Quadratic Variation Process Research Articles

Stocks in the Short Run

The existing literature estimates stock-price volatility accumulated over the trading day. We focus on what happens to volatility within the trading day. Using transactions data from 2001 through 2009, we estimate the path of the quadratic variation process in 5-minute increments day by day for the 30 stocks of the DJIA and for an exchange-traded fund (the SPDR) that tracks the S&P 500. Using a Heston (1993) model, we estimate that 80% of the gap between the level of the volatility process and its asymptotic mean is eliminated within 5-minutes. Roughly two-thirds of daily realized volatility can be explained by a deterministic version of the Heston model that begins the trading day far above its equilibrium and converges to a constant. The remaining third reflects stochastic shocks to volatility arriving after trade begins. The asymptotic mean of the SPDR behaves much like the closing value of the VIX, a volatility index based on the S&P 500 stock index. When standardized by our 5-minute volatility estimates, 5-minute log returns are approximately normally distributed.

Clark–Ocone type formula for non-semimartingales with finite quadratic variation

We provide a suitable framework for the concept of finite quadratic variation for processes with values in a separable Banach space B using the language of stochastic calculus via regularizations, introduced in the case B = R by the second author and P. Vallois. To a real continuous process X we associate the Banach-valued process X ( ⋅ ) , called window process, which describes the evolution of X taking into account a memory τ > 0 . The natural state space for X ( ⋅ ) is the Banach space of continuous functions on [ − τ , 0 ] . If X is a real finite quadratic variation process, an appropriated Itô formula is presented, from which we derive a generalized Clark–Ocone formula for non-semimartingales having the same quadratic variation as Brownian motion. The representation is based on solutions of an infinite-dimensional PDE.

Sequential optimizing strategy in multi-dimensional bounded forecasting games

We propose a sequential optimizing betting strategy in the multi-dimensional bounded forecasting game in the framework of game-theoretic probability of Shafer and Vovk (2001) [10]. By studying the asymptotic behavior of its capital process, we prove a generalization of the strong law of large numbers, where the convergence rate of the sample mean vector depends on the growth rate of the quadratic variation process. The growth rate of the quadratic variation process may be slower than the number of rounds or may even be zero. We also introduce an information criterion for selecting efficient betting items. These results are then applied to multiple asset trading strategies in discrete-time and continuous-time games. In the case of a continuous-time game we present a measure of the jaggedness of a vector-valued continuous process. Our results are examined by several numerical examples.

Large deviations for stochastic differential equations driven by [formula omitted]-Brownian motion

A joint large deviation principle for G -Brownian motion and its quadratic variation process is presented. The rate function is not a quadratic form due to quadratic variation uncertainty. A large deviation principle for stochastic differential equations driven by G -Brownian motion is also established.

Stochastic regression and its application to hedging in finance

In this paper we investigate how to employ stochastic regression to hedge risks in finance, where the risk of a security is measured by its quadratic variation process. Mykland and Zhang used this technique to demonstrate how to reduce the risk of a given security by introducing another security. In this paper, we investigate how to further reduce the remaining unhedgable risk by adding more hedging securities. Some practical guidelines on how to choose those hedging securities in practice is also given.

Nonlinear filtering with signal dependent observation noise

The paper studies the filtering problem for a non-classical frame- work: we assume that the observation equation is driven by a signal dependent noise. We show that the support of the conditional distri- bution of the signal is on the corresponding level set of the derivative of the quadratic variation process. Depending on the intrinsic dimension of the noise, we distinguish two cases: In the first case, the conditional distribution has discrete support and we deduce an explicit represen- tation for the conditional distribution. In the second case, the filtering problem is equivalent to a classical one defined on a manifold and we deduce the evolution equation of the conditional distribution. The re- sults are applied to the filtering problem where the observation noise is an Ornstein-Uhlenbeck process.

Small-time versions of Strassen's law for Lévy processes

We study aspects of the ‘small-time’ behaviour (as t ↓ 0) of a Lévy process X(t), obtaining a very general small-time version of Strassen's almost sure (a.s.) functional law of the iterated logarithm (LIL) for random walks. The class of Lévy processes for which this holds is characterised by an explicit analytic condition on the Lévy measure of X, related to an analogous condition of Kesten for a generalised (large-time) random walk LIL. Both centred and uncentred versions of the small-time result are proved. Subsidiary results concerning functional weak convergence of X(t) to Brownian motion as t ↓ 0 are shown to be equivalent to the main a.s. results. The quadratic variation process of X is considered, and applications via continuous functionals are suggested.

