time and foreign exchange markets the definition of time is still an open question when one deals with high frequency time series if time is simply the calendar time prices can be modeled as continuous random processes and values resulting from transactions or given quotes are discrete samples of this underlying dynamics on the contrary if one takes the business time point of view price dynamics is a discrete random process and time is simply the ordering according to which prices are quoted in the market in this paper we suggest that the business time approach is perhaps a better way of modeling price dynamics than calendar time this conclusion comes from testing probability densities and conditional variances predicted by the two models against the experimental ones the data set we use contains the dem usd exchange quotes provided to us by olsen associates during a period of one year from january to december 1998 in this period 1 620 843 quotes entries in the efx system were recorded

spectral gmm estimation of continuous time processes abstract this paper derives a methodology for the estimation of continuous time stochastic models based on the characteristic function the estimation method does not require discretization of the stochastic process and it is simple to apply in practice the method is essentially generalized method of moments on the complex plane hence it shares the efficiency and distribution properties of gmm estimators we illustrate the method with some applications to relevant estimation problems in continuous time finance we estimate a model of stochastic volatility a jump diffusion model with constant volatility and a model that nests both the stochastic volatility model and the jump diffusion model we find that negative jumps are important to explain skewness and asymmetry in excess kurtosis of the stock return distribution while stochastic volatility is important to capture the overall level of this kurtosis positive jumps are not statistically significant once we allow for stochastic volatility in the model we also estimate a non affine model of stochastic volatility and find that the power of the diffusion coefficient appears to be between one and two rather than the value of one half that leads to the standard affine stochastic volatility model however we find that including jumps into this non affine stochastic volatility model reduces the power of the diffusion coefficient to one half finally we offer an explanation for the observation that the estimate of persistence in stochastic volatility increases dramatically as the frequency of the observed data falls based on a multiple factor stochastic volatility model

term structure models and the zero bound an empirical investigation of japanese yields abstract when japanese short term bond yields were near their zero bound yields on long term bonds showed substantial fluctuation and there was a strong positive relationship between the level of interest rates and yield volatilities risk premiums we explore whether several families of dynamic term structure models that enforce a zero lower bound on short rates imply conditional distributions of japanese bond yields consistent with these patterns multi factor shadow rate and quadratic gaussian models evaluated at their maximum likelihood estimates capture many features of the data furthermore model implied risk premiums track realized excess returns during extended periods of near zero short rates in contrast the conditional distributions implied by non negative affine models do not match their sample counterparts and standard gaussian affine models generate implausibly large negative risk premiums

application of zhangs square root law and herding to financial markets abstract we apply an asymmetric version of kirman s herding model to volatile financial markets in the relation between returns and agent concentration we use the square root law proposed by zhang this can be derived by extending the idea of a critical mean field theory suggested by plerou et al we show that this model is equivalent to the so called 3 2 model of stochastic volatility the description of the unconditional distribution for the absolute returns is in good agreement with the dax independent whether one uses the square root or a conventional linear relation only the statistic of extreme events prefers the former the description of the autocorrelations are in much better agreement for the square root law the volatility clusters are described by a scaling law for the distribution of returns conditional to the value at the previous day in good agreement with the data

asymptotic nonequivalence of garch models and diffusions this paper investigates the statistical relationship of the garch model and its diffusion limit regarding the two types of models as two statistical experiments formed by discrete observations from the models we study their asymptotic equivalence in terms of le cam s deficiency distance to our surprise we are able to show that the garch model and its diffusion limit are asymptotically equivalent only under deterministic volatility with stochastic volatility due to the difference between the structure with respect to noise propagation in their conditional variances their likelihood processes asymptotically behave quite differently and thus they are not asymptotically equivalent this stochastic nonequivalence discredits a general belief that the two types of models are asymptotically equivalent in all respects and warns against the common financial practice that applies statistical inferences derived under the garch model to its diffusion limit

nonparametric specification tests for stochastic volatility models based on volatility density this paper develops a specification test for stochastic volatility models by comparing the nonparametric kernel deconvolution density estimator of an integrated volatility density with its parametric counterpart l2 distance is used to measure the discrepancy the asymptotic null distributions of the test statistics are established and the asymptotic power functions are computed through monte carlo simulations the size and power properties of the test statistics are studied the tests are applied to an empirical example

nonlinear small data scattering for the wave and klein gordon equation on etudie l operateur diffusion qui appartient a la paire d equations u tt au f u 0 et u tt au 0 a represente l operateur j 1 n 2 x j 2 m 2 m r et f u u 1 u ou r 1

algorithms for dominating clique problems we handle in this paper three dominating clique problems namely the decision problem to detect whether a graph has a dominating clique and two optimization versions asking to compute a maximum and a minimum size dominating clique of a graph g if g has a dominating clique for the three problems we propose exact moderately exponential algorithms with worst case running time upper bounds improving those by kratsch and liedloff d kratsch m liedloff an exact algorithm for the minimum dominating clique problem theoret comput sci 385 1 3 2007 226 240 we then study the three problems in sparse and dense graphs also providing improved running time upper bounds finally we propose some exponential time approximation algorithms for the optimization versions

