Lecture 1: Lévy processes

A. E. Kyprianou

Department of Mathematical Sciences, University of Bath

Lévy processes

A process X = Xt : t ≥ 0 defined on a probability space (Ω,F ,P) issaid to be a (one dimensional) Lévy process if it possesses the followingproperties:(i) The paths of X are P-almost surely right continuous with left limits.(ii) P(X0 = 0) = 1.(iii) For 0 ≤ s ≤ t, Xt −Xs is equal in distribution to Xt−s.(iv) For 0 ≤ s ≤ t, Xt −Xs is independent of Xu : u ≤ s.

Some familiar examples(i) Linear Brownian motion σBt − at, t ≥ 0, σ, a ∈ R.(ii) Poisson process with λ, N = Nt : t ≥ 0.(iii) Compound Poisson processes with drift

Nt∑i=1

ξi + ct, t ≥ 0,

where ξi : i ≥ 1 are i.i.d. and c ∈ R.

Lévy processesA process X = Xt : t ≥ 0 defined on a probability space (Ω,F ,P) issaid to be a (one dimensional) Lévy process if it possesses the followingproperties:(i) The paths of X are P-almost surely right continuous with left limits.(ii) P(X0 = 0) = 1.(iii) For 0 ≤ s ≤ t, Xt −Xs is equal in distribution to Xt−s.(iv) For 0 ≤ s ≤ t, Xt −Xs is independent of Xu : u ≤ s.

Nt∑i=1

ξi + ct, t ≥ 0,

Lévy processesA process X = Xt : t ≥ 0 defined on a probability space (Ω,F ,P) issaid to be a (one dimensional) Lévy process if it possesses the followingproperties:(i) The paths of X are P-almost surely right continuous with left limits.(ii) P(X0 = 0) = 1.(iii) For 0 ≤ s ≤ t, Xt −Xs is equal in distribution to Xt−s.(iv) For 0 ≤ s ≤ t, Xt −Xs is independent of Xu : u ≤ s.

Nt∑i=1

ξi + ct, t ≥ 0,

Lévy processes

Note that in the last case of a compound Poisson process with drift, if weassume that E(|ξ1|) =

∫R |x|F (dx) <∞ and choose c = −λ

∫R xF (dx),

then the centred compound Poisson process

Nt∑i=1

ξi − λt∫RxF (dx), t ≥ 0,

is both a Lévy process and a martingale.

Any linear combination of independent Lévy processes is a Lévy process.

Lévy processes

Note that in the last case of a compound Poisson process with drift, if weassume that E(|ξ1|) =

∫R |x|F (dx) <∞ and choose c = −λ

∫R xF (dx),

then the centred compound Poisson process

Nt∑i=1

ξi − λt∫RxF (dx), t ≥ 0,

is both a Lévy process and a martingale.

Any linear combination of independent Lévy processes is a Lévy process.

The Lévy-Khintchine formula

As a consequence of stationary and independent increments it can beshown that any Lévy process X = Xt : t ≥ 0 has the property that, forall t ≥ 0 and θ,

E(eiθXt) = e−Ψ(θ)t

where Ψ(θ) = − logE(eiθX1) is called the characteristic exponent.

Theorem. The function Ψ : R→ C is the characteristic of a Lévy processif and only if

Ψ(θ) = iaθ +1

2σ2θ2 +

∫R(1− eiθx + iθx1(|x|<1))Π(dx).

where σ ∈ R, a ∈ R and Π is a measure concentrated on R\0 whichrespects the integrability condition∫

R(1 ∧ x2)Π(dx) <∞.

Ψ(θ) = iaθ +1

2σ2θ2 +

∫R(1− eiθx + iθx1(|x|<1))Π(dx).

R(1 ∧ x2)Π(dx) <∞.

Ψ(θ) = iaθ +1

2σ2θ2 +

∫R(1− eiθx + iθx1(|x|<1))Π(dx).

R(1 ∧ x2)Π(dx) <∞.

Key examples of L-K formula

For the case of σBt − at,

Ψ(θ) = iaθ +1

2σ2θ2.

