exact solution of a 1d asymmetric exclusion model using a matrix formulation several recent works have shown that the one dimensional fully asymmetric exclusion model which describes a system of particles hopping in a preferred direction with hard core interactions can be solved exactly in the case of open boundaries here the authors present a new approach based on representing the weights of each configuration in the steady state as a product of noncommuting matrices with this approach the whole solution of the problem is reduced to finding two matrices and two vectors which satisfy very simple algebraic rules they obtain several explicit forms for these non commuting matrices which are in the general case infinite dimensional their approach allows exact expressions to be derived for the current and density profiles finally they discuss briefly two possible generalizations of their results the problem of partially asymmetric exclusion and the case of a mixture of two kinds of particles

vicious walks with a wall noncolliding meanders and chiral and bogoliubov de gennes random matrices spatially and temporally inhomogeneous evolution of one dimensional vicious walkers with wall restriction is studied we show that its continuum version is equivalent with a noncolliding system of stochastic processes called brownian meanders here the brownian meander is a temporally inhomogeneous process introduced by yor as a transform of the bessel process that is the motion of radial coordinate of the three dimensional brownian motion represented in spherical coordinates it is proved that the spatial distribution of vicious walkers with a wall at the origin can be described by the eigenvalue statistics of gaussian ensembles of bogoliubov de gennes hamiltonians of the mean field theory of superconductivity which have a particle hole symmetry we report that a time evolution of the present stochastic process is fully characterized by the change of symmetry classes from type c to type ci in the nonstandard classes of random matrix theory of altland and zirnbauer the relation between the noncolliding systems of the generalized meanders of yor which are associated with the even dimensional bessel processes and the chiral random matrix theory is also clarified

stochastic differential equations with reflecting boundary conditions on resout des equations differentielles stochastiques a conditions aux limites reflechissantes par une approche directe basee sur le probleme de skorokhod

structural changes and out of sample prediction of realized range based variance in the stock market abstract this paper aims to examine the effects of structural changes on forecasting the realized range based variance in the stock market considering structural changes in variance in the stock market we develop the har rrv sc model on the basis of the har rrv model subsequently the har rrv and har rrv sc models are used to forecast the realized range based variance of s p 500 index we find that there are many structural changes in variance in the u s stock market and the period after the financial crisis contains more structural change points than the period before the financial crisis the out of sample results show that the har rrv sc model significantly outperforms the har bv model when they are employed to forecast the 1 day 1 week and 1 month realized range based variances which means that structural changes can improve out of sample prediction of realized range based variance the out of sample results remain robust across the alternative rolling fixed window the alternative threshold value in icss algorithm and the alternative benchmark models more importantly we believe that considering structural changes can help improve the out of sample performances of most of other existing har rrv type models in addition to the models used in this paper

minimizing infinite time horizon discounted cost with mean variance and bounded variation controls we consider an infinite time horizon discounted cost minimization problem for a class of ito processes the available controls are drift and diffusion coefficients and the added bounded variation process we show that the value function is twice continuously differentiable and derive an optimal policy which has feedback type drift and diffusion coefficients when the absolute value of the optimal drift grows faster than the running cost function the optimal bounded variation process is identically zero when it grows weaker than the running cost function optimal bounded variation process is a local time type process in this case we relate the control problem with an optimal stopping problem we also establish the abelian limit relations between the value functions of the discounted cost problem and the stationary problem

research on the relationship between the multifractality and long memory of realized volatility in the sseci abstract we examine the multifractal properties of the realized volatility rv and realized bipower variation rbv series in the shanghai stock exchange composite index sseci by using the multifractal detrended fluctuation analysis mf dfa method we find that there exist distinct multifractal characteristics in the volatility series the contributions of two different types of source of multifractality namely fat tailed probability distributions and nonlinear temporal correlations are studied by using the unit root test we also find the strength of the multifractality of the volatility time series is insensitive to the sampling frequency but that the long memory of these series is sensitive