exact simulation of the ornstein uhlenbeck driven stochastic volatility model abstract this paper proposes a novel exact simulation method for the ornstein uhlenbeck driven stochastic volatility model to accomplish this goal our task hinges on properly handling the ornstein uhlenbeck volatility process the major challenge involves conditionally sampling the integral of its square with respect to time given its marginal state as well as its integral with respect to time we thus derive a closed form laplace transform of this conditional distribution via the techniques of changing probability measure as well as analytical extension then we obtain the corresponding conditional cumulative distribution function via fourier transform inversion and finally sample the distribution via the inverse transform method we show that our method achieves a faster convergence rate of root mean square errors comparing with euler discretization method and apply it in the valuation of discretely monitored path dependent options

a model for stock returns and volatility we prove that student s t distribution provides one of the better fits to returns of s p component stocks and the generalized inverse gamma distribution best fits vix and vxo volatility data we further prove that stock returns are best fit by the product distribution of the generalized inverse gamma and normal distributions we find brown noise in vix and vxo time series and explain the mean and the variance of the relaxation times on approach to the steady state distribution

non gaussian garch option pricing models and their diffusion limits this paper investigates the weak convergence of general non gaussian garch models together with an application to the pricing of european style options determined using an extended girsanov principle and a conditional esscher transform as the pricing kernel candidates applying these changes of measure to asymmetric garch models sampled at increasing frequencies we obtain two risk neutral families of processes which converge to different bivariate diffusions which are no longer standard hull white stochastic volatility models regardless of the innovations used the garch implied diffusion limit based on the esscher transform can be obtained by applying the minimal martingale measure under the physical measure however we further show that for skewed garch driving noise the risk neutral diffusion limit of the extended girsanov principle exhibits a non zero market price of volatility risk which is proportional to the market price of the equity risk where the constant of proportionality depends on the skewness and kurtosis of the underlying distribution our theoretical results are further supported by numerical simulations and a calibration exercise to observed market quotes

the focusing energy critical nonlinear schrodinger equation in dimensions five and higher we consider the focusing energy critical nonlinear schr odinger equation iu t delta u u 4 over d 2 u in dimensions d geq 5 we prove that if a maximal lifespan solution u colon i times bbb r d to bbb c obeys sup t in i nabla u t 2 nabla w 2 then it is global and scatters both forward and backward in time here w denotes the ground state which is a stationary solution of the equation in particular if a solution has both energy and kinetic energy less than those of the ground state w at some point in time then the solution is global and scatters we also show that any solution that blows up with bounded kinetic energy must concentrate at least the kinetic energy of the ground state similar results were obtained by kenig and merle for spherically symmetric initial data and dimensions d 3 4 5

volatility forecasting and microstructure noise abstract it is common practice to use the sum of frequently sampled squared returns to estimate volatility yielding the so called realized volatility unfortunately returns are contaminated by market microstructure noise several noise corrected realized volatility measures have been proposed we assess to what extent correction for microstructure noise improves forecasting future volatility using a mixed data sampling midas regression framework we study the population prediction properties of various realized volatility measures assuming i i d microstructure noise next we study optimal sampling issues theoretically when the objective is forecasting and microstructure noise contaminates realized volatility we distinguish between conditional and unconditional optimal sampling schemes and find that conditional optimal sampling seems to work reasonably well in practice

closed form expansion conditional expectation and option valuation enlightened by the theory of watanabe watanabe s 1987 analysis of wiener functionals malliavin calculus and its applications to heat kernels ann probab 15 1 39 for analyzing generalized random variables and its further development in yoshida yoshida n 1992a asymptotic expansions for statistics related to small diffusions j japan statist soc 22 139 159 takahashi takahashi a 1995 essays on the valuation problems of contingent claims ph d thesis haas school of business university of california berkeley takahashi a 1999 an asymptotic expansion approach to pricing contingent claims asia pacific financial markets 6 115 151 as well as kunitomo and takahashi kunitomo n takahashi a 2001 the asymptotic expansion approach to the valuation of interest rate contingent claims math finance 11 1 117 151 kunitomo n takahashi a 2003 on validity of the asymptotic expansion approach in contingent claim analysis ann appl probab 13 3 914 952 etc we focus on a wide range of mult

contemporaneous asymmetry in garch processes abstract the paper introduces a new concept of asymmetry contemporaneous asymmetry in conditional heteroskedasticity models we propose an original class of models aimed to capture the leverage effect contemporaneous asymmetry as well as time varying skewness and kurtosis not only past up and down moves have different impacts on the conditional variance but also positive and negative changes are governed by different conditional variances we give conditions for the existence of a second order and strictly stationary solution the paper also provides consistency results on the quasi maximum likelihood estimation finally an empirical analysis on the french cac 40 stock index is proposed

generalized strichartz inequalities for the wave equation strichartz inequalities for the wave equation u f are estimates of the solution u of the cauchy problem for that equation in the form of space time integral norms in terms of similar norms of the inhomogeneity f and of suitable norms of the initial data

