Quadratic variation

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 is a real-valued stochastic process defined on a probability space $(\Omega,\mathcal{F},\mathbb{P})$ 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. 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 quadratic variation for every 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 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 covariation may be written in terms of the quadratic variation by the polarization identity:

[X,Y]_t=\tfrac{1}{2}([X+Y]_t-[X]_t-[Y]_t).

Finite variation processes

A process X is said to have finite variation if it 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\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.

Itō processes

The quadratic variation of a standard Brownian motion B exists, and is given by [B]_t = t. This generalizes to Itō processes that, 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 chain rule to the Itō 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].

Alternatively this can be written as a Stochastic Differential Equation:

\,d(X_tY_t)=X_{t-}\,dY_t + Y_{t-}\,dX_t+\,dX_t \,dY_t,

where $\,dX_t \,dY_t=\,d[X,Y]_t.$

Martingales

All càdlàg 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 locally square integrable 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 proof of existence of [M] (without using stochastic calculus) is given in Karandikar-Rao (2014).

A useful result for square integrable martingales is the Itō 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 càdlàg square integrable martingale and H is a bounded predictable process, and is often used in the construction of the Itō 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 p > 0.

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.

References

Protter, Philip E. (2004), Stochastic Integration and Differential Equations (2nd ed.), Springer, ISBN 3-540-00313-4
Karandikar, Rajeeva L.; Rao, B. V. (2014). "On quadratic variation of martingales". Proceedings of Indian Academy Of Sciences (Indian Academy of Sciences) 124 (3): 457–469.