construction of constant mean curvature n noids from holomorphic potentials using a weierstrass type representation of constant mean curvature surfaces we give a general method for constructing constant mean curvature n noids of genus 0 from holomorphic potentials where n 3 the ends of these surfaces are embedded and asymptotically approach delaunay surfaces while the surfaces are in general not even almost embedded in particular a 3 parameter family of constant mean curvature trinoids is constructed

multivariate jacobi process with application to smooth transitions abstract we introduce the multivariate jacobi process as a representation for the dynamics of a stochastic discrete probability distribution its domain of application is dynamic analysis of switching regimes in asset return volatility business cycle and corporate credit ratings the paper shows how the multivariate jacobi process is derived from the multivariate cox ingersoll ross cir model by time deformation and presents the main distributional properties for illustration selected continuous time models of prices and returns on financial assets are extended to smooth transitions processes featuring regimes of different volatilities and persistence in this framework the effects of transitions between the regimes on derivative prices and long memory are examined

convergence rates of moving mesh methods for moving boundary partial integro differential equations from regime switching jump diffusion asian option pricing abstract this paper studies the convergence rates of moving mesh methods for a system of moving boundary partial integro differential equations pides which arise in the asian option pricing under the state dependent regime switching jump diffusion models the value function of the asian option under the model is governed by a system of two dimensional pides in this paper the two dimensional pides are recast into a one dimensional moving boundary problem of the pides a moving finite difference method fdm is proposed to solve the one dimensional moving boundary problem and the convergence rates are proved numerical examples are provided to confirm the theoretical results

a unified approach to validating univariate and multivariate conditional distribution models in time series abstract modeling conditional distributions in time series has attracted increasing attention in economics and finance we develop a new class of generalized cramer von mises gcm specification tests for time series conditional distribution models using a novel approach which embeds the empirical distribution function in a spectral framework our tests check a large number of lags and are therefore expected to be powerful against neglected dynamics at higher order lags which is particularly useful for non markovian processes despite using a large number of lags our tests do not suffer much from loss of a large number of degrees of freedom because our approach naturally downweights higher order lags which is consistent with the stylized fact that economic or financial markets are more affected by recent past events than by remote past events unlike the existing methods in the literature the proposed gcm tests cover both univariate and multivariate conditional distribution models in a unified framework they exploit the information in the joint conditional distribution of underlying economic processes moreover a class of easy to interpret diagnostic procedures are supplemented to gauge possible sources of model misspecifications distinct from conventional cm and kolmogorov smirnov ks tests which are also based on the empirical distribution function our gcm test statistics follow a convenient asymptotic n 0 1 distribution and enjoy the appealing nuisance parameter free property that parameter estimation uncertainty has no impact on the asymptotic distribution of the test statistics simulation studies show that the tests provide reliable inference for sample sizes often encountered in economics and finance

fast and stable qr eigenvalue algorithms for generalized companion matrices and secular equations we introduce a class of n n structured matrices which includes three well known classes of generalized companion matrices tridiagonal plus rank one matrices comrade matrices diagonal plus rank one matrices and arrowhead matrices relying on the structure properties of we show that if a then a rq where a qr is the qr decomposition of a this allows one to implement the qr iteration for computing the eigenvalues and the eigenvectors of any a with o n arithmetic operations per iteration and with o n memory storage this iteration applied to generalized companion matrices provides new o n2 flops algorithms for computing polynomial zeros and for solving the associated rational secular equations numerical experiments confirm the effectiveness and the robustness of our approach

dressing cmc n noids the purpose of this paper is to construct cmc n noids with bubbletons we recall a specific class of dressing matrices which we will refer to as simple factor dressing matrices which in many known cases add bubbletons to a cmc surface our new surfaces are obtained by dressing known n noids with embedded delaunay ends by well chosen simple factor dressing matrices in such a way that the dressed surface is also a cmc n noid

real forms of complex surfaces of constant mean curvature it is known that complex constant mean curvature cmc for short immersions in c 3 are natural complexifications of cmc immersions in r 3 in this paper conversely we consider real form surfaces of a complex cmc immersion which are defined from real forms of the twisted sl 2 c loop algebra asl 2 and classify all such surfaces according to the classification of real forms of asl 2 there are seven classes of surfaces which are called integrable surfaces and all integrable surfaces will be characterized by the lorentz harmonicities of their gauss maps into the symmetric spaces s 2 h 2 s 1 1 or the 4 symmetric space sl 2 u 1 we also give a unification to all integrable surfaces via the generalized weierstrass type representation

