider the discrete time process S = {Sn : n ≥ 0} where S0 = 0 and Sn =

n

• i=1

ξi, with {ξi : 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 {Sn−k − Sn : k = 0, 1, ..., n} has the same law as {−Sk : k = 0, 1, ..., n}.

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Lecture 2: The Wiener-Hopf factorisation

Random walks

Random walk: Consider the discrete time process S = {Sn : n ≥ 0} where S0 = 0 and Sn =

n

• i=1

ξi, with {ξi : 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 {Sn−k − Sn : k = 0, 1, ..., n} has the same law as {−Sk : k = 0, 1, ..., n}. Infinite divisibiltiy: Let Γp be a geometrically distributed random variable with parameter p which is independent of the random walk S. The random variable SΓp =

Γp

• i=1

ξi is infinitely divisible.

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Lecture 2: The Wiener-Hopf factorisation

Wiener-Hopf for random walks

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Lecture 2: The Wiener-Hopf factorisation

Wiener-Hopf for random walks

Assume min{F(0, ∞), F(−∞, 0)} > 0, and F has no atoms.

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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, ..., Γp : Sk = max

j=0,1,...,Γp Sj}

and D := inf{k = 0, 1, ..., Γp : Sk = min

j=0,1,...,Γp Sj}.

where Γp is a geometrically distributed random variable with parameter p which is independent of the random walk S.

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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, ..., Γp : Sk = max

j=0,1,...,Γp Sj}

and D := inf{k = 0, 1, ..., Γp : Sk = min

j=0,1,...,Γp Sj}.

where Γp is a geometrically distributed random variable with parameter p which is independent of the random walk S. Set. N = inf{n > 0 : Sn > 0}. In words, the first visit of S to (0, ∞) after time 0.

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Lecture 2: The Wiener-Hopf factorisation

D N G

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Lecture 2: The Wiener-Hopf factorisation

The Wiener-Hopf factorisation

Theorem: SG is independent of SΓp − SG and both are infinitely divisible with the latter equal in distribution to SD.

G

• S

S

G

S G N D

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Lecture 2: The Wiener-Hopf factorisation

Sketch proof

G

(2)

H + H

(1)

H(1)

(1) (2)

N N

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Lecture 2: The Wiener-Hopf factorisation

Sketch proof

D

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Lecture 2: The Wiener-Hopf factorisation

Sketch proof

Let ν be geometrically distributed with parameter P(N > Γp). We have SG =

ν

• i=1

H(i) where {H(i) : i = 1, 2, ...} are independent having the same distribution as SN conditioned on {N ≤ Γp}. The variable SΓp − SG is equal in distribution to SΓp conditional on {Γp < N}. Infinite divisibility follows as a consequence of the fact that SG is a geometric sum of i.i.d. random variables. (This also implies infinite divisibility of SD). Duality implies that the SΓp − SG is equal in distribution to SD, and hence, also infinitely divisible. Said another way: Let SΓp = maxn≤Γp Sn and SΓp = minn≤Γp Sn, then SΓp =d SΓp ⊕ SΓp.

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Lecture 2: The Wiener-Hopf factorisation

The Wiener-Hopf factorisation for Lévy process

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Lecture 2: The Wiener-Hopf factorisation

The Wiener-Hopf factorisation for Lévy process

Let X be any Lévy process and eq be an exponentially distributed random variable which is independent of X. Define Xeq = sup

s≤eq

Xs and Xeq = inf

s≤eq Xs.

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Lecture 2: The Wiener-Hopf factorisation

The Wiener-Hopf factorisation for Lévy process

Let X be any Lévy process and eq be an exponentially distributed random variable which is independent of X. Define Xeq = sup

s≤eq

Xs and Xeq = inf

s≤eq Xs.

Theorem: The random variables Xeq and Xeq − Xeq are independent and infinitely divisible. Moreover Xeq − Xeq =d Xeq. In particular Xeq =d Xeq ⊕ Xeq

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Lecture 2: The Wiener-Hopf factorisation

The Wiener-Hopf factorisation for Lévy process

Let X be any Lévy process and eq be an exponentially distributed random variable which is independent of X. Define Xeq = sup

s≤eq

Xs and Xeq = inf

s≤eq Xs.