efficient method of moments estimation of a stochastic volatility model a monte carlo study abstract we perform an extensive monte carlo study of efficient method of moments emm estimation of a stochastic volatility model emm uses the expectation under the structural model of the score from an auxiliary model as moment conditions we examine the sensitivity to the choice of auxiliary model using arch garch and egarch models for the score as well as nonparametric extensions emm efficiency approaches that of maximum likelihood for larger sample sizes inference is sensitive to the choice of auxiliary model in small samples but robust in larger samples specification tests and t tests show little size distortion

global wellposedness of defocusing critical nonlinear schrodinger equation in the radial case is globally wellposed in time more precisely we obtain a unique solution u u ch1 0 such that for all time u t depends continuously on the data in fact the dependence is even real analytic here moreover there is scattering for t the same statement holds for radial data h s 1 and proves in particular global existence of classical solutions in the radially symmetric case also this issue was open thus this is the analogue for nls of the result for the wave equation with quintic nonlinearity obtained by m struwe str in the radial case and by m grillakis gr s s in general in the case of the wave equation the proof is based on the following two different facts

on existence and scattering with minimal regularity for semilinear wave equations abstract we prove existence and scattering results for semilinear wave equations with low regularity data we also determine the minimal regularity that is needed to ensure local existence and well posedness and we give counterexamples to well posedness more specifically we show that equations of the type u u p with initial data u u t in r n 1 r n have a local solution if p n and we construct counterexamples if p n the existence results rely on mixed norm space time estimates of strichartz type

global well posedness and scattering for the energy critical nonlinear schrodinger equation in r 3 we obtain global well posedness scattering and global l 10 spacetime bounds for energy class solutions to the quintic defocusing schrodinger equa tion in r 1 3 which is energy critical in particular this establishes global existence of classical solutions our work extends the results of bourgain 4 and grillakis 20 which handled the radial case the method is similar in spirit to the induction on energy strategy of bourgain 4 but we perform the induction analysis in both frequency space and physical space simultaneously and replace the morawetz inequality by an interaction variant first used in 12 13 the principal advantage of the interaction morawetz estimate is that it is not localized to the spatial origin and so is better able to handle nonradial solutions in particular this interaction estimate together with an almost conservation argument controlling the movement of l 2 mass in fre quency space rules out the possibility of energy concentration

a martingale approach for testing diffusion models based on infinitesimal operator abstract i develop an omnibus specification test for diffusion models based on the infinitesimal operator the infinitesimal operator based identification of the diffusion process is equivalent to a martingale hypothesis for the processes obtained by a transformation of the original diffusion model my test procedure is then constructed by checking the martingale hypothesis via a multivariate generalized spectral derivative based approach that delivers a n 0 1 asymptotical null distribution for the test statistic the infinitesimal operator of the diffusion process is a closed form function of drift and diffusion terms consequently my test procedure covers both univariate and multivariate diffusion models in a unified framework and is particularly convenient for the multivariate case moreover different transformed martingale processes contain separate information about the drift and diffusion specifications this motivates me to propose a separate inferential test procedure to explore the sources of rejection when a parametric form is rejected simulation studies show that the proposed tests have reasonable size and excellent power performance an empirical application of my test procedure using eurodollar interest rates finds that most popular short rate models are rejected and the drift misspecification plays an important role in such rejections

a new delta expansion for multivariate diffusions via the ito taylor expansion abstract in this paper we develop a new delta expansion approach to deriving analytical approximation to the transition densities of multivariate diffusions using the ito taylor expansion of the conditional expectation of the dirac delta function our approach yields an explicit recursive formulas for the expansion coefficients and is universally applicable for a wide spectrum of models particularly the time inhomogeneous non affine irreducible multivariate diffusions we show that this new approach can be viewed as an extension of ait sahalia 2002 and lee et al 2014 to the case of multivariate models the derived expansions are proved to converge to the true probability density as the observational time interval shrinks the obtained approximations can thereby be used to carry out the maximum likelihood estimation for the diffusions with discretely observed data extensive numerical experiments demonstrate the accuracy and effectiveness of our approach

global well posedness scattering and blow up for the energy critical focusing non linear wave equation we study the energy critical focusing non linear wave equation with data in the energy space in dimensions 3 4 and 5 we prove that for cauchy data of energy smaller than the one of the static solution w which gives the best constant in the sobolev embedding the following alternative holds if the initial data has smaller norm in the homogeneous sobolev space h1 than the one of w then we have global well posedness and scattering if the norm is larger than the one of w then we have break down in finite time

approximating max sat by moderately exponential and parameterized algorithms we study approximation of the max sat problem by moderately exponential algorithms the general goal of the issue of moderately exponential approximation is to catch up on polynomial inapproximability by providing algorithms achieving with worst case running times importantly smaller than those needed for exact computation approximation ratios unachievable in polynomial time we develop several approximation techniques that can be applied to max sat in order to get approximation ratios arbitrarily close to 1

endpoint strichartz estimates we prove an abstract strichartz estimate which implies previously unknown endpoint strichartz estimates for the wave equation in dimension n 4 and the schrodinger equation in dimension n 3 three other applications are discussed local existence for a nonlinear wave equation and strichartz type estimates for more general dispersive equations and for the kinetic transport equation