actions of loop groups on harmonic maps we describe a general framework in which subgroups of the loop group agin act on the space of harmonic maps from s2 to gln this represents a simplification of the action considered by zakharov mikhailov shabat zm zs in that we take the contour for the riemann hilbert problem to be a union of circles however it reduces the basic ingredient to the well known birkhoff decomposition of agin s and this facilitates a rigorous treatment we give various concrete examples of the action and use these to investigate a suggestion of uhlenbeck uh that a limiting version of such an action completion gives rise to her fundamental process of adding a uniton it turns out that this does not occur because completion preserves the energy of harmonic maps however in the special case of harmonic maps from s2 to complex projective space we describe a modification of this completion procedure which does indeed reproduce adding a uniton one aspect of the theory of symmetry groups for diffierential equations is the idea of proliferation of solutions by this is meant that starting with an obvious solution application of symsmetries can lead to further perhaps less obvious solutions in this paper we shall apply the general framework described by zakharov et al zm zs to the harmonic map equation the principal chiral or sigma model of mathematical physics to put this into a wider context we recall that a fundamental discovery in the study of the korteweg de vries and related equations was that the space of solutions admits a certain infinite dimensional symmetry group historically the first manifestation of this via noether s theorem was the appearance of infinitely many conservation laws later this led to an elegant algebraic description of solutions of the equation as the points of an orbit of a representation of an infinite dimensional group see for example wi for a brief description of some of these ideas the existence of previously unnoticed symmetries hidden symmetries for other equations became an interesting possibility and attempts have been made so far with much less success than in the case of the kdv equation to unearth them one such example is the harmonic map equation for maps from a surface d into a compact lie group g or homogeneous space g h the basic observation po uh zm zs which introduces an infinite dimensional group is that harmonic maps from d to g correspond to certain received by the editors february 9 1990 1980 mathematics subject classification 1985 revision primary 58e20 r 1991 american mathematical society 0002 9947 91 1 00 25 per page

finite mixture distributions sequential likelihood and the em algorithm a popular way to account for unobserved heterogeneity is to assume that the data are drawn from a finite mixture distribution a barrier to using finite mixture models is that parameters that could previously be estimated in stages must now be estimated jointly using mixture distributions destroys any additive separability of the log likelihood function we show however that an extension of the em algorithm reintroduces additive separability thus allowing one to estimate parameters sequentially during each maximization step in establishing this result we develop a broad class of estimators for mixture models returning to the likelihood problem we show that relative to full information maximum likelihood our sequential estimator can generate large computational savings with little loss of efficiency

understanding the qr algorithm the qr algorithm is currently the most popular method for finding all eigenvalues of a full matrix while qr is now well understood by specialists in eigenvalue computations this understanding is not being conveyed effectively to the mathematical public many accounts present wilkinson s 1965 convergence proof others establish some of the connections between the qr algorithm the power method and inverse iteration usually much emphasis is rightly placed on the refinements such as shifts of origin which are required to make the algorithm competitive but practically all accounts fail to explain the basic meaning of qr iterations as a consequence the qr algorithm is widely thought to be difficult to understand the purpose of this paper is to try to convince the reader that the opposite is true in fact the qr algorithm is neither more nor less than a clever implementation of simultaneous iteration which is itself a natural easily understood extension of the power method this point

monte carlo methods for estimating smoothing and filtering one and two factor stochastic volatility models one and two factor stochastic volatility models are assessed over three sets of stock returns data sp the problem is to get the shape of the conditional returns distribution right none of the models come close to matching the tails of this distribution including a second factor provides only a relatively small improvement over the single factor models fitting this aspect of the data is important for option pricing and risk management

convergence of algorithms of decomposition type for the eigenvalue problem we develop the theory of convergence of a generic gr algorithm for the matrix eigenvalue problem that includes the qr lr sr and other algorithms as special cases our formulation allows for shifts of origin and multiple gr steps the convergence theory is based on the idea that the gr algorithm performs nested subspace iteration with a change of coordinate system at each step thus the convergence of the gr algorithm depends on the convergence of certain sequences of subspaces it also depends on the quality of the coordinate transformation matrices as measured by their condition numbers we show that with a certain obvious shifting strategy the gr algorithm typically has a quadratic asymptotic convergence rate for matrices possessing certain special types of structure cubic convergence can be achieved

business cycle durations abstract while the development of markov switching extensions to time series modeling has provided a useful way of characterizing business cycle dynamics these models are not without their weaknesses one problem is posed by the fact that since the state space for the unobserved state variables grows with the sample size sampling distributions for maximum likelihood estimates are difficult to establish a second problem is that since the transition probabilities are constant the conditional expected duration of phase is constant this paper extends the model so that the information contained in leading indicator data can be used to forecast transition probabilities these transition probabilities can then be used to calculate expected durations the model is applied to us data to evaluate its ability to explain observed business cycle durations the technical problems encountered with classical techniques are avoided by using bayesian methods gibbs sampling techniques are used to calculate expected posterior durations

