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 (Xt)t≥0 on (Ω,F,P) with values in Rd 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 . . . tn, the random variables Xt0, Xt1 − Xt0 , . . . , Xtn − Xtn−1 are independent.
2. Stationary increments: the law of Xt+h − Xt does not depend on t.
3. Stochastic continuity: ∀ε > 0, limh→0 P(|Xt+h − Xt| ≥ 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, limt↑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 Sn(∆) ≡ Xn∆, we can write Sn(∆) = ∑k=0n−1 Yk where Yk = X(k+1)∆ − Xk∆ 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 Sn(∆).
Choosing n∆ = t, we see that for any t > 0 and any n ≥ 1, Xt = Sn(∆) can be represented as a sum of n independent and identically distributed random variables whose distribution is that of Xt/n: Xt 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 Rd is said to be infinitely divisible if for any integer n ≥ 2, ∃ n independent and identically distributed random variables Y1, …Yn such that Y1 + … + Yn 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 Yk-s, F = μ ∗ μ ∗ ··· ∗ μ is the nth convolution of μ. So an infinitely divisible distribution can also be defined as a distribution F for which the nth 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 Xt is infinitely divisible. This puts a constraint on the possible choices of distributions for Xt: 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 (Xt)t≥0 such that the law of X1 if given by F. Let (Xt)t≥0 be a Lévy process. Then for every t, Xt has an infinitely divisible distribution. Conversely, if F is an infinitely divisible distribution then ∃ a Lévy process (Xt) such that the distribution of X1 is given by F.
2 thoughts on “Cadlag Stochasticities: Lévy Processes. Part 1.”
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