one dimensional partially asymmetric simple exclusion process with open boundaries orthogonal polynomials approach the stationary state of the partially asymmetric simple exclusion process with open boundaries is reconsidered the so called matrix product ansatz is employed this enables us to construct the stationary state in the form of the product of the matrices d and e noticing the fact that the matrix c d e for the model is related to certain q orthogonal polynomials the model is analysed for a wide range of parameters the current and the correlation length are evaluated in the thermodynamic limit it turns out that the phase diagram for the correlation length is richer than that for the totally asymmetric case

handbook of brownian motion facts and formulae i theory i stochastic processes in general ii linear diffusions iii stochastic calculus iv brownian motion v local time as a markov process vi differential systems associated to brownian motion appendix 1 briefly on some diffusions ii tables of distributions of functionals of brownian motion and related processes 1 brownian motion 2 brownian motion with drift 3 reflecting brownian motion 4 bessel process of order 5 bessel process of order 1 2 6 bessel process of order zero 7 ornstein uhlenbeck process 8 radial ornstein uhlenbeck process 9 geometric brownian motion appendix 2 special functions appendix 3 inverse laplace transforms appendix 4 differential equations and their solutions appendix 5 formulae for n fold differentiation

nonparametric regression for locally stationary time series in this paper we study nonparametric models allowing for locally stationary regressors and a regression function that changes smoothly over time these models are a natural extension of time series models with time varying coecients we introduce a kernel based method to estimate the time varying regression function and provide asymptotic theory for our estimates moreover we show that the main conditions of the theory are satised for a large class of nonlinear autoregressive processes with a time varying regression function finally we examine structured models where the regression function splits up into time varying additive components as will be seen estimation in these models does not suer from the curse of dimensionality we complement the technical analysis of the paper by an application to nancial data

binomial determinants paths and hook length formulae abstract we give a combinatorial interpretation for any minor or binomial determinant of the matrix of binomial coefficients this interpretation involves configurations of nonintersecting paths and is related to young tableaux and hook length formulae

volatility forecast comparison using imperfect volatility proxies the use of a conditionally unbiased but imperfect volatility proxy can lead to undesirable outcomes in standard methods for comparing conditional variance forecasts we motivate our study with analytical results on the distortions caused by some widely used loss functions when used with standard volatility proxies such as squared returns the intra daily range or realised volatility we then derive necessary and sufficient conditions on the functional form of the loss function for the ranking of competing volatility forecasts to be robust to the presence of noise in the volatility proxy and derive some useful special cases of this class of robust loss functions the methods are illustrated with an application to the volatility of returns on ibm over the period 1993 to 2003

data based ranking of realised volatility estimators abstract this paper presents new methods for comparing the accuracy of estimators of the quadratic variation of a price process i provide conditions under which the relative accuracy of competing estimators can be consistently estimated as t and show that forecast evaluation tests may be adapted to the problem of ranking these estimators the proposed methods avoid making specific assumptions about microstructure noise and facilitate comparisons of estimators that would be difficult using methods from the extant literature such as those based on different sampling schemes an application to high frequency ibm data between 1996 and 2007 illustrates the new methods

moments of vicious walkers and mobius graph expansions a system of brownian motions in one dimension all started from the origin and conditioned never to collide with each other in a given finite time interval 0 t is studied the spatial distribution of such vicious walkers can be described by using the repulsive eigenvalue statistics of random hermitian matrices and it was shown that the present vicious walker model exhibits a transition from the gaussian unitary ensemble gue statistics to the gaussian orthogonal ensemble goe statistics as the time t goes on from 0 to t in the present paper we characterize this gue to goe transition by presenting the graphical expansion formula for the moments of positions of vicious walkers in the gue limit stackrel ensuremath rightarrow t 0 only the ribbon graphs contribute and the problem is reduced to the classification of orientable surfaces by genus following the time evolution of the vicious walkers however the graphs with twisted ribbons called m obius graphs increase their contribution to our expansion formula and we have to deal with the topology of nonorientable surfaces application of the recent exact result of dynamical correlation functions yields closed expressions for the coefficients in the m obius expansion using the stirling numbers of the first kind