convergence of the shifted qr algorithm for unitary hessenberg matrices this paper shows that for unitary hessenberg matrices the qr algorithm with an exceptional initial value modification of the wilkinson shift gives global convergence moreover the asymptotic rate of convergence is at least cubic higher than that which can be shown to be quadratic only for hermitian tridiagonal matrices under no further assumption a general mixed shift strategy with global convergence and cubic rates is also presented

bayesian inference and state number determination for hidden markov models an application to the information content of the yield curve about inflation this paper is concerned with bayesian inference in hidden markov models focusing on switching regression models we propose a new methodology that delivers a joint estimation of the parameters and the number of regimes that have actually appeared in the studied sample the only prior information that is required on the latter quantity is an upper bound we implement a particle filter algorithm to compute the corresponding estimates applying this methodology to the information content of the yield curve regarding future inflation in four oecd countries we show that the predictive content for given country and combination of maturities is subject to regime switching

maximum likelihood estimation for multivariate observations of markov sources parameter estimation for multivariate functions of markov chains a class of versatile statistical models for vector random processes is discussed the model regards an ordered sequence of vectors as noisy multivariate observations of a markov chain mixture distributions are a special case the foundations of the theory presented here were established by baum petrie soules and weiss a powerful representation theorem by fan is employed to generalize the analysis of baum em et al to a larger class of distributions

forecasting tehran stock exchange volatility markov switching garch approach abstract this paper evaluates several garch models regarding their ability to forecast volatility in tehran stock exchange tse these include garch models with both gaussian and fat tailed residual conditional distribution concerning their ability to describe and forecast volatility from 1 day to 22 day horizon results indicate that ar 2 mrsgarch ged model outperforms other models at one day horizon also the ar 2 mrsgarch ged as well as ar 2 mrsgarch t models outperform other models at 5 day horizon in 10 day horizon three models of ar 2 mrsgarch outperform other models concerning 22 day forecast horizon results indicate no differences between mrsgarch models with that of standard garch models regarding risk management out of sample evaluation 95 var a few models seem to provide reasonable and accurate var estimates at 1 day horizon with a coverage rate close to the nominal level according to the risk management loss functions there is not a uniformly most accurate model

a modification of the dewilde van der veen method for inversion of finite structured matrices abstract we study a class of block structured matrices r r ij i j 1 n with a property that the solution of the corresponding system rx y of linear algebraic equations may be performed for o n arithmetic operations in this paper for finite invertible matrices we analyze in detail factorization and inversion algorithms these algorithms are related to those suggested by p m dewilde and a j van der veen time varying systems and computations kluwer academic publishers new york 1998 for a class of finite and infinite matrices with a small hankel rank the algorithms presented here are more transparent and are a modification of the algorithms from the above reference the approach and the proofs are essentially different from those in the above mentioned reference the paper contains also analysis of complexity and results of numerical experiments

lawson s genus two surface and meromorphic connections we investigate the lawson genus 2 surface by methods from integrable system theory to this surface we derive a globally defined meromorphic dpw potential which is determined up to two unknown functions depending only on the spectral parameter

on the convergence properties of the orthogonal similarity transformations to tridiagonal and semiseparable plus diagonal form in this paper we will compare the convergence properties of three basic reduction methods by placing them in a general framework it covers the reduction to tridiagonal semiseparable and semiseparable plus diagonal form these reductions are often used as the first step in the computation of the eigenvalues and or eigenvectors of arbitrary matrices in this way the calculation of the eigenvalues using for example the qr algorithm reduces in complexity first we will investigate the convergence properties of these three reduction algorithms it will be shown that for the partially reduced matrices at step k of any of these reduction algorithms the lower right k k already reduced sub block will have the lanczos ritz values w r t a certain starting vector it will also be shown that the reductions to semiseparable and to semiseparable plus diagonal form have an extra convergence behavior a special type of subspace iteration is performed on the lower right k k submatrix which contains these ritz values secondly we look in more detail at the behavior of the involved subspace iteration it will be shown that the reduction method can be interpreted as a nested type of multi shift iteration theoretical results will be presented making it possible to predict the convergence behavior of these reduction algorithms also a theoretical bound on the convergence rate is presented finally we illustrate by means of numerical examples how it is possible to tune the convergence behavior such that it can become a powerful tool for certain applications

turbo segmentation of textured images we consider the problem of semi supervised segmentation of textured images existing model based approaches model the intensity field of textured images as a gauss markov random field to take into account the local spatial dependencies between the pixels classical bayesian segmentation consists of also modeling the label field as a markov random field to ensure that neighboring pixels correspond to the same texture class with high probability well known relaxation techniques are available which find the optimal label field with respect to the maximum a posteriori or the maximum posterior mode criterion but these techniques are usually computationally intensive because they require a large number of iterations to converge in this paper we propose a new bayesian framework by modeling two dimensional textured images as the concatenation of two one dimensional hidden markov autoregressive models for the lines and the columns respectively a segmentation algorithm which is similar to turbo decoding in the context of error correcting codes is obtained based on a factor graph approach the proposed method estimates the unknown parameters using the expectation maximization algorithm