volatility regressions with fat tails abstract nowadays a common method to forecast integrated variance is to use the fitted value of a simple ols autoregression of the realized variance however non parametric estimates of the tail index of this realized variance process reveal that its second moment is possibly unbounded in this case the behavior of the ols estimators and the corresponding statistics are unclear we prove that when the second moment of the spot variance is unbounded the slope of the spot variance s autoregression converges to a random variable as the sample size diverges the same result holds when one uses the integrated or realized variance instead of the spot variance we then consider the class of diffusion variance models with an affine drift a class which includes garch and cev processes and we prove that iv estimation with adequate instruments provide consistent estimators of the drift parameters as long as the variance process has a finite first moment regardless of the existence of the second moment in particular for the garch diffusion model with fat tails an iv estimation where the instrument equals the sign of the centered lagged value of the variable of interest provides consistent estimators simulation results corroborate the theoretical findings of the paper

particle filters for continuous likelihood evaluation and maximisation in this paper a method is introduced for approximating the likelihood for the unknown parameters of a state space model the approximation converges to the true likelihood as the simulation size goes to infinity in addition the approximating likelihood is continuous as a function of the unknown parameters under rather general conditions the approach advocated is fast and robust and it avoids many of the pitfalls associated with current techniques based upon importance sampling we assess the performance of the method by considering a linear state space model comparing the results with the kalman filter which delivers the true likelihood we also apply the method to a non gaussian state space model the stochastic volatility model finding that the approach is efficient and effective applications to continuous time finance models and latent panel data models are considered two different multivariate approaches are proposed the neoclassical growth model is considered as an application

asymptotic filtering theory for univariate arch models this paper builds on this earlier work by deriving the asymptotic distribution of the measurement error this allows us to approximate the measurement accuracy of arch conditional variance estimates and compare the efficiency achieved by different arch models we are also able to characterize the relative importance of different kinds of misspecification for example we show that misspecifying conditional means adds only trivially at least asymptotically to measurement error while other factors for example capturing the leverage effect accommodating thick tailed residuals and correctly modelling the variability of the conditional variance process are potentially much more important third we are able to characterize a class of asymptotically optimal arch conditional variance estimates

augmented garch p q process and its diffusion limit abstract a family of parametric garch models defined in terms of an auxiliary process and referred to as the augmented garch process is proposed the strict stationarity of the augmented garch process is characterized and this process is shown to contain many existing parametric garch models the augmented garch process can serve as a general alternative for lagrange multiplier test of many existing garch specifications the diffusion limit of the augmented garch process is shown to contain many bivariate diffusion processes that are commonly used for modeling stochastic volatility in the finance literature this convergence result generalizes that of nelson 1990a to cover a substantially larger class of garch 1 1 models and also extends to the garch p q specification the augmented garch process can be used as a direct approximation to the stochastic volatility models or as the score generator in the efficient method of moments gallant and tauchen 1996 estimation of these models

fractionally integrated generalized autoregressive conditional heteroskedasticity abstract the new class of fractionally integrated generalized autoregressive conditionally heteroskedastic figarch processes is introduced the conditional variance of the process implies a slow hyperbolic rate of decay for the influence of lagged squared innovations unlike i d processes for the mean maximum likelihood estimates mle of the figarch parameters are argued to be t 1 2 consistent the small sample behavior of an approximate mle procedure is assessed through a simulation study which also documents how the estimation of a standard garch model tends to produce integrated or igarch like estimates an empirical example with daily deutschmark u s dollar exchange rates illustrates the practical relevance of the new figarch specification

bayes factors for discrete observations from diffusion processes summary we present an approach to model selection for a time series of data on a fine time scale the underlying process generating the data is modelled as a continuous time stochastic process the underlying continuous processes are assumed to be diffusions with time varying drift and diffusion coefficient several approaches to modelling the diffusion coefficient are described to perform model selection we propose an approximation to the bayes factor that uses only the discrete data we illustrate our approach for several well known processes including brownian motion with drift the ornstein uhlenbeck process a mean reversion process with drift exponential brownian motion and a logistic growth model finally we apply our technique to data from the standard poor s 500 stock index by comparing a random walk to a mean reversion model

model for non gaussian intraday stock returns stock prices are known to exhibit non gaussian dynamics and there is much interest in understanding the origin of this behavior here we present a model that explains the shape and scaling of the distribution of intraday stock price fluctuations called intraday returns and verify the model using a large database for several stocks traded on the london stock exchange we provide evidence that the return distribution for these stocks is non gaussian and similar in shape and that the distribution appears stable over intraday time scales we explain these results by assuming the volatility of returns is constant intraday but varies over longer periods such that its inverse square follows a gamma distribution this produces returns that are student distributed for intraday time scales the predicted results show excellent agreement with the data for all stocks in our study and over all regions of the return distribution

