In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, quadratic variation is used in the analysis of
stochastic process
In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables. Stochastic processes are widely used as mathematical models of systems and phenomena that appea ...
es such as
Brownian motion and other
martingales. Quadratic variation is just one kind of
variation of a process.
Definition
Suppose that
is a real-valued stochastic process defined on a
probability space and with time index
ranging over the non-negative real numbers. Its quadratic variation is the process, written as
, defined as
:
where
ranges over
partitions of the interval