estimating continuous time stochastic volatility models of the short term interest rate abstract we obtain consistent parameter estimates of continuous time stochastic volatility diffusions for the u s risk free short term interest rate sampled weekly over 1954 1995 using the efficient method of moments procedure of gallant and tauchen the preferred model displays mean reversion and incorporates level effects and stochastic volatility in the diffusion function extensive diagnostics indicate that the cox ingersoll ross model with an added stochastic volatility factor provides a good characterization of the short rate process further they suggest that recently proposed garch models fail to approximate the discrete time short rate dynamics while level egarch models perform reasonably well

two fast algorithms for solving diagonal plus semiseparable linear systems in this paper we discuss the structure of the factors of a qr and a urv factorization of a diagonal plus semiseparable matrix the q factor of a qr factorization has the diagonal plus semiseparable structure the ut and v factor of a urv factorization are semiseparable lower hessenberg orthogonal matrices the strictly upper triangular part of the r factor of a qr and of a urv factorization is the strictly upper triangular part of a rank 2 matrix this latter fact provides a tool to construct a fast qr solver and a fast urv solver for linear systems of the form d s x b

asymptotic properties of the maximum likelihood estimator in autoregressive models with markov regime an autoregressive process with markov regime is an autoregressive process for which the regression function at each time point is given by a nonobservable markov chain in this paper we consider the asymptotic properties of the maximum likelihood estimator in a possibly nonstationary process of this kind for which the hidden state space is compact but not necessarily finite consistency and asymptotic normality are shown to follow from uniform exponential forgetting of the initial distribution for the hidden markov chain conditional on the observations

analysis of time series subject to changes in regime abstract this paper introduces an em algorithm for obtaining maximum likelihood estimates of parameters for processes subject to discrete shifts in autoregressive parameters with the shifts themselves modeled as the outcome of a discrete valued markov process the simplicity of the em algorithm permits potential application of the approach to large vector systems

recognition of degraded characters using dynamic bayesian networks in this paper we investigate the application of dynamic bayesian networks dbns to the recognition of degraded characters dbns are an extension of one dimensional hidden markov models hmms which can handle several observation and state sequences in our study characters are represented by the coupling of two hmm architectures into a single dbn model the interacting hmms are a vertical hmm and a horizontal hmm whose observable outputs are the image columns and image rows respectively various couplings are proposed where interactions are achieved through the causal influence between state variables we compare non coupled and coupled models on two tasks the recognition of artificially degraded handwritten digits and the recognition of real degraded old printed characters our models show that coupled architectures perform more accurately on degraded characters than basic hmms the linear combination of independent hmm scores as well as discriminative methods such as support vector machines svms

a monte carlo based approach for pricing credit default swaps with regime switching abstract this paper considers the valuation of a cds credit default swap contract to find out a more accurate cds price we work on an extended merton s model by assuming that the price of the reference asset follows a regime switching black scholes model and moreover the reference asset can default at any time before the expiry time a general pricing formula for the cds containing the unknown no default probability is derived first it is then subsequently shown that the no default probability is equivalent to the price of a down and out binary option written on the same reference asset by simulating the markov chain with the monte carlo technique we obtain an approximation formula for the down and out binary option with the availability of which the calculation of the cds price becomes straightforward finally some numerical experiments are conducted to examine the accuracy of the approximation approach as well as the impacts of the introduction of the regime switching mechanics on the cds price

stationarity of multivariate markov switching arma models in this article we consider multivariate arma models subject to markov switching in these models the parameters are allowed to depend on the state of an unobserved markov chain a natural idea when estimating these models is to impose local stationarity conditions i e stationarity within each regime in this article we show that the local stationarity of the observed process is neither sufficient nor necessary to obtain the global stationarity we derive stationarity conditions and we compute the autocovariance function of this nonlinear process interestingly it turns out that the autocovariance structure coincides with that of a standard arma some examples are proposed to illustrate the stationarity conditions using monte carlo simulations we investigate the consequences of accounting for the stationarity conditions in statistical inference

convergence of the unitary qr algorithm with a unimodular wilkinson shift in applying the qr algorithm to compute the eigenvalues of a unitary hessenberg matrix a projected wilkinson shift of unit modulus is proposed and proved to give global convergence with at least a quadratic asymptotic rate for the qr iteration experimental testing demonstrates that the unimodular shift produces more efficient numerical convergence

