Total variation

In mathematics, especially calculus of variations and the theory of stochastic processes, the variation (also called total variation ) of a function is a measure of the local oscillation behavior of the function. Wherein the stochastic process is a variation of particular importance, since it divides the category of continuous-time processes in two fundamentally different sub- classes: those with and those with a finite infinite variation.

Definition

Let be a function on the real interval. The variation of is defined by

So by the least upper bound ( supremum ), the outvoted all sums that result from an arbitrarily fine subdivision of the interval. (If can not find any real number, the outvoted all the buzz, so the supremum is set to infinity to plus. )

For piecewise monotone, continuous functions, the following theorem holds:

Is in the intervals with each monotonically increasing or decreasing, then for the variation of the equation

The above definition of the variation can be applied to functions that are defined on unbounded intervals, and take on those values ​​in the complex numbers or in normed vector spaces.

Example of a continuous function with infinite variation

We want to show that for the constant on the unit interval function

Applies. Were for each

Then ( make drawing! ) Is

What tends to infinity because of the divergence of the harmonic series.

Application in the calculus of variations

In the calculus of variations often encountered optimization problems of the following type:

Wherein a predetermined set of functions, is about all twice continuously differentiable functions with additional properties such as

Similar problems lead for example to define the spline.

Another reason for the spread of the variation in optimization problems is the following statement: Describes the function of the history of an object in a one-dimensional space over time, then there just at the distance in the period range.

Application in stochastics

In the theory of stochastic processes, the concept of variation plays a special role: An important characterization of processes ( in addition to the division into classes such as Markov, Lévy or Gaussian processes) is almost surely finite or infinite in their capacity, over finite intervals exhibit variation:

• Example of a process almost surely finite variation: for a Poisson process with intensity applies because of the monotony;
• Example of a process almost certain to endless variation: the Wiener process, however, almost certainly has infinite variation on every interval.

For the application of the Wiener process in physics to explain the Brownian motion, this property has fatal consequences: A particle whose motion follows a Wiener process, would travel an infinite distance in each subspace - in stark contradiction to the laws of physics ( the particle would have infinite instantaneous velocity ).

Quadratic variation

Another interesting property of the Wiener process also relates to its variation along: If we replace in the above definition

We come to the concept of quadratic variation

A stochastic process on the interval.

An important result, which is reflected for example in Lemma Itō is the following: If a ( standard ) Wiener process, then for the quadratic variation almost certainly

In general, there are two forms of the quadratic variation / covariation.

Relationship between the two definitions:

Being with and the steady Martin Galt parts are designated.