infinite systems of noncolliding generalized meanders and riemann liouville differintegrals yor s generalized meander is a temporally inhomogeneous modification of the 2 1 dimensional bessel process with 1 in which the inhomogeneity is indexed by kappa in 0 2 nu 1 we introduce the noncolliding particle systems of the generalized meanders and prove that they are pfaffian processes in the sense that any multitime correlation function is given by a pfaffian in the infinite particle limit we show that the elements of matrix kernels of the obtained infinite pfaffian processes are generally expressed by the riemann liouville differintegrals of functions comprising the bessel functions j used in the fractional calculus where orders of differintegration are determined by as special cases of the two parameters the present infinite systems include the quaternion determinantal processes studied by forrester nagao and honner and by nagao which exhibit the temporal transitions between the universality classes of random matrix theory

consistent ranking of volatility models abstract we show that the empirical ranking of volatility models can be inconsistent for the true ranking if the evaluation is based on a proxy for the population measure of volatility for example the substitution of a squared return for the conditional variance in the evaluation of arch type models can result in an inferior model being chosen as best with a probability that converges to one as the sample size increases we document the practical relevance of this problem in an empirical application and by simulation experiments our results provide an additional argument for using the realized variance in out of sample evaluations rather than the squared return we derive the theoretical results in a general framework that is not specific to the comparison of volatility models similar problems can arise in comparisons of forecasting models whenever the predicted variable is a latent variable

singular optimal controls of stochastic recursive systems and hamilton jacobi bellman inequality abstract in this paper we study the optimal singular controls for stochastic recursive systems in which the control has two components the regular control and the singular control under certain assumptions we establish the dynamic programming principle for this kind of optimal singular controls problem and prove that the value function is a unique viscosity solution of the corresponding hamilton jacobi bellman inequality in a given class of bounded and continuous functions at last an example is given for illustration

directed percolation and generalized friendly random walkers the problem of directed percolation dp rst intro duced by broadbent and hammersley 1 continues to at tract interest even though it has so far de ed all attempts atan exact solution even in two dimensions although theproblem was originally formulated statically on a latticewith a preferred direction when the latter is interpreted astime the universal behavior close to the percolation thresh old is also believed to describe the transition from a noise less absorbing state to a noisy active one which occurs ina wide class of stochastic processes 2 it also maps ontoreggeon eld theory which describes high energy diffrac tion scattering in particle physics 3 some time ago arrowsmith mason and essam 4 argued that the pair connectedness probability

a simple r estimation method for semiparametric duration models modeling nonnegative financial variables e g durations between trades traded volumes or asset volatilities is central to a number of studies across financial econometrics and despite the efforts still poses several statistical challenges among them the efficiency aspects of semiparametric estimation in this paper we concentrate on estimation problems in autoregressive conditional duration acd models with unspecified innovation densities exponential quasi likelihood estimators qmle are the usual practice in that context the efficiency of those qmles the only fisher consistent qmles unfortunately rapidly deteriorates away from the reference exponential density a phenomenon that has been emphasized earlier by drost and werker 2003 who propose various semiparametrically efficient procedures to palliate that phenomenon those procedures rely on a general semiparametric approach which typically requires kernel estimation of the underlying innovation density we propose rank based estimators r estimators as a substitute just as the qmle r estimators remain root n consistent irrespective of the underlying density and rely on the choice of a reference density under which they achieve semiparametric efficiency that density however needs not be the exponential one contrary to the semiparametric estimators proposed by drost and werker 2003 r estimators neither require tangent space calculations nor kernel based density estimation numerical results moreover indicate that r estimators based on exponential reference densities uniformly outperform the exponential qmle under such families of innovations as the weibull or burr densities a real data example about modeling the price range of the swiss stock market index concludes the paper

