Quadratic variation

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

Suppose that X_t is a real-valued stochastic process defined on a probability space (Ω,F,P) and with time index t ranging over the non-negative real numbers. Its quadratic variation is the process, written as [X]_t, defined as

$[X]_{t}=\lim _{\Vert P\Vert \rightarrow 0}\sum _{k=1}^{n}(X_{t_{k}}-X_{t_{k-1}})^{2}$

where P ranges over partitions of the interval [0,t] and the norm of the partition P is the mesh.

More generally, the quadratic covariation of two processes X and Y is

$[X,Y]_{t}=\lim _{\Vert P\Vert \to 0}\sum _{k=1}^{n}\left(X_{t_{k}}-X_{t_{k-1}}\right)\left(Y_{t_{k}}-Y_{t_{k-1}}\right).$

The quadratic covariation may be written in terms of the quadratic variation by the polarization identity:

$[X,Y]_{t}={\frac {1}{4}}([X+Y]_{t}-[X-Y]_{t}).$

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 $X_{t}$ may be defined as the increasing process $[X]_{t}$ such that

(i)$[X]_{0}=0$

(ii)$\Delta [X]_{t}=(\Delta X_{t})^{2}\quad \forall t$

(iii)$X_{t}^{2}-[X]_{t}$ 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 $X_{t}$ with continuous sample paths it may be shown to be equivalent to the following definition for a partition

$\pi _{t}=\{0=t_{0}<t_{1}<\cdots <t_{m}=t\}$

whose mesh is defined by

$\delta (\pi _{t})=\max _{k\in [1,m]}|t_{k}-t_{k-1}|$

in terms of which the quadratic-variation process may be defined by

$V_{t}=\lim _{\delta (\pi _{t})\to 0}\sum _{\pi }|X_{t_{k}}-X_{t_{k-1}}|^{2}.$

A related process $\langle X\rangle _{t}$ 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 $[X]_{t}$. 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

$[X]_{t}=[X^{\mathrm {cm} }]_{t}+\sum _{0\leq s\leq t}\Delta X_{s}^{2}$

where $X^{\mathrm {cm} }$ is the canonical continuous martingale in the decomposition of $X$ i.e.

$X_{t}=X_{t}^{\mathrm {cm} }+X_{t}^{\mathrm {dm} }+A_{t}$

where $A$ 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:

$[X,Y]_{t}:={\frac {1}{4}}\left([X+Y]_{t}-[X-Y]_{t}\right)$

Quadratic differentiability

Theorem

If f is continuously differentiable, then $\langle f\rangle (T)=0.$

Proof

Let P be the partition $0=t_{0}<t_{1}<\cdots <t_{n}=T$ where $||P||$ denotes the norm of the partition. Notice that $\left|f'(t)\right|$ is continuous on a compact set $\left[0,T\right]$ and therefore attains a maximum M. Then

${\begin{aligned}\lim _{||P||\to 0}\sum _{k=0}^{n-1}|f(t_{k+1})-f(t_{k})|^{2}&{}=\lim _{||P||\to 0}\sum _{k=0}^{n-1}(f'(t_{k}^{*}))^{2}|t_{k+1}-t_{k}|^{2}\\&{}\leq \lim _{||P||\to 0}\sum _{k=0}^{n-1}M^{2}|t_{k+1}-t_{k}|||P||\\&{}\leq M^{2}T\lim _{||P||\to 0}||P||\,=\,0\end{aligned}}$

where $t_{k}^{*}\in (t_{k},t_{k+1})$ by the mean value theorem.