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Data and methods for A threshold model for local volatility: evidence of leverage and mean reversion effects on historical data [Données et méthodes pour "A threshold model for local volatility: evidence of leverage and mean reversion effects on historical data"]
TL;DR: In this paper, the authors presented the methodology and numerical results for 21 stock prices under the assumption they follow a Drifted Geometric Oscillating Brownian motion model, taking leverage and mean-reversion effects into account.
Abstract: This technical report presents the methodology and the numerical results for 21 stock prices under the assumption they follow a Drifted Geometric Oscillating Brownian motion model. Such a model takes leverage and mean-reversion effects into account. This report completes the article "A threshold model for local volatility: evidence of leverage and mean reversion effects on historical data"
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163 citations
TL;DR: In this paper, a wide variety of inequalities which are established for either stochastic processes or sequences of random variables are presented, as well as relaxation of a spectral gap assumption.
Abstract: Lévy Processes” (eight papers), “III. Empirical Processes” (four papers), and “IV. Stochastic Differential Equations” (four papers). Here are some comments about the individual papers: In I.2 (paper 2 of Part I) the covariance representation method, which relies on Clark’s formula on path spaces, is used to obtain concentration inequalities for functionals of Brownian motion on a manifold, allowing one to obtain tail estimates for this Brownian motion. In I.4 a transportation inequality for the canonical Gaussian measure in R is obtained and applied to Khintchine–Kahane inequalities for norms of random series with nonsymmetric Bernoulli coefficients. In II.1 exponential inequalities for U -statistics of order two are presented; these rely upon the Talagrand inequality for empirical processes but also use martingale type inequalities. In II.2 the unconditional convergence of a Gaussian [and, more generally, independent, identically distributed (iid)] series in a Banach space is studied. Applications to Karhunen–Love representations of Gaussian processes are given. In II.3 estimates of tail properties and moments of multidimensional chaos generated by positive random variables with log concave tails are given. In II.4 a quantitative technique for studying the asymptotic distribution of sequences of Markov processes in infinite dimensions is proposed. The proof relies on the properties of an associated sequence of exponential martingales. In II.5 it is shown that a moving average process driven by a symmetric Lévy process and with a kernel with finite total 2-variation admits an almost surely bounded version. In II.6 a Markovian approach to the entropic convergence in the central limit theorem is presented. The emphasis is on the speed of convergence, as well as relaxing a spectral gap assumption. In II.7 a new version of the Khintchine–Kahane inequality for general Bernoulli random variables is presented with the help of hypercontractive methods. In III.1 necessary and sufficient conditions for the moderate deviations of empirical processes and sums of iid random vectors on a separable Banach space are given. In III.2 exponential concentration inequalities for subadditive functions of independent random variables are obtained. As a consequence, Talagrand’s inequality for empirical processes is refined thanks to further developments of the entropy method introduced by M. Ledoux. In III.3 ratio limit theorems for empirical processes are obtained with the help of concentration inequalities. In III.4 asymptotic distributions of trimmed Wasserstein distances between the true and the empirical distribution function are obtained via weighted approximation results for uniform empirical processes. In IV.1 sharp rates of convergence for splitting-up approximations of stochastic partial differential equations are obtained. The error is estimated in terms of Sobolev’s norm. In IV.4 the existence and uniqueness of a strong solution for a stochastic differential equation driven by a fractional Brownian motion with Hurst index H < 1/2 and with a possibly time-dependent drift which satisfies a suitable integrability condition is obtained. In short, the book presents a wide variety of inequalities which are established for either stochastic processes or sequences of random variables. In 2006 this is still an active domain of research in transportation problems.
141 citations
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TL;DR: McFedries as mentioned in this paper examines new words and phrases that have jumped down from their technological niches and are poised to set up shop in the broader piazza of general language use.
Abstract: 80 IE E E S P E C T R U M • Ju n e 20 02 T echnology is a language-generating machine. With letters, phonemes, prefixes, suffixes, and words as its raw materials, technology constantly manufactures shiny new acronyms, words, and phrases to describe its onrush of new ideas, processes, and products. While technology has always stamped out new words at an impressive clip, most of these terms remain warehoused within the narrow tech communities that defined them. Some of them, however, are shipped out on the linguistic equivalents of planes, trains, and automobiles and get distributed far and wide. A few even morph into general-purpose words and phrases. For example, nuclear technology provided us with such terms as ground zero, fallout, and meltdown; radio spun off flip side, fine-tune, and stereo; aviation contributed push the envelope, automatic pilot, bail out, and gremlin; the car industry donated spark plug, bypass, blow a gasket, and rev up; and even earlier, the railroads gave us derail, sidetrack, streamline, and pick up steam. But though technology has always been a kind of new-word assembly line, what’s different these days is that technology is cranking out fresh terms at a rate that is downright exponential. That’s not because people are making up new words in their spare time, but because we now have more technology than ever. We don’t just have telephones, we have mobile phones, pagers, satellite phones, and wireless devices. We don’t just have computers, we have desktops, servers, notebooks, palmtops, PDAs, and Internet appliances. New gadgets, new technologies, new services, and new ideas stride purposefully down Technology Road, each one pulling a bright red wagon full of newly minted words and phrases. These now have a super-efficient method of propagation: the electronic byways of e-mail, chat rooms, instant messaging, and the Web. In the pre-Internet world, new words would tend to stay within the cultural tributary that coined them; only a few would get swept into the mainstream. Today there is a subculture—the Internet and its adjunct technologies—of hundreds of millions of people, that by definition is part of the mainstream. This means it doesn’t take much for new words and phrases to catch on. Keeping up with this deluge of newfangled technical terms will be the future focus of the Technically Speaking column. I’ll examine new words and phrases that have jumped down from their technological niches and are poised to set up shop in the broader piazza of general language use. This column introduces Paul McFedries as the new author of Technically Speaking, which appears every other month. Readers are invited to correspond with him at techspkg@ieee.org. P E T E R H O R V A T H TECHNICALLY SPEAKING
83 citations
03 Apr 2014
TL;DR: In this paper, the authors investigate hybrid switching diffusions, which are stochastic models whose dynamics switch depending on the state/regime of the system, and apply them to models of rainfall, convection and water vapor, where two states/regimes are considered: precipitation and nonprecipitation.