For the case of a compound Poisson process∑Nti=1 ξi, where the the i.i.d.

variables ξi : i ≥ 1 have common distribution F and the Poisson processof jumps has rate λ,

Ψ(θ) =

∫R(1− eiθx)λF (dx)

For the case of independent linear combinations, letXt = σBt + at+

∑Nti=1 ξi − λ

∫R |x|F (dx)t, where N has rate λ and

ξi : i ≥ 1 have common distribution F satisfying∫R |x|F (dx) <∞

Ψ(θ) = iaθ +1

2σ2θ2 +

∫R(1− eiθx + iθx)λF (dx)

Ψ(θ) = iaθ +1

2σ2θ2.

Ψ(θ) =

∑Nti=1 ξi − λ

Ψ(θ) = iaθ +1

2σ2θ2 +

Ψ(θ) = iaθ +1

2σ2θ2.

Ψ(θ) =

∑Nti=1 ξi − λ

Ψ(θ) = iaθ +1

2σ2θ2 +

Ψ(θ) = iaθ +1

2σ2θ2.

Ψ(θ) =

∑Nti=1 ξi − λ

Ψ(θ) = iaθ +1

2σ2θ2 +

The Lévy-Itô decomposition

Ψ(θ) = iaθ +1

2σ2θ2 +

Ψ(θ) = iaθ +1

2σ2θ2 +

∫R(1− eiθx + iθx1(|x|<1))Π(dx).

Ψ(θ) = iaθ +1

2σ2θ2 +

Ψ(θ) = iaθ +1

2σ2θ2 +

∫R(1− eiθx + iθx1(|x|<1))Π(dx).

Ψ(θ) =

iaθ +

2σ2θ2

∫|x|≥1

(1− eiθx)λ0F0(dx)

+∑n≥0

∫2−(n+1)≤|x|<2−n

(1− eiθx + iθx)λnFn(dx)

where λ0 = Π(R\(−1, 1)) and λn = Π(x : 2−(n+1) ≤ |x| < 2−n)

F0(dx) = λ−10 Π(dx)||x|≥1 and Fn(dx) = λ−1

n Π(dx)|x:2−(n+1)≤|x|<2−n

Ψ(θ) = iaθ +1

2σ2θ2 +

Ψ(θ) = iaθ +1

2σ2θ2 +

∫R(1− eiθx + iθx1(|x|<1))Π(dx).

Suggestive that for any permitted triple (a, σ,Π) the associated Lévy processescan be written as the independent sum

Xt“ = ”at+ σBt +

N0t∑

ξ0i +

∞∑n=1

Nnt∑i=1

ξni −∫

2−(n+1)≤|x|<2−nxλnFn(dx)

where ξni : i ≥ 0 are families of i.i.d. random variables with respectivedistributions Fn and Nn are Poisson processes with respective arrival rates λn

The condition∫R(1∧ x2)Π(dx) <∞ ensures that all these processes "add up".

Brownian motion

0.0 0.2 0.4 0.6 0.8 1.0

−0.1

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Compound Poisson process

0.0 0.2 0.4 0.6 0.8 1.0

−0.4

−0.2

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Brownian motion + compound Poisson process

0.0 0.2 0.4 0.6 0.8 1.0−0.1

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Unbounded variation paths

0.0 0.2 0.4 0.6 0.8 1.0

−0.6

−0.4

−0.2

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Bounded variation paths

0.0 0.2 0.4 0.6 0.8 1.0

−0.4

−0.2

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Bounded vs unbounded variation paths

Paths of a Lévy processes are either almost surely of bounded variationover all finite time horizons or almost surely of unbounded variation overall finite time horizons.

Distinguishing the two cases can be identified from the Lévy-Itôdecomposition:

N0t∑

ξ0i +

∞∑n=1

Nnt∑i=1

ξni −∫

2−(n+1)≤|x|<2−nxΠ(dx)

Bounded variation if and only if

σ = 0 and∫

(−1,1)

|x|Π(dx) <∞

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N0t∑

ξ0i +

∞∑n=1

Nnt∑i=1

ξni −∫

2−(n+1)≤|x|<2−nxΠ(dx)

σ = 0 and∫

(−1,1)

|x|Π(dx) <∞

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N0t∑

ξ0i +

∞∑n=1

Nnt∑i=1

ξni −∫

2−(n+1)≤|x|<2−nxΠ(dx)

σ = 0 and∫

(−1,1)

|x|Π(dx) <∞

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N0t∑

ξ0i +

∞∑n=1

Nnt∑i=1

ξni −∫

2−(n+1)≤|x|<2−nxΠ(dx)

σ = 0 and∫

(−1,1)

|x|Π(dx) <∞

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Infinite divisibility

Suppose that X is an R-valued random variable on (Ω,F ,P), then X isinfinitely divisible if for each n = 1, 2, 3, ...

