Lecture 1: Lvy processes A. E. Kyprianou Department of Mathematical - - PowerPoint PPT Presentation

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Lecture 1: Lévy processes

Lecture 1: Lévy processes

• A. E. Kyprianou

Department of Mathematical Sciences, University of Bath

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Lecture 1: Lévy processes

Lévy processes

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Lecture 1: Lévy processes

Lévy processes

A process X = {Xt : t ≥ 0} defined on a probability space (Ω, F, P) is said to be a (one dimensional) Lévy process if it possesses the following properties:

(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}.

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Lecture 1: Lévy processes

Lévy processes

A process X = {Xt : t ≥ 0} defined on a probability space (Ω, F, P) is said to be a (one dimensional) Lévy process if it possesses the following properties:

(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.

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Lecture 1: Lévy processes

Lévy processes

Note that in the last case of a compound Poisson process with drift, if we assume 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

• R

xF(dx), t ≥ 0, is both a Lévy process and a martingale.

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Lecture 1: Lévy processes

Lévy processes

Note that in the last case of a compound Poisson process with drift, if we assume 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

• R

xF(dx), t ≥ 0, is both a Lévy process and a martingale. Any linear combination of independent Lévy processes is a Lévy process.

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Lecture 1: Lévy processes

The Lévy-Khintchine formula

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Lecture 1: Lévy processes

The Lévy-Khintchine formula

As a consequence of stationary and independent increments it can be shown that any Lévy process X = {Xt : t ≥ 0} has the property that, for all t ≥ 0 and θ, E(eiθXt) = e−Ψ(θ)t where Ψ(θ) = − log E(eiθX1) is called the characteristic exponent.

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Lecture 1: Lévy processes

The Lévy-Khintchine formula

As a consequence of stationary and independent increments it can be shown that any Lévy process X = {Xt : t ≥ 0} has the property that, for all t ≥ 0 and θ, E(eiθXt) = e−Ψ(θ)t where Ψ(θ) = − log E(eiθX1) is called the characteristic exponent.

• Theorem. The function Ψ : R → C is the characteristic of a Lévy process

if 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} which respects the integrability condition

• R

(1 ∧ x2)Π(dx) < ∞.

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Lecture 1: Lévy processes

Key examples of L-K formula

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Lecture 1: Lévy processes

Key examples of L-K formula

For the case of σBt − at, Ψ(θ) = iaθ + 1 2σ2θ2.

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Lecture 1: Lévy processes

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 Nt

i=1 ξi, where the the i.i.d.

variables {ξi : i ≥ 1} have common distribution F and the Poisson process

• f jumps has rate λ,

Ψ(θ) =

• R

(1 − eiθx)λF(dx)

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Lecture 1: Lévy processes

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 Nt

i=1 ξi, where the the i.i.d.

variables {ξi : i ≥ 1} have common distribution F and the Poisson process

• f jumps has rate λ,

Ψ(θ) =

• R

(1 − eiθx)λF(dx) For the case of independent linear combinations, let Xt = σBt + at + Nt

i=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)

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Lecture 1: Lévy processes

The Lévy-Itô decomposition

Ψ(θ) = iaθ + 1 2σ2θ2 +

• R

(1 − eiθx + iθx)λF(dx) Ψ(θ) = iaθ + 1 2σ2θ2 +

• R

(1 − eiθx + iθx1(|x|<1))Π(dx).

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Lecture 1: Lévy processes

The Lévy-Itô decomposition

Ψ(θ) = iaθ + 1 2σ2θ2 +

• R

(1 − eiθx + iθx)λF(dx) Ψ(θ) = iaθ + 1 2σ2θ2 +

• R

(1 − eiθx + iθx1(|x|<1))Π(dx). Ψ(θ) =

• iaθ + 1

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−1

0 Π(dx)|{|x|≥1} and Fn(dx) = λ−1 n Π(dx)|{x:2−(n+1)≤|x|<2−n}

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Lecture 1: Lévy processes

The Lévy-Itô decomposition

Ψ(θ) = iaθ + 1 2σ2θ2 +

• R

(1 − eiθx + iθx)λF(dx) 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 processes can be written as the independent sum Xt“ = ”at + σBt +

N0

t

• i=1

ξ0

i + ∞

• n=1

  

Nn

t

• i=1

ξn

i −

• 2−(n+1)≤|x|<2−n xλnFn(dx)

   where {ξn

i : i ≥ 0} are families of i.i.d. random variables with respective

distributions Fn and N n are Poisson processes with respective arrival rates λn The condition