symmetry of matrix valued stochastic processes and noncolliding diffusion particle systems as an extension of the theory of dyson s brownian motion models for the standard gaussian random matrix ensembles we report a systematic study of hermitian matrix valued processes and their eigenvalue processes associated with the chiral and nonstandard random matrix ensembles in addition to the noncolliding brownian motions we introduce a one parameter family of temporally homogeneous noncolliding systems of the bessel processes and a two parameter family of temporally inhomogeneous noncolliding systems of yor s generalized meanders and show that all of the ten classes of eigenvalue statistics in the altland zirnbauer classification are realized as particle distributions in the special cases of these diffusion particle systems as a corollary of each equivalence in distribution of a temporally inhomogeneous eigenvalue process and a noncolliding diffusion process a stochastic calculus proof of a version of the harish chandra itzykson zuber formula of integral over unitary group is established

the distribution of realized exchange rate volatility using high frequency data on deutschemark and yen returns against the dollar we construct model free estimates of daily exchange rate volatility and correlation that cover an entire decade our estimates termed realized volatilities and correlations are not only model free but also approximately free of measurement error under general conditions which we discuss in detail hence for practical purposes we may treat the exchange rate volatilities and correlations as observed rather than latent we do so and we characterize their joint distribution both unconditionally and conditionally noteworthy results include a simple normality inducing volatility transformation high contemporaneous correlation across volatilities high correlation between correlation and volatilities pronounced and persistent dynamics in volatilities and correlations evidence of long memory dynamics in volatilities and correlations and remarkably precise scaling laws under temporal aggregation

optimal consumption for general diffusions with absorbing and reflecting barriers two stochastic control problems of the storage or inventory type are considered for general diffusion processes in the absorption problem a diffusion process is controlled by subtracting a nondecreasing withdrawal process the controlled process is absorbed if it reaches zero the objective function to be maximized is the expected discounted value of the withdrawals plus a discounted penalty for absorption in the reflection problem the process can be controlled by subtracting a withdrawal process and adding a deposit process and the controlled process must be nonnegative one seeks to maximize an expected discounted weighted difference in withdrawals and deposits the value function is computed and a necessary and sufficient condition for the existence of an optimal policy is given when they exist optimal policies are found to be the minimal processes which keep the controlled process inside an interval

optimal correction problem of a multidimensional stochastic system abstract we consider a stochastic dynamic system which is governed by a multidimensional diffusion process with constant drift and diffusion coefficients the correction corresponds to an additive input which is under control there is no limit on the rate of input into the system the objective is to minimize the expected cumulative cost associated with the position of the system and the amount of control exerted it is proved that hamilton jacobi bellman s equation of the problem has a solution which corresponds to the optimal cost of the problem an existence of optimal policy is proved

on some gaussian ensembles of hermitian matrices a gaussian ensemble of hermitian matrices with an arbitrary ratio of their symmetric and anti symmetric parts is studied similarly a gaussian ensemble of hermitian quaternion matrices with an arbitrary ratio of their self dual and anti self dual parts is studied analytical expressions for the correlation and cluster functions as well as their limits when the order of the matrices is large are derived for both ensembles

ordering of random walks the leader and the laggard we investigate two complementary problems related to maintaining the relative positions of n random walks on the line i the leader problem that is the probability n t that the leftmost particle remains the leftmost as a function of time and ii the laggard problem the probability n t that the rightmost particle never becomes the leftmost we map these ordering problems onto an equivalent n 1 dimensional electrostatic problem from this construction we obtain a very accurate estimate for n t for n 4 the first case that is not exactly solvable 4 t t 4 with 4 0 91342 8 the probability of being the laggard also decays algebraically n t t n we derive 2 1 2 3 3 8 and argue that n n 1 ln n as n

vicious walkers friendly walkers and young tableaux ii with a wall we derive new results for the number of star and watermelon configurations of vicious walkers in the presence of an impenetrable wall by showing that these follow from standard results in the theory of young tableaux and combinatorial descriptions of symmetric functions for the problem of n friendly walkers we derive exact asymptotics for the number of stars and watermelons both in the absence of a wall and in the presence of a wall