Theorem: The random variables Xeq and Xeq − Xeq are independent and infinitely divisible. Moreover Xeq − Xeq =d Xeq. In particular Xeq =d Xeq ⊕ Xeq Analytically speaking: E(eiθXeq ) = q q + Ψ(θ) = E(eiθXeq )E(e

iθXeq ) =: Ψ+ q (θ)Ψ− q (θ).

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Lecture 2: The Wiener-Hopf factorisation

The Wiener-Hopf factorisation for Lévy process

q q + Ψ(θ) = Ψ+

q (θ)Ψ− q (θ)

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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+(−iθ)κ−(iθ), where κ± are so-called Bernstein functions and necessarily take the form κ±(λ) = η± + δ±λ +

• (0,∞)

(1 − e−λx)ν±(dx), with η± ≥ 0 such that η+η− = 0, δ± ≥ 0 and ν± satisfy

• (0,∞)

(1 ∧ x)ν±(dx) < ∞.

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Lecture 2: The Wiener-Hopf factorisation

Wiener-Hopf factorisation, financial and insurance mathematics

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Lecture 2: The Wiener-Hopf factorisation

Wiener-Hopf for spectrally negative Lévy processes

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Lecture 2: The Wiener-Hopf factorisation

Wiener-Hopf for spectrally negative Lévy processes

Work with Laplace exponent instead of characteristic exponent, θ ≥ 0 E(eθXt) = eψ(θ)t where ψ(θ) = −Ψ(−iθ).

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Lecture 2: The Wiener-Hopf factorisation

Wiener-Hopf for spectrally negative Lévy processes

Work with Laplace exponent instead of characteristic exponent, θ ≥ 0 E(eθXt) = eψ(θ)t where ψ(θ) = −Ψ(−iθ). 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).

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Lecture 2: The Wiener-Hopf factorisation

Wiener-Hopf for spectrally negative Lévy processes

Work with Laplace exponent instead of characteristic exponent, θ ≥ 0 E(eθXt) = eψ(θ)t where ψ(θ) = −Ψ(−iθ). 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 supt↑∞ Xt = ∞ (i.e. X does not drift to −∞).

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Lecture 2: The Wiener-Hopf factorisation

Wiener-Hopf for spectrally negative Lévy processes

Work with Laplace exponent instead of characteristic exponent, θ ≥ 0 E(eθXt) = eψ(θ)t where ψ(θ) = −Ψ(−iθ). 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 supt↑∞ Xt = ∞ (i.e. X does not drift to −∞). Theorem [scale functions]: There exists a continuous, non-decreasing function W : [0, ∞) → [0, ∞) satisfying ∞ e−λxW(x)dx = 1 ψ(λ) for λ > ϕ.

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Lecture 2: The Wiener-Hopf factorisation

Wiener-Hopf for spectrally negative Lévy processes

Work with Laplace exponent instead of characteristic exponent, θ ≥ 0 E(eθXt) = eψ(θ)t where ψ(θ) = −Ψ(−iθ). 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 supt↑∞ Xt = ∞ (i.e. X does not drift to −∞). Theorem [scale functions]: There exists a continuous, non-decreasing function W : [0, ∞) → [0, ∞) satisfying ∞ e−λxW(x)dx = 1 ψ(λ) for λ > ϕ. When ϕ = 0 integration by parts shows

• [0,∞)

e−λxW(dx) = 1 κ−(λ), λ > 0.

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Lecture 2: The Wiener-Hopf factorisation

Wiener-Hopf and the ruin problem

u v x

Px(−Xτ−

0 ∈ du, Xτ− 0 − ∈ dv) = {W(x) − W(x − v)} Π(−du − v)dv

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Lecture 2: The Wiener-Hopf factorisation

Wiener-Hopf and the ruin problem

u v x

When Π is absolutely continuous with density π: Px(−Xτ−

0 ∈ du, Xτ− 0 − ∈ dv) = {W(x) − W(x − v)} π(−u − v)dudv

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Lecture 2: The Wiener-Hopf factorisation

Reverse Wiener-Hopf engineering (spectrally negative processes)

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Lecture 2: The Wiener-Hopf factorisation

Reverse Wiener-Hopf engineering (spectrally negative processes)

Rather than trying to Laplace invert 1/κ− for a given spectrally negative Lévy process with exponent ψ, why not ask which Bernstein functions with the property that 1/κ− is Laplace invertible can play the role of κ− in a Wiener-Hopf factorisation ψ(λ) = λκ−(λ).

