Lévy Process as Combination of a Brownian Motion with Drift and Infinite Sum of Independent Compound Poisson Processes: Introduction to Martingales. Part 4.

Every piecewise constant Lévy process Xt0 can be represented in the form for some Poisson random measure with intensity measure of the form ν(dx)dt where ν is a finite measure, defined by

ν(A) = E[#{t ∈ [0,1] : ∆Xt0 ≠ 0, ∆Xt0 ∈ A}], A ∈ B(Rd) —– (1)

Given a Brownian motion with drift γt + Wt, independent from X0, the sum Xt = Xt0 + γt + Wt defines another Lévy process, which can be decomposed as:

Xt = γt + Wt + ∑s∈[0,t] ΔXs = γt + Wt + ∫[0,t]xRd xJX (ds x dx) —– (2)

where JX is a Poisson random measure on [0,∞[×Rd with intensity ν(dx)dt.

Can every Lévy process be represented in this form? Given a Lévy process Xt, we can still define its Lévy measure ν as above. ν(A) is still finite for any compact set A such that 0 ∉ A: if this were not true, the process would have an infinite number of jumps of finite size on [0, T], which contradicts the cadlag property. So ν defines a Radon measure on Rd \ {0}. But ν is not necessarily a finite measure: the above restriction still allows it to blow up at zero and X may have an infinite number of small jumps on [0, T]. In this case the sum of the jumps becomes an infinite series and its convergence imposes some conditions on the measure ν, under which we obtain a decomposition of X.

Let (Xt)t≥0 be a Lévy process on Rd and ν its Lévy measure.

ν is a Radon measure on Rd \ {0} and verifies:

|x|≤1 |x|2 v(dx) < ∞

The jump measure of X, denoted by JX, is a Poisson random measure on [0,∞[×Rd with intensity measure ν(dx)dt.

∃ a vector γ and a d-dimensional Brownian motion (Bt)t≥0 with covariance matrix A such that

Xt = γt + Bt + Xtl + limε↓0 X’εt —– (3)

where

Xtl = ∫|x|≥1,s∈[0,t] xJX (ds x dx)

X’εt = ∫ε≤|x|<1,s∈[0,t] x{JX (ds x dx) – ν(dx)ds}

≡ ∫ε≤|x|<1,s∈[0,t] xJ’X (ds x dx)

The terms in (3) are independent and the convergence in the last term is almost sure and uniform in t on [0,T].

The Lévy-Itô decomposition entails that for every Lévy process ∃ a vector γ, a positive definite matrix A and a positive measure ν that uniquely determine its distribution. The triplet (A,ν,γ) is called characteristic tripletor Lévy triplet of the process Xt. γt + Bt is a continuous Gaussian Lévy process and every Gaussian Lévy process is continuous and can be written in this form and can be described by two parameters: the drift γ and the covariance matrix of Brownian motion, denoted by A. The other two terms are discontinuous processes incorporating the jumps of Xt and are described by the Lévy measure ν. The condition ∫|y|≥1 ν(dy) < ∞ means that X has a finite number of jumps with absolute value larger than 1. So the sum

Xtl = ∑|∆Xs|≥10≤s≤t ∆Xs

contains almost surely a finite number of terms and Xtl is a compound Poisson process. There is nothing special about the threshold ∆X = 1: for any ε > 0, the sum of jumps with amplitude between ε and 1:

Xεt = ∑1>|∆Xs|≥ε0≤s≤t ∆Xs = ∫ε≤|x|≤1,s∈[0,t] xJX(ds x dx) —– (4)

is again a well-defined compound Poisson process. However, contrarily to the compound Poisson case, ν can have a singularity at zero: there can be infinitely many small jumps and their sum does not necessarily converge. This prevents us from making ε go to 0 directly in (4). In order to obtain convergence we have to center the remainder term, i.e., replace the jump integral by its compensated version,

X’εt = ∫ε≤|x|≤1,s∈[0,t] xJ’X (ds x dx) —– (5)

which, is a martingale. While Xε can be interpreted as an infinite superposition of independent Poisson processes, X’εshould be seen as an infinite superposition of independent compensated, i.e., centered Poisson processes to which a central-limit type argument can be applied to show convergence. An important implication of the Lévy-Itô decomposition is that every Lévy process is a combination of a Brownian motion with drift and a possibly infinite sum of independent compound Poisson processes. This also means that every Lévy process can be approximated with arbitrary precision by a jump-diffusion process, that is by the sum of Brownian motion with drift and a compound Poisson process.