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. This limit, if it exists, is defined using convergence in probability.

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:

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

Finite variation processes

A process X is said to have finite variation if it is 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 X has quadratic variation equal to the sum of the squares of the jumps of X. To state this more precisely, the left limit of X_t with respect to t is denoted by X_t-, and the jump of X at time t can be written as ΔX_t = X_t - X_t-. Then, the quadratic variation is given by

$[X]_{t}=\sum _{0<s\leq t}\Delta X_{s}^{2}.$

Itō processes

The quadratic variation of a standard Brownian motion B exists, and is given by [B]_t = t. This generalizes to Itō processes which, by definition, can be expressed in terms of Itō integrals

$X_{t}=X_{0}+\int _{0}^{t}\sigma _{s}\,dB_{s}+\int _{0}^{t}\mu _{s}\,ds,$

where B is a Brownian motion. Any such process has quadratic variation given by

$[X]_{t}=\int _{0}^{t}\sigma _{s}^{2}\,ds.$

Semimartingales

Quadratic variations and covariations of all semimartingales can be shown to exist. They form an important part of the theory of stochastic calculus, appearing in Itō's lemma, which is the generalization of the the chain rule to the Itō integral. The quadratic covariation also appears in the integration by parts formula

$X_{t}Y_{t}=X_{0}Y_{0}+\int _{0}^{t}X_{s-}\,dY_{s}+\int _{0}^{t}Y_{s-}\,dX_{s}+[X,Y]_{t},$

which can be used to compute [X,Y].

Martingales

The construction of quadratic variation processes of martingales is key to the construction of stochastic integrals The quadratic variation [M] of a general local martingale M is the unique increasing cadlag process starting at zero, with jumps Δ[M] = ΔM², and such that M² - [M] is a local martingale.

The proof that this exists is a major hurdle in the development of stochastic calculus. 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.

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.