Estimation of quadratic variation for two-parameter diffusions

In this paper we give a central limit theorem for the weighted quadratic variation process of a two-parameter Brownian motion. As an application, we show that the discretized quadratic variations ∑ i = 1 [ n s ] ∑ j = 1 [ n t ] | Δ i , j Y | 2 of a two-parameter diffusion Y = ( Y ( s , t ) ) ( s , t ) ∈ [ 0 , 1 ] 2 observed on a regular grid G n form an asymptotically normal estimator of the quadratic variation of Y as n goes to infinity.

Wiener integrals, Malliavin calculus and covariance measure structure

We introduce the notion of covariance measure structure for square integrable stochastic processes. We define Wiener integral, we develop a suitable formalism for stochastic calculus of variations and we make Gaussian assumptions only when necessary. Our main examples are finite quadratic variation processes with stationary increments and the bifractional Brownian motion.

Nonsemimartingales: Stochastic differential equations and weak Dirichlet processes

In this paper we discuss existence and uniqueness for a one-dimensional time inhomogeneous stochastic differential equation directed by an $\mathbb F$-semimartingale $M$ and a finite cubic variation process $\xi$ which has the structure $Q + R$ where $Q$ is a finite quadratic variation process and $R$ is {\it{strongly predictable}} in some technical sense: that condition implies in particular that $R$ is \textit{weak Dirichlet}, and it is fulfilled, for instance, when $R$ is independent of $M$. The method is based on a transformation which reduces the {{\it diffusion}} coefficient multiplying $\xi$ to 1. We use generalized Ito and Ito-Wentzell type formulae. A similar method allows to discuss existence and uniqueness theorem when $\xi$ is a Holder continuous process and $\sigma$ is only Holder in space. Using an Ito formula for {\it{reversible}} semimartingales we also show existence of a solution when $\xi$ is a Brownian motion and $\sigma$ is only continuous.

A Differentiation Theory for Itô's Calculus

A peculiar feature of Itô's calculus is that it is an integral calculus that gives no explicit derivative with a systematic differentiation theory counterpart, as in elementary calculus. So, can we define a pathwise stochastic derivative of semimartingales with respect to Brownian motion that leads to a differentiation theory counterpart to Itô's integral calculus? From Itô's definition of his integral, such a derivative must be based on the quadratic variation process. We give such a derivative in this note and we show that it leads to a fundamental theorem of stochastic calculus, a generalized stochastic chain rule that includes the case of convex functions acting on continuous semimartingales, and the stochastic mean value and Rolle's theorems. In addition, it interacts with basic algebraic operations on semimartingales similarly to the way the deterministic derivative does on deterministic functions, making it natural for computations. Such a differentiation theory leads to many interesting applications, some of which we address in an upcoming article.

Quasi sure analysis of local times of anticipating smooth semimartingales

Let X t = ∫ 0 t f s d W s + ∫ 0 t g s d s be the anticipating smooth semimartingale and L t x be its generalized local time. In this paper, we give some estimates about the quasi sure property of X t and its quadratic variation process 〈 X 〉 t . We also study the fractional smoothness of L t x and prove that the quadratic variation process of L t x can be constructed as the quasi sure limit of the form ∑ Δ n ( L t a i + 1 n − L t a i n ) 2 , where Δ n = ( a i n , a i + 1 n ) is a sequence of subdivisions of [ a , b ] , a i n = i ( b − a ) / 2 n + a , i = 0 , 1 , … , 2 n .

On bifractional Brownian motion

This paper is devoted to analyzing several properties of the bifractional Brownian motion introduced by Houdré and Villa. This process is a self-similar Gaussian process depending on two parameters H and K and it constitutes a natural generalization of fractional Brownian motion (which is obtained for K = 1 ). Here, we adopt the strategy of stochastic calculus via regularization. Of particular interest to us is the case H K = 1 2 . In this case, the process is a finite quadratic variation process with bracket equal to a constant times t and it has the same order of self-similarity as standard Brownian motion. It is a short-memory process even though it is neither a semimartingale nor a Dirichlet process.