Abstract: This paper investigates stochastic models whose dynamics switch depending on the state/regime of the system. Such models have been called “hybrid switching diffusions” and exhibit “sliding dynamics” with noise. Here the aim is an application to models of rainfall, convection, and water vapor, where two states/regimes are considered: precipitation and nonprecipitation. Regime changes are modeled with a “trigger function,” and four trigger models are considered: deterministic triggers (i.e., Heaviside function) and stochastic triggers (finite-state Markov jump process), with either a single threshold for regime transitions or two distinct thresholds (allowing for hysteresis). These triggers are idealizations of those used in convective parameterizations of global climate models, and they are investigated here in a model for a single atmospheric column. Two types of results are presented here. First, exact statistics are presented for all four models, and a comparison indicates how the trigger choice influen...
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References
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Proceedings Article•
01 Jan 1973
TL;DR: The classical maximum likelihood principle can be considered to be a method of asymptotic realization of an optimum estimate with respect to a very general information theoretic criterion to provide answers to many practical problems of statistical model fitting.
Abstract: In this paper it is shown that the classical maximum likelihood principle can be considered to be a method of asymptotic realization of an optimum estimate with respect to a very general information theoretic criterion. This observation shows an extension of the principle to provide answers to many practical problems of statistical model fitting.
18,539 citations
TL;DR: In this paper, the authors present a set of stylized empirical facts emerging from the statistical analysis of price variations in various types of financial markets, including distributional properties, tail properties and extreme fluctuations, pathwise regularity, linear and nonlinear dependence of returns in time and across stocks.
Abstract: We present a set of stylized empirical facts emerging from the statistical analysis of price variations in various types of financial markets. We first discuss some general issues common to all statistical studies of financial time series. Various statistical properties of asset returns are then described: distributional properties, tail properties and extreme fluctuations, pathwise regularity, linear and nonlinear dependence of returns in time and across stocks. Our description emphasizes properties common to a wide variety of markets and instruments. We then show how these statistical properties invalidate many of the common statistical approaches used to study financial data sets and examine some of the statistical problems encountered in each case.
2,994 citations
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TL;DR: In this paper, the random walk model is strongly rejected for the entire sample period (1962-1985) and for all sub-periods for a variety of aggregate returns indexes and size-sorted portfolios.
Abstract: In this paper, we test the random walk hypothesis for weekly stock market returns by comparing variance estimators derived from data sampled at different frequencies. The random walk model is strongly rejected for the entire sample period (1962-1985) and for all sub-periods for a variety of aggregate returns indexes and size-sorted portfolios. Although the rejections are largely due to the behavior of small stocks, they cannot be ascribed to either the effects of infrequent trading or time-varying volatilities. Moreover, the rejection of the random walk cannot be interpreted as supporting a mean-reverting stationary model of asset prices, but is more consistent with a specific nonstationary alternative hypothesis.
2,920 citations
TL;DR: The authors examined the relation between the variance of equity returns and several explanatory variables and found that equity variances have a strong positive association with both financial leverage and, contrary to the predictions of the options literature, interest rates.
2,469 citations
TL;DR: In this paper, the moments and the asymptotic distribution of the realized volatility error were derived under the assumption of a rather general stochastic volatility model, and the difference between realized volatility and the discretized integrated volatility (which is called actual volatility) were estimated.
Abstract: Summary. The availability of intraday data on the prices of speculative assets means that we can use quadratic variation-like measures of activity in financial markets, called realized volatility, to study the stochastic properties of returns. Here, under the assumption of a rather general stochastic volatility model, we derive the moments and the asymptotic distribution of the realized volatility error—the difference between realized volatility and the discretized integrated volatility (which we call actual volatility). These properties can be used to allow us to estimate the parameters of stochastic volatility models without recourse to the use of simulation-intensive methods.
2,207 citations