a semiparametric stochastic volatility model abstract in this paper the correlation structure in the classical leverage stochastic volatility sv model is generalized based on a linear spline in the new model the correlation between the return and volatility innovations is time varying and depends nonparametrically on the type of news arrived to the market theoretical properties of the proposed model are examined the model estimation and comparison are conducted by bayesian methods the performance of the estimates are examined in simulations the new model is fitted to daily and weekly us data and compared with the classical sv and garch models in terms of their in sample and out of sample performances empirical results suggest evidence in favor of the proposed model in particular the new model finds strong evidence of time varying leverage effect in individual stocks when the classical model fails to identify the leverage effect

testing whether the underlying continuous time process follows a diffusion an infinitesimal operator based approach abstract we develop a nonparametric test to check whether a process can be represented by a stochastic differential equation driven only by a brownian motion our testing procedure utilizes the infinitesimal operator based martingale characterization combined with a generalized spectral approach such a testing procedure is feasible and convenient because the infinitesimal operator of the diffusion process has a closed form expression the proposed test is applicable to both univariate and multivariate processes and has an n 0 1 limit distribution under the diffusion hypothesis simulation and empirical studies show that the proposed test has reasonable performance in small samples

filtering for partially observed diffusion and its applications abstract this paper provides an analytic method of filtering for partially observed diffusions which can be also used for parameter estimation with the quasi maximum likelihood method the filtering is shown to have consistency in a weak sense in addition using the stochastic volatility models a comparative simulation study is carried out to see how well the proposed method numerically works the performance of the proposed method is basically better than that of the extended kalman filtering

maximum likelihood estimation of partially observed diffusion models this paper develops a maximum likelihood ml method to estimate partially observed diffusion models based on data sampled at discrete times the method combines two techniques recently proposed in the literature in two separate steps in the first step the closed form approach of ait sahalia 2008 is used to obtain a highly accurate approximation to the joint transition probability density of the latent and the observed states in the second step the efficient importance sampling technique of richard and zhang 2007 is used to integrate out the latent states thereby yielding the likelihood function using both simulated and real data we show that the proposed ml method works better than alternative methods the new method does not require the underlying diffusion to have an affine structure and does not involve infill simulations therefore the method has a wide range of applicability and its computational cost is moderate

dispersion of small amplitude solutions of the generalized korteweg de vries equation abstract we study the long time behavior of small solutions of the initial value problem for the generalized korteweg de vries equation t u x 3 u x f u 0 gkdv u x 0 g x for the case where f w w s with s 1 4 23 57 3 8625 our results imply that if g l11 g l22 is sufficiently small then sup r 1 t 1 3 u t l in particular the solution tends to zero in the supremum norm the proofs make use of duhamel s formula and dispersion estimates for the linear propagator as well as chain and leibniz rules for fractional derivatives of compositions d f u lp and products d fg lp 0

temporal aggregation of multivariate garch processes this paper derives results for the temporal aggregation of multivariate garch 1 1 processes in the general vector specification it is shown that the class of weak multivariate garch 1 1 processes is closed under temporal aggregation fourth moment characteristics turn out to be crucial for the low frequency dynamics for both stock and flow variables in some aspects the aggregation characteristics of multivariate garch processes are shown to be different from those of vector autoregressive moving average processes a numerical example illustrates some of the results

market based estimation of stochastic volatility models abstract we propose a method for estimating stochastic volatility models by adapting the hjm approach to the case of volatility derivatives we characterize restrictions that observed variance swap dynamics have to satisfy to prevent arbitrage opportunities when the drift of variance swap rates are affine under the pricing measure we obtain closed form expressions for those restrictions and formulas for forward variance curves using data on the s p500 index and variance swap rates on different time to maturities we find that linear mean reverting one factor models provide inaccurate representation of the dynamics of the variance swap rates while two factor models significantly outperform the former both in and out of sample

the global cauchy problem for the critical non linear wave equation abstract we study the global cauchy problem for the non linear wave equation p 1 0 for the critical value p n 2 n 2 in space dimension n 3 we identify a weak space time integrability property stip of the solutions and prove that it is sufficient to ensure the uniqueness of weak solutions the global existence of finite energy solutions with the naturally associated stip and the global existence of regular solutions with some n dependent restrictions on the regularity for spherically symmetric solutions we prove that the previous crucial stip follows from the morawetz inequality actually in a much stronger form than necessary thereby proving that all the previous results hold in the spherically symmetric case

filtering and forecasting with misspecified arch models i getting the right variance with the wrong model this paper investigates the properties of the conditional covariance estimates generated by a misspecified arch model for example suppose that we observe a diffusion process at discrete time intervals of length h for each h we use a garch 1 1 model to estimate the instantaneous conditional covariance matrix of the diffusion under mild regularity conditions the difference between these conditional covariance estimates and the true conditional covariance converges to zero in probability as h 0 many other arch models for example exponential arch have similar consistency properties this may well account for the success of arch models in short term forecasting using high frequency data since even misspecified arch models can produce good estimates of volatility