value at risk via mixture distributions reconsidered value at risk var has evolved as one of the most prominent measures of downside risk in financial markets zhang and cheng m h zhang q s cheng an approach to var for capital markets with gaussian mixture applied mathematics and computation 168 2005 1079 1085 proposed an approach to var for daily returns based on gaussian mixtures which have become rather popular in empirical economics and finance since the seminal paper of hamilton j d hamilton a new approach to the economic analysis of nonstationary time series and the business cycle econometrica 57 2 1989 357 384 however they do not conduct tests to assess the accuracy of the mixture implied var measures recently guidolin and timmermann m guidolin a timmermann term structure of risk under alternative econometric specifications journal of econometrics 131 2006 285 308 showed that markov mixture models do well in measuring var at a monthly frequency but the results may not hold for daily returns due to their more pronounced non gaussian features this paper provides an extensive application of various markov mixture models to var for daily returns of major european stock markets including out of sample backtesting to accommodate the properties of daily returns we consider both gaussian and student s t mixtures and we compare the performance of both uni and multivariate models under different parameter updating schemes we find that a univariate mixture of two student s t distributions performs best overall however by the example of the recent turmoil in financial markets we also highlight a weak point of the approach

specification testing in markov switching time series models this paper develops a series of specification tests of markov switching time series models tests for omitted autocorrelation omitted arch misspecification of the markovian dynamics and omitted explanatory variables are proposed all of the tests can be constructed as a natural byproduct of the routine used to calculate the smoothed probability that a given observation came from a particular regime and do not require estimation of additional parameters the paper performs monte carlo analysis of the tests and briefly illustrates their use with an empirical application

serial correlation in latent discrete variable models abstract we consider the problems of estimation and testing in models with serially correlated discrete latent variables a particular case of this is the time series regression model in which a discrete explanatory variable is measured with error test statistics are derived for detecting serial correlation in such a model we then show that the likelihood function can be evaluated by a recurrence relation and thus maximum likelihood estimation is computationally feasible an illustrative example of these methods is given followed by a brief discussion of their applicability to a markov model of switching regressions

a series form solution for pricing variance and volatility swaps with stochastic volatility and stochastic interest rate abstract in this paper we present analytical pricing formulae for variance and volatility swaps when both of the volatility and interest rate are assumed to be stochastic and follow a cir cox ingersoll ross process forming a heston cir hybrid model the solutions are written in a series form with a theoretical proof of their convergence ensuring the accuracy of the determined swap prices the application of the formulae in practice is also demonstrated through the designed numerical experiments

estimation and comparison of multiple change point models this paper provides a new bayesian approach for models with multiple change points the centerpiece of the approach is a formulation of the change point model in terms of a latent discrete state variable that indicates the regime from which a particular observation has been drawn this state variable is specified to evolve according to a discrete time discrete state markov process with the transition probabilities constrained so that the state variable can either stay at the current value or jump to the next higher value this parameterization exactly reproduces the change point model the model is estimated by markov chain monte carlo methods using an approach that is based on chib 1996 this methodology is quite valuable since it allows for the fitting of more complex change point models than was possible before methods for the computation of bayes factors are also developed all the techniques are illustrated using simulated and real data sets

alternative algorithms for the estimation of dynamic factor mimic and varying coefficient regression models this paper provides a general approach to the formulation and estimation of dynamic unobserved component models after introducing the general model two methods for estimating the unknown parameters are presented both are algorithms for maximizing the likelihood function the first is based on the method of scoring the second is the em algorithm a derivative free method each iteration of em requires a kalman filter and smoother followed by straightforward regression calculations the paper suggests using the em methods to quickly locate a neighborhood of the maximum scoring can then be used to pinpoint the maximum and calculate the information matrix

a new exact solution for pricing european options in a two state regime switching economy in this study we derive a new exact solution for pricing european options in a two state regime switching economy two coupled black scholes partial differential equations pdes under the regime switching are solved using the fourier transform method a key feature of the newly derived solution is its simplicity in the form of a single integral with a real integrand which leads to great computational efficiency in comparison with other closed form solutions previously presented in the literature numerical examples are provided to demonstrate some interesting results obtained from our pricing formula

a lattice based approach to option and bond valuation under mean reverting regime switching diffusion processes abstract nowadays the pricing of financial instruments under continuous time markov switching models have received a widespread attention from researchers and practitioners in the finance industry lattice based approaches are amongst the most widely used approaches to solve the pricing problem in this context recently yuen and yang 2010 have proposed a simple and fast recombining trinomial tree method to handle the case of regime switching geometric brownian motion processes in this paper we generalize their approach to the regime switching exponential mean reverting case with state dependent switching rates and derive the necessary conditions for the positivity of conditional branching probabilities we employ the hull and white s tree building procedure to limit tree growth away from the long run mean of the process we use the proposed lattice framework to price contingent claims of european american and barrier type and demonstrate its applicability in pricing default free zero coupon bonds in the proposed tree structure the number of nodes is substantially decreased and the computational cost is effectively reduced compared to the usual approaches extensive numerical experiments illustrate the efficiency and flexibility of the proposed scheme

