Lévy process

In probability theory, a Lévy process, named after the French mathematician Paul Lévy, is a stochastic process with independent, stationary increments: it represents the motion of a point whose successive displacements are random and independent, and statistically identical over different time intervals of the same length.A Lévy process may thus be viewed as the continuous-time analog of a random walk.If X = ( X t ) t ≥ 0 {displaystyle X=(X_{t})_{tgeq 0}} is a Lévy process, then its characteristic function ϕ X ( θ ) {displaystyle phi _{X}( heta )} is given by In probability theory, a Lévy process, named after the French mathematician Paul Lévy, is a stochastic process with independent, stationary increments: it represents the motion of a point whose successive displacements are random and independent, and statistically identical over different time intervals of the same length.A Lévy process may thus be viewed as the continuous-time analog of a random walk. The most well known examples of Lévy processes are the Wiener process, often called the Brownian motion process, and the Poisson process. Aside from Brownian motion with drift, all other proper (that is, not deterministic) Lévy processes have discontinuous paths. A stochastic process X = { X t : t ≥ 0 } {displaystyle X={X_{t}:tgeq 0}} is said to be a Lévy process if it satisfies the following properties: If X {displaystyle X} is a Lévy process then one may construct a version of X {displaystyle X} such that t ↦ X t {displaystyle tmapsto X_{t}} is almost surely right continuous with left limits. A continuous-time stochastic process assigns a random variable Xt to each point t ≥ 0 in time. In effect it is a random function of t. The increments of such a process are the differences Xs − Xt between its values at different times t < s. To call the increments of a process independent means that increments Xs − Xt and Xu − Xv are independent random variables whenever the two time intervals do not overlap and, more generally, any finite number of increments assigned to pairwise non-overlapping time intervals are mutually (not just pairwise) independent. To call the increments stationary means that the probability distribution of any increment Xt − Xs depends only on the length t − s of the time interval; increments on equally long time intervals are identically distributed. If X {displaystyle X} is a Wiener process, the probability distribution of Xt − Xs is normal with expected value 0 and variance t − s. If X {displaystyle X} is the Poisson process, the probability distribution of Xt − Xs is a Poisson distribution with expected value λ(t − s), where λ > 0 is the 'intensity' or 'rate' of the process. The distribution of a Lévy process has the property of infinite divisibility: given any integer 'n', the law of a Lévy process at time t can be represented as the law of n independent random variables, which are precisely the increments of the Lévy process over time intervals of length t/n, which are independent and identically distributed by assumptions 2 and 3. Conversely, for each infinitely divisible probability distribution F {displaystyle F} , there is a Lévy process X {displaystyle X} such that the law of X 1 {displaystyle X_{1}} is given by F {displaystyle F} .