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

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Lecture 1: Lévy processes

Brownian motion

0.0 0.2 0.4 0.6 0.8 1.0 −0.1 0.0 0.1 0.2

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Lecture 1: Lévy processes

Compound Poisson process

0.0 0.2 0.4 0.6 0.8 1.0 −0.4 −0.2 0.0 0.2 0.4 0.6 0.8

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Lecture 1: Lévy processes

Brownian motion + compound Poisson process

0.0 0.2 0.4 0.6 0.8 1.0 −0.1 0.0 0.1 0.2 0.3 0.4

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Lecture 1: Lévy processes

Unbounded variation paths

0.0 0.2 0.4 0.6 0.8 1.0 −0.6 −0.4 −0.2 0.0 0.2 0.4 0.6

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Lecture 1: Lévy processes

Bounded variation paths

0.0 0.2 0.4 0.6 0.8 1.0 −0.4 −0.2 0.0 0.2

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Lecture 1: Lévy processes

Bounded vs unbounded variation paths

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Lecture 1: Lévy processes

Bounded vs unbounded variation paths

Paths of a Lévy processes are either almost surely of bounded variation

• ver all finite time horizons or almost surely of unbounded variation over

all finite time horizons.

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Lecture 1: Lévy processes

Bounded vs unbounded variation paths

Paths of a Lévy processes are either almost surely of bounded variation

• ver all finite time horizons or almost surely of unbounded variation over

all finite time horizons. Distinguishing the two cases can be identified from the Lévy-Itô decomposition: Xt“ = ”at + σBt +

N0

t

• i=1

ξ0

i + ∞

• n=1

  

Nn

t

• i=1

ξn

i −

• 2−(n+1)≤|x|<2−n xΠ(dx)

  

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Lecture 1: Lévy processes

Bounded vs unbounded variation paths

Paths of a Lévy processes are either almost surely of bounded variation

• ver all finite time horizons or almost surely of unbounded variation over

all finite time horizons. Distinguishing the two cases can be identified from the Lévy-Itô decomposition: Xt“ = ”at + σBt +

N0

t

• i=1

ξ0

i + ∞

• n=1

  

Nn

t

• i=1

ξn

i −

• 2−(n+1)≤|x|<2−n xΠ(dx)

   Bounded variation if and only if σ = 0 and

• (−1,1)

|x|Π(dx) < ∞

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Lecture 1: Lévy processes

Infinite divisibility

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Lecture 1: Lévy processes

Infinite divisibility

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

d

= X(1,n) + · · · + X(n,n) where {X(i,n) : i = 1, ..., n} are independent and identically distributed and the equality is in distribution.

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Lecture 1: Lévy processes

Infinite divisibility

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

d

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

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Lecture 1: Lévy processes

Infinite divisibility

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

d

= X(1,n) + · · · + X(n,n) where {X(i,n) : i = 1, ..., n} are independent and identically distributed and the equality is in distribution. Said another way, if µ is the characteristic function of X then for each n = 1, 2, 3, ... we have that µ = (µn)n where µn is the the characteristic function 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 1

n − X0)

from which stationary and independent increments implies infinite divisibility.

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Lecture 1: Lévy processes

Infinite divisibility

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

d

= X(1,n) + · · · + X(n,n) where {X(i,n) : i = 1, ..., n} are independent and identically distributed and the equality is in distribution. Said another way, if µ is the characteristic function of X then for each n = 1, 2, 3, ... we have that µ = (µn)n where µn is the the characteristic function 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 1

n − X0)

from which stationary and independent increments implies infinite divisibility. This goes part way to explaining why E(eiθXt) = e−Ψ(θ)t = [E(eiθX1)]t

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Lecture 1: Lévy processes

Lévy processes in finance and insurance

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Lecture 1: Lévy processes

Financial modelling: Share value, a day and a year

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Lecture 1: Lévy processes

Financial modelling

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Lecture 1: Lévy processes

Financial modelling

Black Scholes model of a risky asset: St = exp{x + σBt − at}, t ≥ 0 (Initial value S0 = ex). Criticised at many levels.

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Lecture 1: Lévy processes

Financial modelling

Black Scholes model of a risky asset: St = exp{x + σBt − at}, t ≥ 0 (Initial value S0 = ex). Criticised at many levels. Lévy model of a risky asset: St = exp{x + Xt}, t ≥ 0. Does better than Black-Scholes on some issues (eg infinitely divisibility instead of Gaussian increments) but still generally viewed as a statistically poor fit with real data.

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Lecture 1: Lévy processes

Financial modelling

Black Scholes model of a risky asset: St = exp{x + σBt − at}, t ≥ 0 (Initial value S0 = ex). Criticised at many levels. Lévy model of a risky asset: St = exp{x + Xt}, t ≥ 0. Does better than Black-Scholes on some issues (eg infinitely divisibility instead of Gaussian increments) but still generally viewed as a statistically poor fit with real data. Modelling with a Lévy process means choosing the triplet (a, σ, Π).