arch models as diffusion approximations abstract this paper investigates the convergence of stochastic difference equations e g arch to stochastic differential equations as the length of the discrete time intervals between observations goes to zero these results are applied to the garch 1 1 model of bollerslev 1986 and to the ar 1 exponential arch model of nelson 1989 in their continuous time limits the conditional variance processes in these models have stationary distributions that are inverted gamma and lognormal respectively in addition a class of diffusion approximations based on the exponential arch model is developed

do high frequency measures of volatility improve forecasts of return distributions many finance questions require the predictive distribution of returns we propose a bivariate model of returns and realized volatility rv and explore which features of that time series model contribute to superior density forecasts over horizons of 1 to 60 days out of sample this term structure of density forecasts is used to investigate the importance of the intraday information embodied in the daily rv estimates the functional form for log rv dynamics the timing of information availability and the assumed distributions of both return and log rv innovations we find that a joint model of returns and volatility that features two components for log rv provides a good fit to s p 500 and ibm data and is a significant improvement over an egarch model estimated from daily returns

generalized autoregressive conditional heteroskedasticity a natural generalization of the arch autoregressive conditional heteroskedastic process introduced in engle 1982 to allow for past conditional variances in the current conditional variance equation is proposed stationarity conditions and autocorrelation structure for this new class of parametric models are derived maximum likelihood estimation and testing are also considered finally an empirical example relating to the uncertainty of the inflation rate is presented

an exact algorithm for minimum distortion embedding let g be an unweighted connected graph on n vertices we show that an embedding of the shortest path metric of g into the line with minimum distortion can be found in time 5 n o n this is the first algorithm breaking the trivial n barrier

bootstrap specification tests for diffusion processes this paper discusses specification tests for diffusion processes in the one dimensional case our proposed test is closest to the nonparametric test of ait sahalia rev financ stud 9 1996 385 however we compare cdfs instead of densities in the multidimensional and or multifactor case our proposed test is based on comparison of the empirical cdf of actual data and the empirical cdf of simulated data asymptotically valid critical values are obtained using an empirical process version of the block bootstrap which accounts for parameter estimation error an example based on a simple version of the cox et al econometrica 53 1985 385 model is outlined and related monte carlo experiments are carried out

nonlinear garch model and 1 f noise abstract auto regressive conditionally heteroskedastic arch family models are still used by practitioners in business and economic policy making as a conditional volatility forecasting models furthermore arch models still are attracting an interest of the researchers in this contribution we consider the well known garch 1 1 process and its nonlinear modifications reminiscent of ngarch model we investigate the possibility to reproduce power law statistics probability density function and power spectral density using arch family models for this purpose we derive stochastic differential equations from the garch processes in consideration we find the obtained equations to be similar to a general class of stochastic differential equations known to reproduce power law statistics we show that linear garch 1 1 process has power law distribution but its power spectral density is brownian noise like however the nonlinear modifications exhibit both power law distribution and power spectral density of the 1 f form including 1 f noise

high frequency approximation of solutions to critical nonlinear wave equations this work is devoted to the description of bounded energy sequences of solutions to the equation 1 u u 4 0 in inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i up to remainder terms small in energy norm and in every strichartz norm the proof relies on scattering theory for 1 and on a structure theorem for bounded energy sequences of solutions to the linear wave equation in particular we infer the existence of an a priori estimate of strichartz norms of solutions to 1 in terms of their energy

minimal model of financial stylized facts in this work we propose a statistical characterization of a linear stochastic volatility model featuring inverse gamma stationary distribution for the instantaneous volatility we detail the derivation of the moments of the return distribution revealing the role of the inverse gamma law in the emergence of fat tails and of the relevant correlation functions we also propose a systematic methodology for estimating the parameters and we describe the empirical analysis of the standard poor s 500 index daily returns confirming the ability of the model to capture many of the established stylized facts as well as the scaling properties of empirical distributions over different time horizons

exact and superpolynomial approximation algorithms for the densest k subgraph problem abstract the densest k subgraph problem is a generalization of the maximum clique problem in which we are given a graph and a positive integer k and we search among all the subsets of k vertices of the input graph for the subset which induces the maximum number of edges densest k subgraph is a well known optimization problem with various applications as for example in the design of public encryption schemes the evaluation of certain financial derivatives the identification of communities with similar characteristics etc in this paper we first present algorithms for finding exact solutions for densest k subgraph which improve upon the standard exponential time complexity of an exhaustive enumeration by creating a link between the computation of an optimum for this problem to the computation of other graph parameters such as dominating set vertex cover longest path etc an fpt algorithm is also proposed which considers as a parameter the size of the minimum vertex cover finally we present several approximation algorithms which run in moderately exponential or parameterized time describing trade offs between complexity and approximability in contrast with most of the algorithms in the bibliography our algorithms need only polynomial space

exact and approximate bandwidth in this paper we gather several improvements in the field of exact and approximate exponential time algorithms for the bandwidth problem for graphs with treewidth t we present an o n o t 2 n exact algorithm moreover for any two positive integers k 2 r 1 we present a 2kr 1 approximation algorithm that solves bandwidth for an arbitrary input graph in o k n k 1 r time and polynomial space where by o we denote the standard big o notation but omitting polynomial factors finally we improve the currently best known exact algorithm for arbitrary graphs with an o 4 383 n time and space algorithm in the algorithms for the small treewidth we develop a technique based on the fast fourier transform parallel to the fast subset convolution techniques introduced by bjorklund et al this technique can be also used as a simple method of finding a chromatic number of all subgraphs of a given graph in o 2 n time and space what matches the best known results