a new approximate likelihood estimator for arma filtered hidden markov models hidden markov models hmms are successfully applied in various fields of time series analysis colored noise e g due to filtering violates basic assumptions of the model although it is well known how to consider autoregressive ar filtering there is no algorithm to take into account moving average ma filtering in parameter estimation exactly we present an approximate likelihood estimator for ma filtered hmm that is generalized to deal with an autoregressive moving average arma filtered hmm the approximation order of the likelihood calculation can be chosen therefore we obtain a sequence of estimators for the hmm parameters as well as for the filter coefficients the recursion equations for an efficient algorithm are derived from exact expressions for the forward iterations by simulations we show that the derived estimators are unbiased in filter situations where standard hmm s are not able to recover the true dynamics special implementation strategies together with small approximations yield further acceleration of the algorithm

rational qr iteration without inversion in this manuscript a new method will be presented for performing a qr iteration with a i a i 1 qr without explicit inversion of the factor a i 1 a qr method driven by a rational function is attractive since convergence can occur at both sides of the matrix each step of this new iteration consists of two substeps in the explicit version first an rq factorization of the initial matrix a i rq will be computed followed by a qr factorization of the matrix a i q h the factorization of a i q h can be computed in an intelligent manner exploiting properties of the already known rq factorization of a i the similarity transformation yielding the qr step is defined by the unitary factor q in the qr factorization of the transformed matrix a i q h examples will be given illustrating how to efficiently compute the factorization for some specific classes of matrices the novelties of this approach with respect to these matrix classes will be discussed

the qr algorithm for unitary hessenberg matrices abstract let h be an n n unitary right hessenberg matrix with positive subdiagonal elements using what we call the schur parameterization of h we show how one step of the shifted qr algorithm for h can be carried out in o n arithmetic operations coupled with the shift strategy of eberlein and huang 3 this will permit computation of the spectrum of h to machine precision in o n 2 operations one potential application is the computation of gauss szego quadrature formulas 12 given the associated schur parameters 7 the weights can also be computed by direct analogy with 6

moving mesh methods for pricing asian options with regime switching in this paper we study moving mesh implicit finite difference methods for pricing asian options with regime switching the price of asian options with regime switching follows a system of partial differential equations pdes with moving boundaries based on quadratic interpolation between two consecutive levels of computational solutions the moving mesh finite difference methods are constructed to solve the resulted system of pdes with moving boundaries the moving mesh finite difference schemes studied in this paper include implicit euler schemes rannacher schemes and crank nicolson schemes the convergence rates for these schemes are obtained numerical examples are provided to confirm the theoretical results

a spectral curve approach to lawson symmetric cmc surfaces of genus 2 minimal and cmc surfaces in s 3 can be treated via their associated family of flat sl 2 mathbb c connections in this the paper we parametrize the moduli space of flat sl 2 mathbb c connections on the lawson minimal surface of genus 2 which are equivariant with respect to certain symmetries of lawson s geometric construction the parametrization uses hitchin s abelianization procedure to write such connections explicitly in terms of flat line bundles on a complex 1 dimensional torus this description is used to develop a spectral curve theory for the lawson surface this theory applies as well to other cmc and minimal surfaces with the same holomorphic symmetries as the lawson surface but different riemann surface structure additionally we study the space of isospectral deformations of compact minimal surface of genus g ge 2 and prove that it is generated by simple factor dressing

estimation of markov regime switching regression models with endogenous switching abstract following hamilton 1989 a new approach to the economic analysis of nonstationary time series and the business cycle econometrica 57 357 384 estimation of markov regime switching regressions typically relies on the assumption that the latent state variable controlling regime change is exogenous we relax this assumption and develop a parsimonious model of endogenous markov regime switching inference via maximum likelihood estimation is possible with relatively minor modifications to existing recursive filters the model nests the exogenous switching model yielding straightforward tests for endogeneity in monte carlo experiments maximum likelihood estimates of the endogenous switching model parameters were quite accurate even in the presence of certain model misspecifications as an application we extend the volatility feedback model of equity returns given in turner et al 1989 a markov model of heteroskedasticity risk and learning in the stock market journal of financial economics 25 3 22 to allow for endogenous switching

markov switching models with endogenous explanatory variables ii a two step mle procedure abstract this paper proposes a two step maximum likelihood estimation mle procedure to deal with the problem of endogeneity in markov switching regression models a joint estimation procedure provides us with an asymptotically most efficient estimator but it is not always feasible due to the curse of dimensionality in the matrix of transition probabilities a two step estimation procedure which ignores potential correlation between the latent state variables suffers less from the curse of dimensionality and it provides a reasonable alternative to the joint estimation procedure in addition our monte carlo experiments show that the two step estimation procedure can be more efficient than the joint estimation procedure in finite samples when there is zero or low correlation between the latent state variables