nonintersecting brownian motions on the half line and discrete gaussian orthogonal polynomials we study the distribution of the maximal height of the outermost path in the model of n nonintersecting brownian motions on the half line as n showing that it converges in the proper scaling to the tracy widom distribution for the largest eigenvalue of the gaussian orthogonal ensemble this is as expected from the viewpoint that the maximal height of the outermost path converges to the maximum of the airy2 process minus a parabola our proof is based on riemann hilbert analysis of a system of discrete orthogonal polynomials with a gaussian weight in the double scaling limit as this system approaches saturation we consequently compute the asymptotics of the free energy and the reproducing kernel of the corresponding discrete orthogonal polynomial ensemble in the critical scaling in which the density of particles approaches saturation both of these results can be viewed as dual to the case in which the mean density of eigenvalues in a random matrix model is vanishing at one point

singular control of stochastic linear systems with recursive utility we formulate a class of singular stochastic control problem with recursive utility where the cost function is determined by a backward stochastic differential equation some characteristics of the value function of the control problem are obtained by the method of approximation via penalization and the optimal control process is constructed

volatility estimators and the inverse range process in a random volatility random walk and wiener processes abstract the purpose of this paper is to study the mean the variance the probability distribution and the hazard rate of the inverse range process of an a priori unknown volatility random walk motivation for this process arises when it is necessary to obtain statistics that pertain to a process volatility in addition to the usual variance statistics as a result range process statistics are indicated as an additional source of information in the study of processes volatility examples and applications are considered

noncolliding brownian motion and determinantal processes a system of one dimensional brownian motions bms conditioned never to collide with each other is realized as i dyson s bm model which is a process of eigenvalues of hermitian matrix valued diffusion process in the gaussian unitary ensemble gue and as ii the h transform of absorbing bm in a weyl chamber where the harmonic function h is the product of differences of variables the vandermonde determinant the karlin mcgregor formula gives determinantal expression to the transition probability density of absorbing bm we show from the karlin mcgregor formula if the initial state is in the eigenvalue distribution of gue the noncolliding bm is a determinantal process in the sense that any multitime correlation function is given by a determinant specified by a matrix kernel by taking appropriate scaling limits spatially homogeneous and inhomogeneous infinite determinantal processes are derived we note that the determinantal processes related with noncolliding particle systems have a feature in common such that the matrix kernels are expressed using spectral projections of appropriate effective hamiltonians on the common structure of matrix kernels continuity of processes in time is proved and general property of the determinantal processes is discussed

analysis of the efficiency of the shanghai stock market a volatility perspective abstract by applying the rolling window method we investigate the efficiency of the shanghai stock market through the dynamic changes of local hurst exponents based on multifractal detrended fluctuation analysis we decompose the realized volatility into continuous sample paths and jump components and analyze their long range correlations of decomposing components our results reveal that the efficiency of the shanghai stock market improved greatly based on the time varying hurst exponents

a method of integration over matrix variables the integral over twon n hermitan matricesz g c dadbexp tr a 2 b 2 2cab g n a 4 b 4 is evaluated in the limit of largen for this purpose use is made of the theory of diffusion equation and that of orthogonal polynomials with a non local weight the above integral arises in the study of the planar approximation to quantum field theory

families of vicious walkers we consider a generalization of the vicious walker problem in which n random walkers in rd are grouped into p families using field theoretic renormalization group methods we calculate the asymptotic behaviour of the probability that no pairs of walkers from different families have met up to time t for d 2 this is constant but for d 2 it decays as a power t which we compute to e2 in an expansion in e 2 d the second order term depends on the ratios of the diffusivities of the different families in two dimensions we find a logarithmic decay ln t and compute exactly

