L´ evy processes: definitions and main properties

Definition 5.1.2. An R-valued process {ξt}t≥0, with ξ0 = 0, defined on a probability

space (Ω, F , P) is said to be a L´evy process if it satisfies the following conditions: for every s, t ≥ 0, ξt+s− ξs is independent of the σ-algebra Fsξ generated by {ξu}0≤u≤s; for

every s, t ≥ 0, the random variables ξt+s− ξs and ξt have the same law; and the process

{ξ_t} is continuous in probability—that is to say, for fixed t, and for all  > 0, it holds that P (|ξt− ξu| > ) → 0 as u → t.

There are many examples of L´evy processes that have been named and studied in depth. In what follows we shall be looking at Brownian motion, the Poisson processes, the gamma process, the variance gamma process, the negative binomial process, the inverse Gaussian process, the normal inverse Gaussian process, and the generalised hyperbolic process.

Due to the independent and stationary increments properties, it should be evident that for a L´evy process {ξt} we have:

E h eiλξti₌ E h eiλξ1it_{. (5.2)} It follows that:

Proposition 5.1.1. If {ξt} is a L´evy process, the random variable ξtis infinitely divisible

for all t ≥ 0.

Definition 5.1.3. By a L´evy measure ν(dz) we mean a positive measure defined on R\{0} satisfying

Z

R\{0}

The L´evy measure associated with a L´evy process has the following interpretation: if B is a measurable subset of R\{0}, then ν(B) is the rate at which jumps arrive for which the jump size lies in B. Suppose we consider the sets defined for n ∈ N by

Bn= {z ∈ R | 1/n ≤ |z| ≤ 1}. (5.4)

Let ν(dz) be the L´evy measure associated with a L´evy process {ξt}. Then if ν(Bn) → ∞

for large n, we say that {ξt} is a process of infinite activity, meaning that the rate of

arrival of small jumps is unbounded; and if ν(R\{0}) < ∞ we say that {ξt} has finite

activity, since the process has finite number of jumps in any finite time interval. With these observations in hand, we introduce the so-called L´evy-Khintchine formula for a L´evy process.

Proposition 5.1.2. Let {ξt} be a L´evy process taking values in R. Then for each t > 0,

the random variable ξt is infinitely divisible and its characteristic function is given for

λ ∈ R by

E [exp (iλξt)] = exp

 t  iλp −1 2qλ 2₊ Z R  eiλz− 1 − iλz1{|z| < 1}ν(dz)  (5.5)

where p ∈ R and q ≥ 0 are constants, and ν is a L´evy measure on R\{0}. A L´evy process admits a so-called L´evy-Itˆo decomposition of the following form: Proposition 5.1.3. An R-valued L´evy process {ξt} can be decomposed in the form

ξt= Xt(1)+ X (2) t + X

(3)

t , (5.6)

where X_t(1) is a Brownian motion with drift, X_t(2) is a compound Poisson process with jump sizes greater than or equal to unity, and X_t(3) is a L´evy process with jump sizes less than unity. The processes X_t(1), X_t(2), and X_t(3) are independent.

By a Brownian motion with drift we mean a process of the form {qBt+ pt}, where {Bt}

is a standard Brownian motion.

Definition 5.1.4. The function φ : R → C defined by

E [exp (iλξt)] = exp (−tφ(λ)) (5.7)

Definition 5.1.5. If Eeαξt _{< ∞ for α in some non-trivial connected region of R}

containing the origin, the function ψ(α) defined by

E [exp (αξt)] = exp (tψ(α)) (5.8)

on that region of R is called the L´evy exponent, or Laplace exponent, or cumulant func- tion, of the L´evy process {ξt}.

It follows that ψ(α) = −φ(−iα). In particular, as a consequence of Proposition5.1.2we have φ(λ) = −ipλ + 1 2qλ 2₋ Z _ eiλz− 1 − iλz1{|z| < 1}ν(dz) (5.9) and ψ(α) = pα +1 2qα 2₊ Z (eαz_{− 1 − αz1{|z| < 1}) ν(dz).} (5.10)

Remark 5.1.1. Bearing in mind the finite and infinite activity properties mentioned earlier, if we set q = 0, we can make the following observations. In the finite activity case, the process {ξt} is a compound Poisson process with “drift”. In the infinite activity

case, if R

|z|≤1|z|ν(dz) < ∞, the paths of the process {ξt} are of bounded variation on

any finite time interval. Again, in the infinite activity case, if R_|z|≤1|z|ν(dz) = ∞, the paths of the process {ξt} are no longer of bounded variation on any finite time interval.

Remark 5.1.2. The term “L´evy process” (which, according to C¸ inlar 2011, was coined by Paul A. Meyer) refers to the French mathematician Paul L´evy (1886-1971). We summarise from Schoutens (2003) the main facts of L´evy’s career: L´evy was born in Paris and studied at the ´Ecole Polytechnique. He obtained a doctoral degree in mathematics from the University of Paris, and became a professor at the ´Ecole des Mines in Paris at the age of twenty-seven. He made numerous contributions to the theory of stochastic processes, and is generally regarded as one of the most prominent figures of modern probability theory. During the first world war, L´evy served in the artillery, and carried out mathematical analysis concerning defence against air attacks. In 1963 he was elected to an honorary membership of the London Mathematical Society, and in 1964 he was elected to the Acad´emie des Sciences.