rational krylov sequence methods for eigenvalue computation abstract algorithms to solve large sparse eigenvalue problems are considered a new class of algorithms which is based on rational functions of the matrix is described the lanczos method the arnoldi method the spectral transformation lanczos method and rayleigh quotient iteration all are special cases but there are also new algorithms which correspond to rational functions with several poles in the simplest case a basis of a rational krylov subspace is found in which the matrix eigenvalue problem is formulated as a linear matrix pencil with a pair of hessenberg matrices

pricing variance and volatility swaps with stochastic volatility stochastic interest rate and regime switching abstract in this paper we propose a two factor heston cir hybrid model for the pricing of variance and volatility swaps by introducing the second regime switching factor into the heston cir hybrid model while this model is closer to reality taking advantages of the heston stochastic volatility cir stochastic interest rate and regime switching it has a more complicated structure and thus leads to extra difficulty in finding analytical solutions albeit difficult we have still managed to present analytical pricing formulae for variance and volatility swaps based on the derived forward characteristic function in a series form the series solutions are accompanied by a radius of convergence to ensure its safe application and their fast convergence demonstrated through numerical experiments facilitates the implementation in practice

coarse classification of constant mean curvature cylinders we give a coarse classification of constant mean curvature cmc immersions of cylinders into r 3 via the loop group method particularly for this purpose we consider double loop groups and a new type of potentials which are meromorphic 1 forms on riemann surfaces

the qr iteration method for hermitian quasiseparable matrices of an arbitrary order abstract the qr iteration method for tridiagonal matrices is in the heart of one classical method to solve the general eigenvalue problem in this paper we consider the more general class of quasiseparable matrices that includes not only tridiagonal but also companion comrade unitary hessenberg and semiseparble matrices a fast qr iteration method exploiting the hermitian quasiseparable structure and thus generalizing the classical tridiagonal scheme is presented the algorithm is based on an earlier work y eidelman and i gohberg a modification of the dewilde van der veen method for inversion of finite structured matrices linear algebra appl 343 344 2002 419 450 and it applies to the general case of hermitian quasiseparable matrices of an arbitrary order the algorithm operates on generators i e a linear set of parameters defining the quasiseparable matrix and the storage and the cost of one iteration are only linear the results of some numerical experiments are presented an application of this method to solve the general eigenvalue problem via quasiseparable matrices will be analyzed separately elsewhere

markov switching models with endogenous explanatory variables the maximum likelihood estimation of a markov switching regression model based on the hamilton filter is not valid in the presence of endogenous explanatory variables however we show that there exists an appropriate transformation of the model that allows us to directly employ the hamilton filter the transformed model explicitly allows for a vector of bias correction terms as additional regressors and the new disturbance term is uncorrelated with all the regressors in the transformed model within this framework a quasi maximum likelihood estimation procedure is presented a procedure to test for endogeneity based on the wald statistic or the likelihood ratio statistic is also presented

multivariate contemporaneous threshold autoregressive models this paper proposes a contemporaneous threshold multivariate smooth transition autoregressive c mstar model in which the regime weights depend on the ex ante probabilities that latent regime specific variables exceed certain threshold values a key feature of the model is that the transition function depends on all the parameters of the model as well as on the data since the mixing weights are also a function of the regime specific noise covariance matrix the model can account for contemporaneous regime specific co movements of the variables the stability and distributional properties of the proposed model are discussed as well as issues of estimation testing and forecasting the practical usefulness of the c mstar model is illustrated by examining the relationship between us stock prices and interest rates

nonlinearities and regimes in conditional correlations with different dynamics abstract new parameterizations of the dynamic conditional correlation dcc model and of the regime switching dynamic correlation rsdc model are introduced such that these models provide a specific dynamics for each correlation they imply a nonlinear autoregressive form of dependence on lagged correlations and are based on properties of the hadamard exponential matrix the new models are applied to a data set of twenty stock market indices and a data set of the thirty dow jones components comparing them to the classical dcc and rsdc models the empirical results show that the new models improve their classical versions in terms of several criteria

markov switching model selection using kullback leibler divergence abstract in markov switching regression models we use kullback leibler kl divergence between the true and candidate models to select the number of states and variables simultaneously specifically we derive a new information criterion markov switching criterion msc which is an estimate of kl divergence msc imposes an appropriate penalty to mitigate the over retention of states in the markov chain and it performs well in monte carlo studies with single and multiple states small and large samples and low and high noise we illustrate the usefulness of msc via applications to the u s business cycle and to media advertising