Xd= X(1,n) + · · ·+X(n,n)

where X(i,n) : i = 1, ..., n are independent and identically distributedand the equality is in distribution.Said another way, if µ is the characteristic function of X then for eachn = 1, 2, 3, ... we have that µ = (µn)n where µn is the the characteristicfunction of some R-valued random variable.For any Lévy process:

Xt = (Xt −Xt (n−1)n

) + (Xt(n−1)n

−Xt(n−2)n

) + · · ·+ (Xt 1n−X0)

from which stationary and independent increments implies infinitedivisibility.This goes part way to explaining why

E(eiθXt) = e−Ψ(θ)t = [E(eiθX1)]t

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Xd= X(1,n) + · · ·+X(n,n)

where X(i,n) : i = 1, ..., n are independent and identically distributedand the equality is in distribution.

Said another way, if µ is the characteristic function of X then for eachn = 1, 2, 3, ... we have that µ = (µn)n where µn is the the characteristicfunction of some R-valued random variable.For any Lévy process:

) + (Xt(n−1)n

−Xt(n−2)n

) + · · ·+ (Xt 1n−X0)

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Xd= X(1,n) + · · ·+X(n,n)

where X(i,n) : i = 1, ..., n are independent and identically distributedand the equality is in distribution.Said another way, if µ is the characteristic function of X then for eachn = 1, 2, 3, ... we have that µ = (µn)n where µn is the the characteristicfunction of some R-valued random variable.

For any Lévy process:

) + (Xt(n−1)n

−Xt(n−2)n

) + · · ·+ (Xt 1n−X0)

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Xd= X(1,n) + · · ·+X(n,n)

) + (Xt(n−1)n

−Xt(n−2)n

) + · · ·+ (Xt 1n−X0)

from which stationary and independent increments implies infinitedivisibility.

This goes part way to explaining why

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Xd= X(1,n) + · · ·+X(n,n)

) + (Xt(n−1)n

−Xt(n−2)n

) + · · ·+ (Xt 1n−X0)

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Lévy processes in finance and insurance

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Financial modelling: Share value, a day and a year

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Financial modelling

Black Scholes model of a risky asset:

St = expx+ σBt − at, t ≥ 0

(Initial value S0 = ex). Criticised at many levels.

Lévy model of a risky asset:

St = expx+Xt, t ≥ 0.

Does better than Black-Scholes on some issues (eg infinitely divisibilityinstead of Gaussian increments) but still generally viewed as a statisticallypoor fit with real data.

Modelling with a Lévy process means choosing the triplet (a, σ,Π).

The inclusion of σ is a choice of the inclusion of ‘noise’ and the choice ofΠ models jump structure and a can be used to deal with so-called riskneutrality: The existence of a measure P under which X is a Lévy processsatisfying

E(eXT ) = eqT in other words −Ψ(−i) = q.

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Financial modelling

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Financial modelling

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Financial modelling

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Financial modelling

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Some favourite Lévy processes in finance

The Kou model: η1, η2, λ > 0, p ∈ (0, 1),

σ2 ≥ 0 and Π(dx) = λpη1e−η1x1(x>0)dx+ λ(1− p)η2e−η2|x|1(x<0)dx

Ψ(θ) = iaθ +1

2σ2θ2 − λpiθ

η1 − iθ+λ(1− p)iθη2 + iθ

The KoBoL/CGMY model:

σ2 ≥ 0 and Π(dx) = Ce−Mx

x1+Y1(x>0)dx+ C

e−G|x|

|x|1+Y1(x<0)dx

Ψ(θ) = iaθ +1

2σ2θ2 + CΓ(−Y )(M − iθ)Y −MY + (G+ iθ)Y −GY

Spectrally negative Lévy processes:

σ2 ≥ 0 and Π(0,∞) = 0.