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Lecture 2: The Wiener-Hopf factorisation

Reverse Wiener-Hopf engineering (spectrally negative processes)

Rather than trying to Laplace invert 1/κ− for a given spectrally negative Lévy process with exponent ψ, why not ask which Bernstein functions with the property that 1/κ− is Laplace invertible can play the role of κ− in a Wiener-Hopf factorisation ψ(λ) = λκ−(λ). Theorem [K.& Hubalek, Vigon]: the Bernstein function κ(λ) := η + δλ +

• (0,∞)

(1 − e−λx)ν(dx) is a Wiener-Hopf factor of a spectrally negative Lévy process which does not drift to −∞ if and only if δ ≥ 0 and ν is absolutely continuous with non-increasing density.

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Lecture 2: The Wiener-Hopf factorisation

Reverse Wiener-Hopf engineering (spectrally negative processes)

Rather than trying to Laplace invert 1/κ− for a given spectrally negative Lévy process with exponent ψ, why not ask which Bernstein functions with the property that 1/κ− is Laplace invertible can play the role of κ− in a Wiener-Hopf factorisation ψ(λ) = λκ−(λ). Theorem [K.& Hubalek, Vigon]: the Bernstein function κ(λ) := η + δλ +

• (0,∞)

(1 − e−λx)ν(dx) is a Wiener-Hopf factor of a spectrally negative Lévy process which does not drift to −∞ if and only if δ ≥ 0 and ν is absolutely continuous with non-increasing density. Example: Let c > 0, ν ≥ 0 and θ ∈ (0, 1), κ(λ) = cλΓ(ν + λ) Γ(ν + λ + θ) for which η = δ = 0 and ν(x, ∞) = c Γ(θ)e−x(ν+θ−1) (ex − 1)θ−1 . W(x) = Γ(ν + θ) cΓ(ν) + θ cΓ(1 − θ) ∞ (x ∧ z) ez(1−ν) (ez − 1)1+θ dz.

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Lecture 2: The Wiener-Hopf factorisation

Wiener-Hopf and the general first passage problem

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Lecture 2: The Wiener-Hopf factorisation

Wiener-Hopf and the general first passage problem

Theorem: For all α, β, x > 0 we have E

• e

−ατ+

x −βXτ+ x 1(τ+ x <∞)

• =

E

• e−βXeα 1(Xeα >x)
• E
• e−βXeα
• .

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Lecture 2: The Wiener-Hopf factorisation

Wiener-Hopf and the general first passage problem

Theorem: For all α, β, x > 0 we have E

• e

−ατ+

x −βXτ+ x 1(τ+ x <∞)

• =

E

• e−βXeα 1(Xeα >x)
• E
• e−βXeα
• .

Note that E

• e−βXeα 1(Xeα >x)
• =

E

• e−βXeα 1(τ+

x <eα)

• =

E

• 1(τ+

x <eα)e

−βXτ+

x E

• e

−β(Xeα −Xτ+

x

)

• Fτ+

x

• .

Now, conditionally on Fτ+

x ∩ {τ +

x < eα} the random variables Xeα − Xτ+

x

and Xeα have the same distribution thanks to the lack of memory property

• f eα and the strong Markov property. Hence, we have the factorization

E

• e−βXeα 1(Xeα >x)
• = E
• e

−ατ+

x −βXτ+ x

• E
• e−βXeα
• .

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Lecture 2: The Wiener-Hopf factorisation

e e X X x

x

x

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Lecture 2: The Wiener-Hopf factorisation

Meromorphic Lévy processes and the Wiener-Hopf factorisation

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Lecture 2: The Wiener-Hopf factorisation

Meromorphic Lévy processes [Kuznetsov, K., Pardo Ann.Appl.Probab. 2012]

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Lecture 2: The Wiener-Hopf factorisation

Meromorphic Lévy processes [Kuznetsov, K., Pardo Ann.Appl.Probab. 2012]

A Lévy process is said to belong to the Meromorphic class (M-class), if and only if the Lévy measure Π(dx) has a density with respect to the Lebesgue measure, given by π(x) = I{x>0}

• n≥1

anρne−ρnx + I{x<0}

• n≥1

ˆ anˆ ρneˆ

ρnx,

(1) where all the coefficients an, ˆ an, ρn, ˆ ρn are positive, the sequences {ρn}n≥1 and {ˆ ρn}n≥1 are stricly increasing, and ρn → +∞ and ˆ ρn → +∞ as n → +∞.