Comment: A Selective Overview of Nonparametric Methods in Financial Econometrics

These comments concentrate on two issues arising from Fan’s overview. The first concerns the importance of finite sample estimation bias relative to the specification and discretization biases that are emphasized in Fan’s discussion. Past research and simulations given here both reveal that finite sample effects can be more important than the other two effects when judged from either statistical or economic viewpoints. Second, we draw attention to a very different nonparametric technique that is based on computing an empirical version of the quadratic variation process. This technique is not mentioned by Fan but has many advantages and has accordingly attracted much recent attention in financial econometrics and empirical applications.

Stochastic differential equations with fractal noise

AbstractStochastic differential equations in ℝn with random coefficients are considered where one continuous driving process admits a generalized quadratic variation process. The latter and the other driving processes are assumed to possess sample paths in the fractional Sobolev space Wβ2 for some β > 1/2. The stochastic integrals are determined as anticipating forward integrals. A pathwise solution procedure is developed which combines the stochastic Itô calculus with fractional calculus via norm estimates of associated integral operators in Wα 2 for 0 < α < 1. Linear equations are considered as a special case. This approach leads to fast computer algorithms basing on Picard's iteration method. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Two inequalities for iterated stochastic integrals

Let \( M = (M_{t}, {\mathcal F}_{t})_{t \geq 0} \) be a continuous local martingale with quadratic variation process \( \langle M \rangle \) and \( M_{0} = 0 \) . In this paper, we consider the corresponding sequence of the iterated stochastic integrals \( I_{n}(M) = (I_{n}(t, M), {\mathcal F}_{t})(n \geq 0) \) , defined inductively by

Strong approximation of continuous local martingales by simple random walks

The aim of this paper is to represent any continuous local martingale as an almost sure limit of a nested sequence of simple, symmetric random walk, time changed by a discrete quadratic variation process. One basis of this is a similar construction of Brownian motion. The other major tool is a representation of continuous local martingales given by Dambis, Dubins and Schwarz (DDS) in terms of Brownian motion time-changed by the quadratic variation. Rates of convergence (which are conjectured to be nearly optimal in the given setting) are also supplied. A necessary and sufficient condition for the independence of the random walks and the discrete time changes or equivalently, for the independence of the DDS Brownian motion and the quadratic variation is proved to be the symmetry of increments of the martingale given the past, which is a reformulation of an earlier result by Ocone [8].

Strong approximation of continuous local martingales by simple random walks

The aim of this paper is to represent any continuous local martingale as an almost sure limit of a nested sequence of simple, symmetric random walks, time changed by a discrete quadratic variation process. One basis of this is a similar construction of Brownian motion. The other major tool is a representation of continuous local martingales given by Dambis, Dubins and Schwarz (DDS) in terms of Brownian motion time-changed by the quadratic variation. Rates of convergence (which are conjectured to be nearly optimal in the given setting) are also supplied. A necessary and sufficient condition for the independence of the random walks and the discrete time changes or, equivalently, for the independence of the DDS Brownian motion and the quadratic variation is proved to be the symmetry of increments of the martingale given the past, which is a reformulation of an earlier result by Ocone.

Maximal inequalities for a series of continuous local martingales

Let {Xj=(Xtj,ℱt),j≥1} be a sequence of continuous local martingales and {〈Xj〉} the corresponding sequence of their quadratic variation processes and let Hn(x,y),n=1,2,… be the Hermite polynomials with parametric variable y. In this paper, we consider the series ∑j=1∞Hn2(Xj,〈Xj〉) of the continuous local martingales Hn(Xj,〈Xj〉)=(Hn(Xtj,〈Xj〉t),ℱt)t≥0, j=1,2,…, and its discrete analogue, and obtain some maximal inequalities.

Itô's formula for C 1,λ -functions of a càdlàg process and related calculus

This article develops a framework of stochastic calculus with respect to a cadlag finite quadratic variation process. We apply it to the study of a generalization of a semimartingale driven SDE studied by Kurtz, Pardoux and Protter [KPP]. We prove an Ito's formula for functions f(X) of a semimartingale with jumps when f has weak smoothness properties. Examples of X for which this formula is valid are time reversible semimartingales and solutions of [KPP] equations driven by Levy processes, provided the sum of the absolute values of the jumps, raised to the power 1 + λ, is a.s. finite, where λ takes values between 0 and 1.