determinantal process starting from an orthogonal symmetry is a pfaffian process when the number of particles n is finite the noncolliding brownian motion bm and the noncolliding squared bessel process with index 1 besq are determinantal processes for arbitrary fixed initial configurations in the present paper we prove that if initial configurations are distributed with orthogonal symmetry they are pfaffian processes in the sense that any multitime correlation functions are expressed by pfaffians the 2 2 skew symmetric matrix valued correlation kernels of the pfaffians processes are explicitly obtained by the equivalence between the noncolliding bm and an appropriate dilatation of a time reversal of the temporally inhomogeneous version of noncolliding bm with finite duration in which all particles start from the origin n 0 and by the equivalence between the noncolliding besq and that of the noncolliding squared generalized meander starting from n 0

measuring volatility with the realized range realized variance being the summation of squared intra day returns has quickly gained popularity as a measure of daily volatility following parkinson 1980 the extreme value method for estimating the variance of the rate of return journal of business 53 61 65 we replace each squared intra day return by the high low range for that period to create a novel and more efficient estimator called the realized range in addition we suggest a bias correction procedure to account for the effects of microstructure frictions based upon scaling the realized range with the average level of the daily range simulation experiments demonstrate that for plausible levels of non trading and bid ask bounce the realized range has a lower mean squared error than the realized variance including variants thereof that are robust to microstructure noise empirical analysis of the s p500 index futures and the s p100 constituents confirms the potential of the realized range

bounds for the critical line of the theta contact processes with 1 or theta or 2 we study a family of the one dimensional contact processes introduced by durrett and griffeath 1983 which is parametrized by theta for each theta or 1 there is a unique critical value lambda c theta so that any process becomes extinct with probability 1 for lambda lambda c theta in this paper we give rigorous upper and lower bounds for the critical line lambda lambda c theta for 1 or theta or 2 in order to obtain the upper bound we extend the holley liggett argument which was originally given for the case theta 2 the basic contact process we construct a new class of attractive renewal measures with positive densities and the upper bound of lambda c theta is given as the largest root of a cubic equation theta lambda 3 3 theta 2 lambda 2 3 2 theta lambda 2 theta 0 recently liggett 1991 reported an upper bound of the critical value for the case theta 1 the threshold contact process by a modified version of the holley liggett argument our result includes these previous results as special cases

brownian motion ensembles correlation functions of determinantal processes we calculate the n point correlation function for a large class of brownian motion ensembles of random matrix theory the corresponding fokker planck equation describes a determinantal process in the theory of matrix valued stochastic differential equations and can be mapped onto a schodinger equation of non interacting electrons in one dimension the correlation functions are obtained explicitly via a suitable generalization of the method of biorthogonal functions

econometric analysis of realized volatility and its use in estimating stochastic volatility models 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

gaussian ensembles of random hermitian matrices intermediate between orthogonal and unitary ones a gaussian ensemble of hermitian matrices depending on a parameter is considered when 0 the ensemble is gaussian orthogonal and when 1 it is gaussian unitary an analytic expression for then level correlation and cluster functions is given for anyn and 0 1 this ensemble is of relevance in the study of time reversal symmetry breaking of nuclear interactions

the euler characteristic of the moduli space of curves let fg 1 g 1 be the mapping class group consisting of all isotopy classes of base point and orientation preserving homeomorphisms of a closed oriented surface f of genus g let 1 be its euler characteristic in the sense of wall that is z f i fgl f l z e f where f is any torsion free subgroup of finite index in f 1 and e is a contractible space on which f acts freely and properly discontinuously an example of such a space is the teichmiiller space 1 and g f can be interpreted as the orbifold euler characteristic of f i join the moduli space of curves of genus g with base point the purpose of this paper is to prove the following formula for f i