...and others... Variance Gamma, Meixner, Hyperbolic Lévy processes,β-Lévy processes, θ-Lévy processes, Hypergeometric Lévy processes, .....

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Ψ(θ) = iaθ +1

2σ2θ2 − λpiθ

x1+Y1(x>0)dx+ C

e−G|x|

|x|1+Y1(x<0)dx

Ψ(θ) = iaθ +1

σ2 ≥ 0 and Π(0,∞) = 0.

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Ψ(θ) = iaθ +1

2σ2θ2 − λpiθ

x1+Y1(x>0)dx+ C

e−G|x|

|x|1+Y1(x<0)dx

Ψ(θ) = iaθ +1

σ2 ≥ 0 and Π(0,∞) = 0.

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Ψ(θ) = iaθ +1

2σ2θ2 − λpiθ

x1+Y1(x>0)dx+ C

e−G|x|

|x|1+Y1(x<0)dx

Ψ(θ) = iaθ +1

σ2 ≥ 0 and Π(0,∞) = 0.

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Ψ(θ) = iaθ +1

2σ2θ2 − λpiθ

x1+Y1(x>0)dx+ C

e−G|x|

|x|1+Y1(x<0)dx

Ψ(θ) = iaθ +1

σ2 ≥ 0 and Π(0,∞) = 0.

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Need to know for option pricing

Note: X is a Markov process. Can work with Px to mean P(·|X0 = x).Standard theory dictates that the price of a European-type option overtime horizon [0, T ] as a function of the initial value of the stock

Ex(e−qT (f(eXT ))

where q ≥ 0 is the discounting rate.More exotic financial derivatives require an understanding of first passageproblems.

The value of an American put with strike K > 0:

Ex(e−qτ−a (K − e

Xτ−a )+)

where τ−a = inft > 0 : Xt < a for some a ∈ R.Barrier option (up and out call with strike K > 0):

Ex(e−qt(eXt −K)+1(sups≤tXs≤b))

for some b > logK.More generally, complex instruments such as credit-default swaps andconvertible contingencies are built upon the key mathematical ingredient

Px( infs≤t

Xs > 0)

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Ex(e−qT (f(eXT ))

where q ≥ 0 is the discounting rate.

More exotic financial derivatives require an understanding of first passageproblems.

Xτ−a )+)

Px( infs≤t

Xs > 0)

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Ex(e−qT (f(eXT ))

Xτ−a )+)

Px( infs≤t

Xs > 0)

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Ex(e−qT (f(eXT ))

Xτ−a )+)

where τ−a = inft > 0 : Xt < a for some a ∈ R.

Barrier option (up and out call with strike K > 0):

Px( infs≤t

Xs > 0)

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Ex(e−qT (f(eXT ))

Xτ−a )+)

for some b > logK.

More generally, complex instruments such as credit-default swaps andconvertible contingencies are built upon the key mathematical ingredient

Px( infs≤t

Xs > 0)

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Ex(e−qT (f(eXT ))

Xτ−a )+)

Px( infs≤t

Xs > 0)

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Insurance mathematics

The first passage problem also occurs in the related setting of insurancemathematics.

The classical risk insurance ruin problem sees the wealth of an insuranceproblem modelled by the so-called Cramér-Lundberg process:

Xt := x+ ct−Nt∑i=1

with the understanding that x is the initial wealth, c is the rate at whichpremiums are collected and Nt : t ≥ 0 is a Poisson process describingthe arrival of the i.i.d. claims ξi : i ≥ 0.This is nothing but a spectrally negative Lévy process.

A classical field of study, so called Gerber-Shiu, theory, concerns the studyof the joint law of

τ−0 , Xτ−0and X

τ−0 −,

the time of ruin, the deficit at ruin and the wealth prior to ruin.

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with the understanding that x is the initial wealth, c is the rate at whichpremiums are collected and Nt : t ≥ 0 is a Poisson process describingthe arrival of the i.i.d. claims ξi : i ≥ 0.

This is nothing but a spectrally negative Lévy process.

τ−0 −,

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τ−0 −,

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τ−0 −,

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We are interested in

Ex(e−qτ−0 ;−X

τ−0∈ du, X

τ−0 −∈ dv).

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