Independent increments

In probability theory, independent increments are a property of stochastic processes and random measures. Most of the time, a process or random measure has independent increments by definition, which underlines their importance. Some of the stochastic processes that by definition possess independent increments are the Wiener process, all Lévy processes, all additive process and the Poisson point process. Let be a stochastic process. In most cases, or . Then the stochastic process has independent increments if and only if for every and any choice with the random variables are stochastically independent. A random measure has got independent increments if and only if the random variables are stochastically independent for every selection of pairwise disjoint measurable sets and every . Let be a random measure on and define for every bounded measurable set the random measure on as Then is called a random measure with independent S-increments, if for all bounded sets and all the random measures are independent. Independent increments are a basic property of many stochastic processes and are often incorporated in their definition.

About this result

This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.

Related concepts (4)

Related lectures (11)

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, in which displacements in pairwise disjoint time intervals are independent, and displacements in different time intervals of the same length have identical probability distributions. A Lévy process may thus be viewed as the continuous-time analog of a random walk.

Infinite divisibility (probability)

In probability theory, a probability distribution is infinitely divisible if it can be expressed as the probability distribution of the sum of an arbitrary number of independent and identically distributed (i.i.d.) random variables. The characteristic function of any infinitely divisible distribution is then called an infinitely divisible characteristic function. More rigorously, the probability distribution F is infinitely divisible if, for every positive integer n, there exist n i.i.d. random variables Xn1, .

Independent increments

Levy Flights and Central Limit Theorem

Covers Levy flights, Central Limit Theorem, and Mesoscopic Master Equation with transition rates in an assurance system.

Brownian Motion: Einstein Derivation

Covers the derivation of Brownian motion by Einstein and Langevin equations, including the Chapman Kolmogorov equation.

Stochastic Differential Equations MOOC: Advanced statistical physics

Covers Stochastic Differential Equations, Wiener increment, Ito's lemma, and white noise integration in financial modeling.