Quadratic variation: Difference between revisions
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In [[mathematics]], '''quadratic variation''' is |
In [[mathematics]], '''quadratic variation''' is used in the analysis of [[stochastic process]]es such as [[Wiener process|Brownian motion]] and [[Martingale (probability theory)|martingale]]s. Quadratic variation is just one kind of variation of a process (see [[function variation]]). |
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==Definition== |
==Definition== |
Revision as of 01:00, 12 June 2008
In mathematics, quadratic variation is used in the analysis of stochastic processes such as Brownian motion and martingales. Quadratic variation is just one kind of variation of a process (see function variation).
Definition
The quadratic variation of a function f on the interval [0, T] is defined as
where P ranges over partitions of the interval [0,T] and the norm of the partition is the mesh. More generally, the quadratic covariation of two functions f and g on the interval [0,T] is
Many authors denote the quadratic variation of f by [f,f] instead of . The quadratic covariation may be written in terms of the quadratic variation by the polarization identity:
Quadratic variations of semimartingales
The construction of quadratic variation processes of semimartingales is key to the construction of stochastic integrals The quadratic variation of a general L2 bounded martingale may be defined as the increasing process such that
- (i)
- (ii)
- (iii) is a uniformly integrable martingale.
The proof that such a process may be constructed and is unique is a major hurdle in the development of stochastic calculus. However for a process with continuous sample paths it may be shown to be equivalent to the following definition for a partition
whose mesh is defined by
in terms of which the quadratic-variation process may be defined by
A related process is historically sometimes used as the basis of the integral; this process is defined as a previsible process satisfying the first and third conditions above. It can be shown that this process is the previsible projection of . While much of the theory can be developed from this, this approach to the theory of stochastic integration of discontinuous processes proves to be the wrong starting point.
This definition is extended to semimartingales by defining
where is the canonical continuous martingale in the decomposition of i.e.
where is of finite variation.
The definition of the quadratic variation process gives rise immediately to the definition of the covariation process can be defined by the polarization identity:
Quadratic differentiability
Theorem
If f is continuously differentiable, then
Proof
Let P be the partition where denotes the norm of the partition. Notice that is continuous on a compact set and therefore attains a maximum M. Then
where by the mean value theorem.