Every piecewise constant Lévy process X_{t}^{0} 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] : ∆X_{t}^{0} ≠ 0, ∆X_{t}^{0} ∈ A}], A ∈ B(R^{d}) —– (1)

Given a Brownian motion with drift γt + W_{t}, independent from X^{0}, the sum X_{t} = X_{t}^{0} + γt + W_{t} defines another Lévy process, which can be decomposed as:

X_{t} = γt + W_{t} + ∑_{s∈[0,t]} ΔX_{s} = γt + W_{t} + ∫_{[0,t]xRd} xJ_{X} (ds x dx) —– (2)

where J_{X} is a Poisson random measure on [0,∞[×R^{d} with intensity ν(dx)dt.

Can every Lévy process be represented in this form? Given a Lévy process X_{t}, 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 R^{d} \ {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 (X_{t})_{t≥0} be a Lévy process on R^{d} and ν its Lévy measure.

ν is a Radon measure on R^{d} \ {0} and verifies:

∫_{|x|≤1} |x|^{2} v(dx) < ∞

The jump measure of X, denoted by J_{X}, is a Poisson random measure on [0,∞[×R^{d} with intensity measure ν(dx)dt.

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

X_{t} = γ_{t} + B_{t} + X_{t}^{l} + lim_{ε↓0} X’^{ε}_{t} —– (3)

where

X_{t}^{l} = ∫_{|x|≥1,s∈[0,t]} xJ_{X} (ds x dx)

X’^{ε}_{t} = ∫_{ε≤|x|<1,s∈[0,t]} x{J_{X} (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 X_{t}. 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 X_{t} 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

X_{t}^{l} = ∑^{|∆Xs|≥1}_{0≤s≤t} ∆X_{s}

contains almost surely a finite number of terms and X_{t}^{l} 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:

X^{ε}_{t} = ∑^{1>|∆Xs|≥ε}_{0≤s≤t} ∆X_{s} = ∫_{ε≤|x|≤1,s∈[0,t]} xJ_{X}(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’^{ε}_{t} = ∫_{ε≤|x|≤1,s∈[0,t]} xJ’_{X} (ds x dx) —– (5)

which, is a martingale. While X^{ε} can be interpreted as an infinite superposition of independent Poisson processes, X’^{ε}_{t }should 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.