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.

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 cądląg 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\le t}\Delta X_s^2.

The proof that continuous finite variation processes have zero quadratic variation follows from the following inequality. Here, P is a partition of the interval [0,t], and V_t(X) is the variation of X over [0,t].

\begin{align} \sum_{k=1}^n(X_{t_k}-X_{t_{k-1}})^2&\le\max_{k\le n}|X_{t_k}-X_{t_{k-1}}|\sum_{k=1}^n|X_{t_k}-X_{t_{k-1}}|\\ &\le\max_{|u-v|\le\Vert P\Vert}|X_u-X_v|V_t(X) \end{align}

By the continuity of X, this vanishes in the limit as \Vert P\Vert goes to zero.

It? processes

The quadratic variation of a standard Brownian motion B exists, and is given by [B]_t = t. This generalizes to It? processes which, by definition, can be expressed in terms of It? 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 It?'s lemma, which is the generalization of the chain rule to the It? integral. The quadratic covariation also appears in the integration by parts formula

X_tY_t=X_0Y_0+\int_0^tX_{s-}\,dY_s + \int_0^tY_{s-}\,dX_s+[X,Y]_t,

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

Martingales

All cądląg martingales, and local martingales have well defined quadratic variation, which follows from the fact that such processes are examples of semimartingales. It can be shown that the quadratic variation [M] of a general local martingale M is the unique right-continuous and increasing process starting at zero, with jumps Δ[M] = ΔM², and such that M² - [M] is a local martingale.

A useful result for square integrable martingales is the It? isometry, which can be used to calculate the variance of Ito integrals,

\mathbb{E}\left(\left(\int_0^t H\,dM\right)^2\right) = \mathbb{E}\left(\int_0^tH^2\,d[M]\right).

This result holds whenever M is a cądląg square integrable martingale and H is a bounded predictable process, and is often used in the construction of the It? integral.

Another important result is the Burkholder-Davis-Gundy inequality. This gives bounds for the maximum of a martingale in terms of the quadratic variation. For a local martingale M starting at zero, with maximum denoted by M_t^* ≡sup_s≤t|M_s|, and any real number p ≥1, the inequality is

c_p\mathbb{E}([M]_t^{p/2})\le \mathbb{E}((M^*_t)^p)\le C_p\mathbb{E}([M]_t^{p/2}).

Here, c_p < C_p are constants depending on the choice of p, but not depending on the martingale M or time t used. If M is a continuous local martingale, then the Burkholder-Davis-Gundy inequality holds for any positive value of p.

An alternative process, the predictable quadratic variation is sometimes used for locally square integrable martingales. This is written as <M>_t, and is defined to be the unique right-continuous and increasing predictable process starting at zero such that M² - <M> is a local martingale. Its existence follows from the Doob-Meyer decomposition theorem and, for continuous local martingales, it is the same as the quadratic variation.

See also

• Total variation, Bounded variation

References

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