Cadlag Stochasticities: Lévy Processes. Part 1.

A compound Poisson process with a Gaussian distribution of jump sizes, and a jump diffusion of a Lévy process with Gaussian component and finite jump intensity.

A cadlag stochastic process (X_t)_t≥0 on (Ω,F,P) with values in R^d such that X₀ = 0 is called a Lévy process if it possesses the following properties:

1. Independent increments: for every increasing sequence of times t₀ . . . t_n, the random variables X_t0, X_t1 − X_t0 , . . . , X_tn − X_tn−1 are independent.

2. Stationary increments: the law of X_t+h − X_t does not depend on t.

3. Stochastic continuity: ∀ε > 0, lim_h→0 P(|X_t+h − X_t| ≥ ε) = 0.

A sample function x on a well-ordered set T is cadlag if it is continuous from the right and limited from the left at every point. That is, for every t₀ ∈ T, t ↓ t₀ implies x(t) → x(t₀), and for t ↑ t₀, lim_t↑t0 x(t)exists, but need not be x(t₀). A stochastic process X is cadlag if almost all its sample paths are cadlag.

The third condition does not imply in any way that the sample paths are continuous, and is verified by the Poisson process. It serves to exclude processes with jumps at fixed (nonrandom) times, which can be regarded as “calendar effects” and means that for given time t, the probability of seeing a jump at t is zero: discontinuities occur at random times.

If we sample a Lévy process at regular time intervals 0, ∆, 2∆, . . ., we obtain a random walk: defining S_n(∆) ≡ X_n∆, we can write S_n(∆) = ∑_k=0ⁿ⁻¹ Y_k where Y_k = X_(k+1)∆ − X_k∆ are independent and identically dependent random variables whose distribution is the same as the distribution of X_∆. Since this can be done for any sampling interval ∆ we see that by specifying a Lévy process one can specify a whole family of random walks S_n(∆).

Choosing n∆ = t, we see that for any t > 0 and any n ≥ 1, X_t = S_n(∆) can be represented as a sum of n independent and identically distributed random variables whose distribution is that of X_t/n: X_t can be “divided” into n independent and identically distributed parts. A distribution having this property is said to be infinitely divisible.

A probability distribution F on R^d is said to be infinitely divisible if for any integer n ≥ 2, ∃ n independent and identically distributed random variables Y₁, …Y_n such that Y₁ + … + Y_n has distribution F.

Since the distribution of independent and identically distributed sums is given by convolution of the distribution of the summands, denoting by μ the distribution of Y_k^-s, F = μ ∗ μ ∗ ··· ∗ μ is the n^th convolution of μ. So an infinitely divisible distribution can also be defined as a distribution F for which the n^th convolution root is still a probability distribution, for any n ≥ 2.

Thus, if X is a Lévy process, for any t > 0 the distribution of X_t is infinitely divisible. This puts a constraint on the possible choices of distributions for X_t: whereas the increments of a discrete-time random walk can have arbitrary distribution, the distribution of increments of a Lévy process has to be infinitely divisible.

The most common examples of infinitely divisible laws are: the Gaussian distribution, the gamma distribution, α-stable distributions and the Poisson distribution: a random variable having any of these distributions can be decomposed into a sum of n independent and identically distributed parts having the same distribution but with modified parameters. Conversely, given an infinitely divisible distribution F, it is easy to see that for any n ≥ 1 by chopping it into n independent and identically distributed components we can construct a random walk model on a time grid with step size 1/n such that the law of the position at t = 1 is given by F. In the limit, this procedure can be used to construct a continuous time Lévy process (X_t)_t≥0 such that the law of X₁ if given by F. Let (X_t)_t≥0 be a Lévy process. Then for every t, X_t has an infinitely divisible distribution. Conversely, if F is an infinitely divisible distribution then ∃ a Lévy process (X_t) such that the distribution of X₁ is given by F.

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