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Lecture 2: The Wiener-Hopf factorisation

Meromorphic Lévy processes [Kuznetsov, K., Pardo Ann.Appl.Probab. 2012]

A Lévy process is said to belong to the Meromorphic class (M-class), if and only if the Lévy measure Π(dx) has a density with respect to the Lebesgue measure, given by π(x) = I{x>0}

• n≥1

anρne−ρnx + I{x<0}

• n≥1

ˆ anˆ ρneˆ

ρnx,

(1) where all the coefficients an, ˆ an, ρn, ˆ ρn are positive, the sequences {ρn}n≥1 and {ˆ ρn}n≥1 are stricly increasing, and ρn → +∞ and ˆ ρn → +∞ as n → +∞. We allow the case of a finite number summands (on either or both sides of the origin) with obvious modifications to the above. Gaussian and linear component are unconstrained.

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Lecture 2: The Wiener-Hopf factorisation

Meromorphic Lévy processes [Kuznetsov, K., Pardo Ann.Appl.Probab. 2012]

A Lévy process is said to belong to the Meromorphic class (M-class), if and only if the Lévy measure Π(dx) has a density with respect to the Lebesgue measure, given by π(x) = I{x>0}

• n≥1

anρne−ρnx + I{x<0}

• n≥1

ˆ anˆ ρneˆ

ρnx,

(1) where all the coefficients an, ˆ an, ρn, ˆ ρn are positive, the sequences {ρn}n≥1 and {ˆ ρn}n≥1 are stricly increasing, and ρn → +∞ and ˆ ρn → +∞ as n → +∞. We allow the case of a finite number summands (on either or both sides of the origin) with obvious modifications to the above. Gaussian and linear component are unconstrained. To ensure that

• R x2π(x)dx converges we need to impose the additional

constraint that

• n≥1

anρ−2

n

+

• n≥1

ˆ anˆ ρ−2

n

< ∞

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Lecture 2: The Wiener-Hopf factorisation

Meromorphic Lévy processes (Theorem: equivalent definition)

(i) The characteristic exponent Ψ(z) is a meromorphic function which has poles at points {−iρn, iˆ ρn}n≥1, where ρn and ˆ ρn are positive real numbers. (ii) For q ≥ 0 function q + Ψ(z) has roots at points {−iζn, iˆ ζn}n≥1 where ζn and ˆ ζn are nonnegative real numbers (strictly positive if q > 0). We will write ζn(q), ˆ ζn(q) if we need to stress the dependence on q. (iii) The roots and poles of q + Ψ(iz) satisfy the following interlacing condition ... − ρ2 < −ζ2 < −ρ1 < −ζ1 < 0 < ˆ ζ1 < ˆ ρ1 < ˆ ζ2 < ˆ ρ2 < ... (iv) The Wiener-Hopf factors are expressed as convergent infinite products, E

• e−zXeq
• =
• n≥1

1 +

z ρn

1 +

z ζn

E

• e

zXeq

• =
• n≥1

1 +

z ˆ ρn

1 +

z ˆ ζn

.

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Lecture 2: The Wiener-Hopf factorisation

Hyper-exponential jumps

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Lecture 2: The Wiener-Hopf factorisation

Hyper-exponential jumps

The density of the Lévy measure is π(x) = 1{x>0}

N

• i=1

aiρie−ρix + 1{x<0}

ˆ N

• i=1

ˆ aiˆ ρieˆ

ρix,

where ai, ˆ ai, ρi and ˆ ρi are positive numbers.

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Lecture 2: The Wiener-Hopf factorisation

Hyper-exponential jumps

The density of the Lévy measure is π(x) = 1{x>0}

N

• i=1

aiρie−ρix + 1{x<0}

ˆ N

• i=1

ˆ aiˆ ρieˆ

ρix,

where ai, ˆ ai, ρi and ˆ ρi are positive numbers. Including Gaussian and linear drift, one can verify that the characteristic exponent is a rational function and that hyper-exponential Lévy processes have finite activity jumps and paths of bounded variation unless σ > 0.