an exponential time 2 approximation algorithm for bandwidth the bandwidth of a graph g on n vertices is the minimum b such that the vertices of g can be labeled from 1 to n such that the labels of every pair of adjacent vertices differ by at most b in this paper we present a 2 approximation algorithm for the bandwidth problem that takes worst case o 1 9797 n o 3 0 6 2 1 7 n time and uses polynomial space this improves both the previous best 2 and 3 approximation algorithms of cygan et al which have o 3 n and o 2 n worst case running time bounds respectively our algorithm is based on constructing bucket decompositions of the input graph a bucket decomposition partitions the vertex set of a graph into ordered sets called buckets of almost equal sizes such that all edges are either incident to vertices in the same bucket or to vertices in two consecutive buckets the idea is to find the smallest bucket size for which there exists a bucket decomposition the algorithm uses a divide and conquer strategy along with dynamic programming to achieve the improved time bound

alternative models for stock price dynamics abstract this paper evaluates the role of various volatility specifications such as multiple stochastic volatility sv factors and jump components in appropriate modeling of equity return distributions we use estimation technology that facilitates nonnested model comparisons and use a long data set which provides rich information about the conditional and unconditional distribution of returns we consider two broad families of models 1 the multifactor loglinear family and 2 the affine jump family both classes of models have attracted much attention in the derivatives and econometrics literatures there are various tradeoffs in considering such diverse specifications if pure diffusion sv models are chosen over jump diffusions it has important implications for hedging strategies if logarithmic models are chosen over affine ones it may seriously complicate option pricing comparing many different specifications of pure diffusion multifactor models and jump diffusion models we find that 1 log linear models have to be extended to two factors with feedback in the mean reverting factor 2 affine models have to have a jump in returns stochastic volatility or probably both models 1 and 2 are observationally equivalent on the data set in hand in either 1 or 2 the key is that the volatility can move violently as we obtain models with comparable empirical fit one must make a choice based on arguments other than statistical goodness of fit criteria the considerations include facility to price options to hedge and parsimony the affine specification with jumps in volatility might therefore be preferred because of the closed form derivatives prices

closing the garch gap continuous time garch modeling abstract it is the purpose of this paper to build a bridge between continuous time models which are central in the modern finance literature and weak garch processes in discrete time which often provide parsimonious descriptions of the observed data the properties of continuous time processes which exhibit garch type behavior at all discrete frequencies will be discussed several examples of such processes illustrate the general theory the class of continuous time garch models can be divided into two subclasses in the first group garch diffusions the sample paths are smooth and in the other group garch jump diffusions the sample paths are erratic a simple complete characterization of both types is given in terms of the kurtosis of the observed discrete time data these two groups of garch processes can be described by three and four coefficients respectively explicit formulas of all implied discrete time weak garch parameters are available moreover knowledge of the discrete time garch parameters at only one frequency completely determines the continuous time coefficients of the garch process so in estimating a continuous time garch process it suffices to estimate the discrete time garch parameters for the available data frequency the analysis carries over to models with an autoregressive component

stochastic interest rate volatility modeling with a continuous time garch 1 1 model in this work we develop a continuous time garch 1 1 cogarch 1 1 model driven by a nig levy process in order to analyze the volatility characteristics of turkish interest rates to our knowledge this is the first work considering nig cogarch modeling of interest rate data that utilizes the indirect inference method for parameter estimation the discrete time garch 1 1 model has been used as a skeleton for building the nig cogarch 1 1 model daily interest rates on the turkish two year maturity treasury bond for the period between 02 01 2006 and 31 12 2010 have been used for the analysis the empirical results show that the nig cogarch 1 1 model successfully captures the volatility clustering and heavy tailed behavior of the interest rate returns and yields better in sample estimations for conditional volatility in terms of forecast error statistics than the discrete time model

reconsidering the continuous time limit of the garch 1 1 process in this note we reconsider the continuous time limit of the garch 1 1 process let k and p2 denote respectively the cumulative returns and the volatility processes we consider the continuous time approximation of the couple k p2 we show that by choosing di erent parameterizations as a function of the discrete interval h we can obtain either a degenerate or a non degenerate di usion limit we then show that garch 1 1 processes can be obtained as euler approximations of degenerate di usions while any euler approximation of a non degenerate di usion is a stochastic volatility process 2000 elsevier science s a all rights reserved

temporal aggregation of volatility models in this paper we consider temporal aggregation of volatility models we introduce semiparametric volatility models termed square root stochastic autoregressive volatility sr sarv which are characterized by autoregressive dynamics of the stochastic variance our class encompasses the usual garch models and various asymmetric garch models moreover our stochastic volatility models are characterized by multiperiod conditional moment restrictions in terms of observables the sr sarv class is a natural extension of the class of weak garch models this extension has four advantages i we do not assume that fourth moments are finite ii we allow for asymmetries skewness leverage effect that are excluded from weak garch models iii we derive conditional moment restrictions and iv our framework allows us to study temporal aggregation of igarch models

