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

Lecture 1: Lévy processes Lecture 1: Lévy processes A. E. Kyprianou Department of Mathematical Sciences, University of Bath 1/ 22

Lecture 1: Lévy processes Lévy processes 2/ 22

Lecture 1: Lévy processes Lévy processes A process X = { X t : 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 ( X 0 = 0) = 1 . (iii) For 0 ≤ s ≤ t, X t − X s is equal in distribution to X t − s . (iv) For 0 ≤ s ≤ t, X t − X s is independent of { X u : u ≤ s } . 2/ 22

Lecture 1: Lévy processes Lévy processes A process X = { X t : 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 ( X 0 = 0) = 1 . (iii) For 0 ≤ s ≤ t, X t − X s is equal in distribution to X t − s . (iv) For 0 ≤ s ≤ t, X t − X s is independent of { X u : u ≤ s } . Some familiar examples (i) Linear Brownian motion σB t − a t , t ≥ 0 , σ, a ∈ R . (ii) Poisson process with λ , N = { N t : t ≥ 0 } . (iii) Compound Poisson processes with drift N t � ξ i + ct, t ≥ 0 , i =1 where { ξ i : i ≥ 1 } are i.i.d. and c ∈ R . 2/ 22

Lecture 1: Lévy processes Lévy processes Note that in the last case of a compound Poisson process with drift, if we � � assume that E ( | ξ 1 | ) = R | x | F (d x ) < ∞ and choose c = − λ R xF (d x ) , then the centred compound Poisson process N t � � ξ i − λt xF (d x ) , t ≥ 0 , R i =1 is both a Lévy process and a martingale. 3/ 22

Lecture 1: Lévy processes Lévy processes Note that in the last case of a compound Poisson process with drift, if we � � assume that E ( | ξ 1 | ) = R | x | F (d x ) < ∞ and choose c = − λ R xF (d x ) , then the centred compound Poisson process N t � � ξ i − λt xF (d x ) , t ≥ 0 , R i =1 is both a Lévy process and a martingale. Any linear combination of independent Lévy processes is a Lévy process. 3/ 22

Lecture 1: Lévy processes The Lévy-Khintchine formula 4/ 22

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 = { X t : t ≥ 0 } has the property that, for all t ≥ 0 and θ , E (e i θX t ) = e − Ψ( θ ) t where Ψ( θ ) = − log E (e i θX 1 ) is called the characteristic exponent . 4/ 22

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 = { X t : t ≥ 0 } has the property that, for all t ≥ 0 and θ , E (e i θX t ) = e − Ψ( θ ) t where Ψ( θ ) = − log E (e i θX 1 ) is called the characteristic exponent . Theorem. The function Ψ : R → C is the characteristic of a Lévy process if and only if Ψ( θ ) = i a θ + 1 � 2 σ 2 θ 2 + (1 − e i θx + i θx 1 ( | x | < 1) )Π(d x ) . R where σ ∈ R , a ∈ R and Π is a measure concentrated on R \{ 0 } which respects the integrability condition � (1 ∧ x 2 )Π(d x ) < ∞ . R 4/ 22

Lecture 1: Lévy processes Key examples of L-K formula 5/ 22

Lecture 1: Lévy processes Key examples of L-K formula For the case of σB t − a t , Ψ( θ ) = i a θ + 1 2 σ 2 θ 2 . 5/ 22

Lecture 1: Lévy processes Key examples of L-K formula For the case of σB t − a t , Ψ( θ ) = i a θ + 1 2 σ 2 θ 2 . For the case of a compound Poisson process � N t i =1 ξ i , where the the i.i.d. variables { ξ i : i ≥ 1 } have common distribution F and the Poisson process of jumps has rate λ , � (1 − e i θx ) λF (d x ) Ψ( θ ) = R 5/ 22

Lecture 1: Lévy processes Key examples of L-K formula For the case of σB t − a t , Ψ( θ ) = i a θ + 1 2 σ 2 θ 2 . For the case of a compound Poisson process � N t i =1 ξ i , where the the i.i.d. variables { ξ i : i ≥ 1 } have common distribution F and the Poisson process of jumps has rate λ , � (1 − e i θx ) λF (d x ) Ψ( θ ) = R For the case of independent linear combinations, let X t = σB t + a t + � N t � i =1 ξ i − λ R | x | F (d x ) t, where N has rate λ and � { ξ i : i ≥ 1 } have common distribution F satisfying R | x | F (d x ) < ∞ Ψ( θ ) = i a θ + 1 � 2 σ 2 θ 2 + (1 − e i θx + i θx ) λF (d x ) R 5/ 22