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Lecture 2: The Wiener-Hopf factorisation

Hyper-exponential jumps

The density of the Lévy measure is π(x) = 1{x>0}

N

• i=1

aiρie−ρix + 1{x<0}

ˆ N

• i=1

ˆ aiˆ ρieˆ

ρix,

where ai, ˆ ai, ρi and ˆ ρi are positive numbers. Including Gaussian and linear drift, one can verify that the characteristic exponent is a rational function and that hyper-exponential Lévy processes have finite activity jumps and paths of bounded variation unless σ > 0. Note that this class has been looked at by many other authors in the past and historically is starts life as the Kou process in the context of mathematical finance.

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Lecture 2: The Wiener-Hopf factorisation

β-family [Kuznetsov Ann.Appl.Probab. 2010]

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Lecture 2: The Wiener-Hopf factorisation

β-family [Kuznetsov Ann.Appl.Probab. 2010]

The characteristic exponent (Ψ(θ) = − log E(eiθX1), θ ∈ R) is given by Ψ(θ) = iaz + 1 2σ2z2 + c1 β1

• B(α1, 1 − λ1) − B(α1 − iθ

β1 , 1 − λ1)

• + c2

β2

• B(α2, 1 − λ2) − B(α2 + iθ

β2 , 1 − λ2)

• where B(x, y) = Γ(x)Γ(y)/Γ(x + y) is the Beta function, with parameter

range a ∈ R, σ, ci, αi, βi > 0 and λ1, λ2 ∈ (0, 3) \ {1, 2}.

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Lecture 2: The Wiener-Hopf factorisation

β-family [Kuznetsov Ann.Appl.Probab. 2010]

The characteristic exponent (Ψ(θ) = − log E(eiθX1), θ ∈ R) is given by Ψ(θ) = iaz + 1 2σ2z2 + c1 β1

• B(α1, 1 − λ1) − B(α1 − iθ

β1 , 1 − λ1)

• + c2

β2

• B(α2, 1 − λ2) − B(α2 + iθ

β2 , 1 − λ2)

• where B(x, y) = Γ(x)Γ(y)/Γ(x + y) is the Beta function, with parameter

range a ∈ R, σ, ci, αi, βi > 0 and λ1, λ2 ∈ (0, 3) \ {1, 2}. The corresponding Lévy measure Π has density π(x) = c1 e−α1β1x (1 − e−β1x)λ1 1{x>0} + c2 eα2β2x (1 − eβ2x)λ2 1{x<0}. The β-class of Lévy processes includes another recently introduced family

• f Lévy processes known as Lamperti-stable processes.

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Lecture 2: The Wiener-Hopf factorisation

Hypergeometric Lévy processes [K. Pardo, Rivero Ann.Appl.Probab. 2010] and [Pardo

and Kuznetsov 2012]

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Lecture 2: The Wiener-Hopf factorisation

Hypergeometric Lévy processes [K. Pardo, Rivero Ann.Appl.Probab. 2010] and [Pardo

and Kuznetsov 2012]

The characteristic exponent (Ψ(θ) = E(eiθX1), θ ∈ R) is given by Ψ(θ) = Γ(1 − β + γ − iθ) Γ(1 − β + iθ) Γ(ˆ β + ˆ γ + iθ) Γ(ˆ β + iθ) where (β, γ, ˆ β, ˆ γ) belong to the admissible range {β ≤ 1, γ ∈ (0, 1), ˆ β ≥ 0, ˆ γ ∈ (0, 1)}.

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Lecture 2: The Wiener-Hopf factorisation

Hypergeometric Lévy processes [K. Pardo, Rivero Ann.Appl.Probab. 2010] and [Pardo

and Kuznetsov 2012]

The characteristic exponent (Ψ(θ) = E(eiθX1), θ ∈ R) is given by Ψ(θ) = Γ(1 − β + γ − iθ) Γ(1 − β + iθ) Γ(ˆ β + ˆ γ + iθ) Γ(ˆ β + iθ) where (β, γ, ˆ β, ˆ γ) belong to the admissible range {β ≤ 1, γ ∈ (0, 1), ˆ β ≥ 0, ˆ γ ∈ (0, 1)}. The Lévy density is given by π(x) = −

Γ(η) Γ(η−ˆ γ)Γ(−γ)e−(1−β+γ)x 2F1(1 + γ, η; η − ˆ

γ; e−x) if x > 0 −

Γ(η) Γ(η−γ)Γ(−ˆ γ)e( ˆ β+ˆ γ)x 2F1(1 + ˆ

γ, η; η − γ; e−x) if x < 0 where η = 1 − β + γ + ˆ β + ˆ γ.