Quadratic variation

Summary

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 1-variation for every

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https://en.wikipedia.org/wiki/Quadratic_variation

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On the rate of convergence in the martingale central limit theorem

Jean-Christophe Mourrat

Consider a discrete-time martingale, and let V-2 be its normalized quadratic variation. As V-2 approaches 1, and provided that some Lindeberg condition is satisfied, the distribution of the rescaled martingale approaches the Gaussian distribution. For any p >= 1, (Ann. Probab. 16 (1988) 275-299) gave a bound on the rate of convergence in this central limit theorem that is the sum of two terms, say A(p) + B-p, where up to a constant, A(p) = parallel to V-2 - 1 parallel to(p/(2p+1))(p). Here we discuss the optimality of this term, focusing on the restricted class of martingales with bounded increments. In this context, (Ann. Probab. 10 (1982) 672-688) sketched a strategy to prove optimality for p = 1. Here we extend this strategy to any p >= 1, thereby justifying the optimality of the term A(p). As a necessary step, we also provide a new bound on the rate of convergence in the central limit theorem for martingales with bounded increments that improves on the term B-p, generalizing another result of (Ann. Probab. 10 (1982) 672-688).

Int Statistical Inst2013

We compare the forecasts of Quadratic Variation given by the Realized Volatility (RV) and the Two Scales Realized Volatility (TSRV) computed from high frequency data in the presence of market microstructure noise, under several different dynamics for the volatility process and assumptions on the noise. We show that TSRV largely outperforms RV, whether looking at bias, variance, RMSE or out-of-sample forecasting ability. An empirical application to all DJIA stocks confirms the simulation results.

2008

Upper bounds for the reach-avoid probability via robust optimization

John Lygeros

We consider finite horizon reach-avoid problems for discrete time stochastic systems. Our goal is to construct upper bound functions for the reach-avoid probability by means of tractable convex optimization problems. We achieve this by restricting attention to the span of Gaussian radial basis functions and imposing structural assumptions on the transition kernel of the stochastic processes as well as the target and safe sets of the reach-avoid problem. In particular, we require the kernel to be written as a Gaussian mixture density with each mean of the distribution being affine in the current state and input and the target and safe sets to be written as intersections of quadratic inequalities. Taking advantage of these structural assumptions, we formulate a recursion of semidefinite programs where each step provides an upper bound to the value function of the reach- avoid problem. The upper bounds provide a performance metric to which any suboptimal control policy can be compared, and can themselves be used to construct suboptimal control policies. We illustrate via numerical examples that even if the resulting bounds are conservative, the associated control policies achieve higher reach-avoid probabilities than heuristic controllers for problems of large state-input space dimensions (more than 20). The results presented in this paper, far exceed the limits of current approximation methods for reach-avoid problems in the specific class of stochastic systems considered.

2015