dynamic integration of time and state domain methods for volatility estimation time and state domain methods are two common approaches to nonparametric prediction whereas the former uses data predominantly from recent history the latter relies mainly on historical information combining these two pieces of valuable information is an interesting challenge in statistics we surmount this problem by dynamically integrating information from both the time and state domains the estimators from these two domains are optimally combined based on a data driven weighting strategy which provides a more efficient estimator of volatility asymptotic normality is separately established for the time domain the state domain and the integrated estimators by comparing the efficiency of the estimators we demonstrate that the proposed integrated estimator uniformly dominates the other two estimators the proposed dynamic integration approach is also applicable to other estimation problems in time series extensive simulations are conducted to demonstrate that the newly proposed procedure outper

nonlinearity and temporal dependence nonlinearities in the drift and diffusion coefficients influence temporal dependence in diffusion models we study this link using three measures of temporal dependence mixing mixing and mixing stationary diffusions that are mixing have mixing coefficients that decay exponentially to zero when they fail to be mixing they are still mixing and mixing but coefficient decay is slower than exponential for such processes we find transformations of the markov states that have finite variances but infinite spectral densities at frequency zero the resulting spectral densities behave like those of stochastic processes with long memory finally we show how state dependent poisson sampling alters the temporal dependence

local scale models state space alternative to integrated garch processes state space alternative to autoregressive conditional heteroskedasticity models are proposed the initial model which is labelled the gaussian local scale model has a measurement density which is gaussian conditional on the unobservable precision the precision is assumed to be a gamma variable which evolves by being scaled by a beta variable the resulting forecast is a student s t random variable with a scale which is approximately an exponentially weighted moving average ewma of the squares of the past observations the degrees of freedom of the student s t distribution is controlled by the size of the discount parameter of the ewma procedure the gaussianity of the measurement density is potentially inadequate when the model is applied to heavy tailed finance data instead this assumption can be replaced by an exponential power density which allows the observed excess kurtosis to be modelled the choice of the exponential power means that the model still maintains conjugacy so allowing the derivation of an exact filter and likelihood function this model is called the generalised local scale model it has been used to model two exchange rate series

sharp adaptive estimation of the drift function for ergodic diffusions the global estimation problem of the drift function is considered for a large class of ergodic diffusion processes the unknown drift s is supposed to belong to a nonparametric class of smooth functions of order k 1 but the value of k is not known to the statistician a fully data driven procedure of estimating the drift function is proposed using the estimated risk minimization method the sharp adaptivity of this procedure is proven up to an optimal constant when the quality of the estimation is measured by the integrated squared error weighted by the square of the invariant density

efficient estimation in semiparametric garch models abstract it is well known that financial data sets exhibit conditional hereroskedasticity garch type models are often used to model this phenomenon since the distribution of the rescaled innovations is generally far from a normal distribution a semiparametric approach is advisable several publications observed that adaptive estimation of the euclidean parameters is not possible in the usual parametrization when the distribution of the rescaled innovations is the efficient score functions in the parametric model of the autoregression parameters are orthogonal to the tangent space generated by the nuisance parameter thus suggesting that adaptive estimation of the autoregression parameters is possible indeed we construct adaptive and hence efficient estimators in a general garch in mean type context including integrated garch models our analysis is based on a general lan theorem for time series models published elsewhere in contrast to recent literature about arch models we do not need moment condition

the surprise element jumps in interest rates abstract that information surprises result in discontinuous interest rates is no surprise to participants in the bond markets we develop a class of poisson gaussian models of the fed funds rate to capture surprise effects and show that these models offer a good statistical description of short rate behavior and are useful in understanding many empirical phenomena jump poisson processes capture empirical features of the data which would not be captured by gaussian models and there is strong evidence that existing gaussian models would be well enhanced by jump and arch type processes the analytical and empirical methods in the paper support many applications such as testing for fed intervention effects which are shown to be an important source of surprise jumps in interest rates the jump model is shown to mitigate the non linearity of interest rate drifts so prevalent in pure diffusion models day of week effects are modelled explicitly and the jump model provides evidence of bond market overreaction rejecting the martingale hypothesis for interest rates jump models mixed with markov switching processes predicate that conditioning on regime is important in determining short rate behavior

global well posedness and scattering for the defocusing energy supercritical cubic nonlinear wave equation abstract in this paper we consider the defocusing cubic nonlinear wave equation u t t u u 2 u 0 in the energy supercritical regime in dimensions d 6 with no radial assumption on the initial data we prove that if a solution satisfies an a priori bound in the critical homogeneous sobolev space throughout its maximal interval of existence that is u l t h x s c h x s c 1 then the solution is global and it scatters our analysis is based on the methods of the recent works of kenig and merle 2008 21 and killip and visan 2010 26 27 treating the energy supercritical nonlinear schrodinger and wave equations