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.
Definition Suppose that X t {\displaystyle X_{t}} is a real-valued stochastic process defined on a probability space ( Ω , F , P ) {\displaystyle (\Omega ,{\mathcal {F}},\mathbb {P} )} and with time index t {\displaystyle t} ranging over the non-negative real numbers. Its quadratic variation is the process, written as [ X ] t {\displaystyle [X]_{t}} , defined as
[ X ] t = lim ‖ P ‖ → 0 ∑ k = 1 n ( X t k − X t k − 1 ) 2 {\displaystyle [X]_{t}=\lim _{\Vert P\Vert \rightarrow 0}\sum _{k=1}^{n}(X_{t_{k}}-X_{t_{k-1}})^{2}}
where P {\displaystyle P} ranges over partitions of the interval [ 0 , t ] {\displaystyle [0,t]} and the norm of the partition P {\displaystyle P} 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 t > 0 {\displaystyle t>0} 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 X {\displaystyle X} and Y {\displaystyle Y} is
[ X , Y ] t = lim ‖ P ‖ → 0 ∑ k = 1 n ( X t k − X t k − 1 ) ( Y t k − Y t k − 1 ) . {\displaystyle [X,Y]_{t}=\lim _{\Vert P\Vert \to 0}\sum _{k=1}^{n}\left(X_{t_{k}}-X_{t_{k-1}}\right)\left(Y_{t_{k}}-Y_{t_{k-1}}\right).}
The covariation may be written in terms of the quadratic variation by the polarization identity:
[ X , Y ] t = 1 2 ( [ X + Y ] t − [ X ] t − [ Y ] t ) . {\displaystyle [X,Y]_{t}={\tfrac {1}{2}}([X+Y]_{t}-[X]_{t}-[Y]_{t}).}
Notation: the quadratic variation is also notated as ⟨ X ⟩ t {\displaystyle \langle X\rangle _{t}} or ⟨ X , X ⟩ t {\displaystyle \langle X,X\rangle _{t}} .
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