Lecture 1: Lévy processes The Lévy-Itô decomposition Ψ( θ ) = i a θ + 1 � 2 σ 2 θ 2 + (1 − e i θx + i θx ) λF (d x ) R Ψ( θ ) = i a θ + 1 � 2 σ 2 θ 2 + (1 − e i θx + i θx 1 ( | x | < 1) )Π(d x ) . R 6/ 22

Lecture 1: Lévy processes The Lévy-Itô decomposition Ψ( θ ) = i a θ + 1 � 2 σ 2 θ 2 + (1 − e i θx + i θx ) λF (d x ) R Ψ( θ ) = i a θ + 1 � 2 σ 2 θ 2 + (1 − e i θx + i θx 1 ( | x | < 1) )Π(d x ) . R �� � � � i a θ + 1 2 σ 2 θ 2 (1 − e i θx ) λ 0 F 0 (d x ) Ψ( θ ) = + | x |≥ 1 �� � 2 − ( n +1) ≤| x | < 2 − n (1 − e i θx + i θx ) λ n F n (d x ) � + n ≥ 0 where λ 0 = Π( R \ ( − 1 , 1)) and λ n = Π( { x : 2 − ( n +1) ≤ | x | < 2 − n } ) F 0 (d x ) = λ − 1 0 Π(d x ) | {| x |≥ 1 } and F n (d x ) = λ − 1 n Π(d x ) | { x :2 − ( n +1) ≤| x | < 2 − n } 7/ 22

Lecture 1: Lévy processes The Lévy-Itô decomposition Ψ( θ ) = i a θ + 1 � 2 σ 2 θ 2 + (1 − e i θx + i θx ) λF (d x ) R Ψ( θ ) = i a θ + 1 � 2 σ 2 θ 2 + (1 − e i θx + i θx 1 ( | x | < 1) )Π(d x ) . R Suggestive that for any permitted triple ( a , σ, Π) the associated Lévy processes can be written as the independent sum N 0  N n  ∞ t t �   � ξ 0 � � ξ n X t “ = ” a t + σB t + i + i − 2 − ( n +1) ≤| x | < 2 − n xλ n F n (d x ) i =1 n =1  i =1  where { ξ n i : i ≥ 0 } are families of i.i.d. random variables with respective distributions F n and N n are Poisson processes with respective arrival rates λ n � R (1 ∧ x 2 )Π(d x ) < ∞ ensures that all these processes "add up". The condition 8/ 22

Lecture 1: Lévy processes Brownian motion 0.2 0.1 0.0 − 0.1 0.0 0.2 0.4 0.6 0.8 1.0 9/ 22

Lecture 1: Lévy processes Compound Poisson process 0.8 0.6 0.4 0.2 0.0 − 0.2 − 0.4 0.0 0.2 0.4 0.6 0.8 1.0 10/ 22

Lecture 1: Lévy processes Brownian motion + compound Poisson process 0.4 0.3 0.2 0.1 0.0 − 0.1 0.0 0.2 0.4 0.6 0.8 1.0 11/ 22

Lecture 1: Lévy processes Unbounded variation paths 0.6 0.4 0.2 0.0 − 0.2 − 0.4 − 0.6 0.0 0.2 0.4 0.6 0.8 1.0 12/ 22

Lecture 1: Lévy processes Bounded variation paths 0.2 0.0 − 0.2 − 0.4 0.0 0.2 0.4 0.6 0.8 1.0 13/ 22

Lecture 1: Lévy processes Bounded vs unbounded variation paths 14/ 22

Lecture 1: Lévy processes Bounded vs unbounded variation paths Paths of a Lévy processes are either almost surely of bounded variation over all finite time horizons or almost surely of unbounded variation over all finite time horizons. 14/ 22

Lecture 1: Lévy processes Bounded vs unbounded variation paths Paths of a Lévy processes are either almost surely of bounded variation over 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: N 0   N n ∞ t t �   � ξ 0 � � ξ n X t “ = ” a t + σB t + i + i − 2 − ( n +1) ≤| x | < 2 − n x Π(d x ) i =1 n =1  i =1  14/ 22

Lecture 1: Lévy processes Bounded vs unbounded variation paths Paths of a Lévy processes are either almost surely of bounded variation over 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: N 0   N n ∞ t t �   � ξ 0 � � ξ n X t “ = ” a t + σB t + i + i − 2 − ( n +1) ≤| x | < 2 − n x Π(d x ) i =1 n =1  i =1  Bounded variation if and only if � σ = 0 and | x | Π(d x ) < ∞ ( − 1 , 1) 14/ 22

Lecture 1: Lévy processes Infinite divisibility 15/ 22

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 , ... d X = X (1 ,n ) + · · · + X ( n,n ) where { X ( i,n ) : i = 1 , ..., n } are independent and identically distributed and the equality is in distribution. 15/ 22

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 , ... d X = 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 = ( µ n ) n where µ n is the the characteristic function of some R -valued random variable. 15/ 22