realized range based estimation of integrated variance abstract we provide a set of probabilistic laws for estimating the quadratic variation of continuous semimartingales with the realized range based variance a statistic that replaces every squared return of the realized variance with a normalized squared range if the entire sample path of the process is available and under a set of weak conditions our statistic is consistent and has a mixed gaussian limit whose precision is five times greater than that of the realized variance in practice of course inference is drawn from discrete data and true ranges are unobserved leading to downward bias we solve this problem to get a consistent mixed normal estimator irrespective of non trading effects this estimator has varying degrees of efficiency over realized variance depending on how many observations that are used to construct the high low the methodology is applied to taq data and compared with realized variance our findings suggest that the empirical path of quadratic variation is also estimated better with the realized range based variance

scaling limit of vicious walks and two matrix model we consider the diffusion scaling limit of the one dimensional vicious walker model of fisher and derive a system of nonintersecting brownian motions the spatial distribution of n particles is studied and it is described by use of the probability density function of eigenvalues of nxn gaussian random matrices the particle distribution depends on the ratio of the observation time t and the time interval t in which the nonintersecting condition is imposed as t t is going on from 0 to 1 there occurs a transition of distribution which is identified with the transition observed in the two matrix model of pandey and mehta despite of the absence of matrix structure in the original vicious walker model in the diffusion scaling limit accumulation of contact repulsive interactions realizes the correlated distribution of eigenvalues in the multimatrix model as the particle distribution

continuous record asymptotics for rolling sample variance estimators it is widely known that conditional covariances of asset returns change over time researchers doing empirical work have adopted many strategies for accommodating conditional heteroskedasticity among the popular strategies are a chopping the available data into short blocks of time and assuming homoskedasticity within the blocks b performing one sided rolling regressions in which only data from say the preceding five year period is used to estimate the conditional covariance of returns at a given date and c performing two sided rolling regressions in which covariances are estimated for each date using say five years of lags and five years of leads another model garch amounts to a one sided weighted rolling regression we develop continuous record asymptotic approximations for the measurement error in conditional variances and covariances when using these methods we derive asymptotically optimal window lengths for standard rolling regressions and optimal weights for weighted rolling regressions as an empirical example we estimate volatility on the s p 500 stock index using daily data from 1928 to 1990

regularity of the value function for a two dimensional singular stochastic control problem it is desired to control a two dimensional brownian motion by adding a possibly singularly continuous process to it so as to minimize an expected infinite horizon discounted running cost the hamilton jacobi bellman characterization of the value function v is a variational inequality which has a unique twice continuously differentiable solution the optimal control process is constructed by solving the skorokhod problem of reflecting the two dimensional brownian motion along a free boundary in the nabla v direction

localization transition of d friendly walkers friendly walkers is a stochastic model obtained from independent one dimensional simple random walks s k j j 0 k 1 2 d by introducing non crossing condition and reward for collisions characterized by parameters here the reward for collisions is described as follows if at a given time n a site in is occupied by exactly m 2 walkers then the site increases the probabilistic weight for the walkers by multiplicative factor exp m 1 we study the localization transition of this model in terms of the positivity of the free energy and describe the location and the shape of the critical surface in the d 1 dimensional space for the parameters

the properties and mechanism of long term memory in nonparametric volatility abstract recent empirical literature documents the presence of long term memory in return volatility but the mechanism of the existence of long term memory is still unclear in this paper we investigate the origin and properties of long term memory with nonparametric volatility using high frequency time series data of the chinese shanghai composite stock price index we perform detrended fluctuation analysis dfa on three different nonparametric volatility estimators with different sampling frequencies for the same volatility series the hurst exponents reduce as the sampling time interval increases but they are still larger than 1 2 which means that no matter how the interval changes it still cannot change the existence of long memory rrv presents a relatively stable property on long term memory and is less influenced by sampling frequency rv and rbv have some evolutionary trends depending on time intervals which indicating that the jump component has no significant impact on the long term memory property this suggests that the presence of long term memory in nonparametric volatility can be contributed to the integrated variance component considering the impact of microstructure noise rbv and rrv still present long term memory under various time intervals we can infer that the presence of long term memory in realized volatility is not affected by market microstructure noise our findings imply that the long term memory phenomenon is an inherent characteristic of the data generating process not a result of microstructure noise or volatility clustering