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Lecture 1: Lévy processes

Financial modelling

Black Scholes model of a risky asset: St = exp{x + σBt − at}, t ≥ 0 (Initial value S0 = ex). Criticised at many levels. Lévy model of a risky asset: St = exp{x + Xt}, t ≥ 0. Does better than Black-Scholes on some issues (eg infinitely divisibility instead of Gaussian increments) but still generally viewed as a statistically poor 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 of Π models jump structure and a can be used to deal with so-called risk neutrality: The existence of a measure P under which X is a Lévy process satisfying E(eXT ) = eqT in other words − Ψ(−i) = q.

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Lecture 1: Lévy processes

Some favourite Lévy processes in finance

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Lecture 1: Lévy processes

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)iθ η2 + iθ

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Lecture 1: Lévy processes

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)iθ η2 + iθ The KoBoL/CGMY model: σ2 ≥ 0 and Π(dx) = C e−Mx x1+Y 1(x>0)dx + C e−G|x| |x|1+Y 1(x<0)dx Ψ(θ) = iaθ + 1 2σ2θ2 + CΓ(−Y ){(M − iθ)Y − M Y + (G + iθ)Y − GY }

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Lecture 1: Lévy processes

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)iθ η2 + iθ The KoBoL/CGMY model: σ2 ≥ 0 and Π(dx) = C e−Mx x1+Y 1(x>0)dx + C e−G|x| |x|1+Y 1(x<0)dx Ψ(θ) = iaθ + 1 2σ2θ2 + CΓ(−Y ){(M − iθ)Y − M Y + (G + iθ)Y − GY } Spectrally negative Lévy processes: σ2 ≥ 0 and Π(0, ∞) = 0.

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Lecture 1: Lévy processes

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)iθ η2 + iθ The KoBoL/CGMY model: σ2 ≥ 0 and Π(dx) = C e−Mx x1+Y 1(x>0)dx + C e−G|x| |x|1+Y 1(x<0)dx Ψ(θ) = iaθ + 1 2σ2θ2 + CΓ(−Y ){(M − iθ)Y − M Y + (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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Lecture 1: Lévy processes

Need to know for option pricing

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Lecture 1: Lévy processes

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 over time 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.

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Lecture 1: Lévy processes

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 over time 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 passage problems.

20/ 22

Lecture 1: Lévy processes

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 over time 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 passage problems.

The value of an American put with strike K > 0: Ex(e−qτ−

a (K − e

Xτ−

a )+)

where τ −

a = inf{t > 0 : Xt < a} for some a ∈ R.

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Lecture 1: Lévy processes

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 over time 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 passage problems.

The value of an American put with strike K > 0: Ex(e−qτ−

a (K − e

Xτ−

a )+)

where τ −

a = inf{t > 0 : Xt < a} for some a ∈ R.

Barrier option (up and out call with strike K > 0): Ex(e−qt(eXt − K)+1(sups≤t Xs≤b)) for some b > log K.

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Lecture 1: Lévy processes

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 over time 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 passage problems.

The value of an American put with strike K > 0: Ex(e−qτ−

a (K − e

Xτ−

a )+)

where τ −

a = inf{t > 0 : Xt < a} for some a ∈ R.

Barrier option (up and out call with strike K > 0): Ex(e−qt(eXt − K)+1(sups≤t Xs≤b)) for some b > log K. More generally, complex instruments such as credit-default swaps and convertible contingencies are built upon the key mathematical ingredient Px(inf

s≤t Xs > 0)

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Lecture 1: Lévy processes

Insurance mathematics

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

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Lecture 1: Lévy processes

Insurance mathematics

The first passage problem also occurs in the related setting of insurance mathematics. The classical risk insurance ruin problem sees the wealth of an insurance problem modelled by the so-called Cramér-Lundberg process: Xt := x + ct −

Nt

• i=1

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

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Lecture 1: Lévy processes

Insurance mathematics

The first passage problem also occurs in the related setting of insurance mathematics. The classical risk insurance ruin problem sees the wealth of an insurance problem modelled by the so-called Cramér-Lundberg process: Xt := x + ct −

Nt

• i=1

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

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Lecture 1: Lévy processes

Insurance mathematics

The first passage problem also occurs in the related setting of insurance mathematics. The classical risk insurance ruin problem sees the wealth of an insurance problem modelled by the so-called Cramér-Lundberg process: Xt := x + ct −

Nt

• i=1

ξi, with the understanding that x is the initial wealth, c is the rate at which premiums are collected and {Nt : t ≥ 0} is a Poisson process describing the 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 study

• f the joint law of

τ −

0 , Xτ−

0 and Xτ− 0 −,

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

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Lecture 1: Lévy processes

Ruin

u v x

We are interested in Ex(e−qτ−

0 ; −Xτ− 0 ∈ du, Xτ− 0 − ∈ dv).