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Concept
Quadratic variation
Summary
In mathematics, quadratic variation is used in the analysis of stochastic processes such as Brownian motion and other martingales. Quadratic variation is just one kind of variation of a process.
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 and the norm of the partition is the mesh. This limit, if it exists, is defined using convergence in probability. Note that a process may be of finite quadratic variation in the sense of the definition given here and its paths be nonetheless almost surely of infinite 1-variation for every in the classical sense of taking the supremum of the sum over all partitions; this is in particular the case for Brownian motion.
More generally, the covariation (or cross-variance) of two processes and is
The covariation may be written in terms of the quadratic variation by the polarization identity:
Notation: the quadratic variation is also notated as or .
A process is said to have finite variation if it has bounded variation over every finite time interval (with probability 1). Such processes are very common including, in particular, all continuously differentiable functions. The quadratic variation exists for all continuous finite variation processes, and is zero.
This statement can be generalized to non-continuous processes. Any càdlàg finite variation process has quadratic variation equal to the sum of the squares of the jumps of . To state this more precisely, the left limit of with respect to is denoted by , and the jump of at time can be written as . Then, the quadratic variation is given by
The proof that continuous finite variation processes have zero quadratic variation follows from the following inequality. Here, is a partition of the interval , and is the variation of over .
By the continuity of , this vanishes in the limit as goes to zero.
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