regime switching dynamic correlations for asymmetric and fat tailed conditional returns abstract a non gaussian multivariate regime switching dynamic correlation model for financial asset returns is proposed it incorporates the multivariate generalized hyperbolic law for the conditional distribution of returns all model parameters are estimated consistently using a new two stage expectation maximization algorithm that also allows for incorporation of shrinkage estimation via quasi bayesian priors it is shown that use of markov switching correlation dynamics not only leads to highly accurate risk forecasts but also potentially reduces the regulatory capital requirements during periods of distress in terms of portfolio performance the new regime switching model delivers consistently higher sharpe ratios and smaller losses than the equally weighted portfolio and all competing models finally the regime forecasts are employed in a new dynamic risk control strategy that avoids most losses during the financial crisis and vastly improves risk adjusted returns

binomial markov switching multifractal model with skewed t innovations and applications to chinese ssec index abstract this paper presents the binomial markov switching multifractal bmsm model of asset returns with skewed t innovations bmsm skewed t for short which considers the fat tails skewness and multifractality in asset returns simultaneously the parameters of bmsm skewed t model can be estimated by maximum likelihood ml methods and volatility forecasting can be accomplished via bayesian updating in order to evaluate the performance of bmsm skewed t model bmsm model with normal innovations bmsm n bmsm model with student t innovations bmsm t and garch 1 1 models garch n garch t and garch skewed t are chosen for comparison through empirical studies on shanghai stock exchange composite index ssec we find that for sample estimation bmsm models outperform the garch 1 1 models through bic and aic rules and bmsm skewed t performs the best among all the models due to its fat tails skewness and multifractality in addition bmsm skewed t model dominates other models at most forecasting horizons for out of sample volatility forecasts in terms of mse mae and spa test

nonparametric time variant frequency response function estimates using arbitrary excitations the time variant frequency response function tv frf uniquely characterises the dynamic behaviour of a linear time variant ltv system this paper proposes a method for estimating nonparametrically the dynamic part of the tv frf from known input noisy output observations the arbitrary time variation of the tv frf is modelled by legendre polynomials in opposition to existing solutions the proposed method is applicable to arbitrary inputs

bayesian methods for hidden markov models markov chain monte carlo mcmc sampling strategies can be used to simulate hidden markov model hmm parameters from their posterior distribution given observed data some mcmc methods used in practice for computing likelihood conditional probabilities of hidden states and the most likely sequence of states can be improved by incorporating established recursive algorithms the most important of these is a set of forward backward recursions calculating conditional distributions of the hidden states given observed data and model parameters i show how to use the recursive algorithms in an mcmc context and demonstrate mathematical and empirical results showing a gibbs sampler using the forward backward recursions mixes more rapidly than another sampler often used for hmms iintroduce an augmented variables technique for obtaining unique state labels in hmms and finite mixture models i show how recursive computing allows the statistically efficient use of mcmc output when estimating the hidden states i

prescribing curvature on compact surfaces with conical singularities we study the berger nirenberg problem on surfaces with conical singularities i e we discuss conditions under which a function on a riemann surface is the gaussian curvature of some conformal metric with a prescribed set of singularities of conical types

algorithms for the geronimus transformation for orthogonal polynomials on the unit circle abstract let l be a positive definite bilinear functional on the unit circle defined on p n the space of polynomials of degree at most n then its geronimus transformation l is defined by l p q l z p z z q z for all p q p n c given l there are infinitely many such l which can be described by a complex free parameter the hessenberg matrix that appears in the recurrence relations for orthogonal polynomials on the unit circle is unitary and can be factorized using its associated schur parameters recent results show that the unitary hessenberg matrices associated with l and l respectively are related by a qr step where all the matrices involved are of order n 1 for the analogue on the real line of this so called spectral transformation the tridiagonal jacobi matrices associated with the respective functionals are related by an lr step in this paper we derive algorithms that compute the new schur parameters after applying a geronimus transformation we present two forward algorithms and one backward algorithm the qr step between unitary hessenberg matrices plays a central role in the derivation of each of the algorithms where the main idea is to do the inverse of a qr step making use of the special structure of unitary hessenberg matrices all the algorithms are efficient and need only o n flops we present several numerical experiments to analyse the accuracy and to explain the behaviour of the algorithms

specification testing when score test statistics are identically zero we investigate the problem of previous specification testing when the score vector evaluated at the restricted mle is identically zero several econometric examples are provided a general test procedure which generalizes the geometric principle of the score test is proposed the wald and the likelihood ratio tests are also analyzed even under such irregularities the usual asymptotic distribution of the likelihood ratio statistics is still valid however the wald type statistics need to be modified the generalized score the likelihood ratio and the modified wald tests are shown to be asymptotically equivalent the asymptotic efficiency of the tests is derived

the geometric cauchy problem for surfaces with lorentzian harmonic gauss maps the geometric cauchy problem for a class of surfaces in a pseudo riemannian manifold of dimension 3 is to find the surface which contains a given curve with a prescribed tangent bundle along the curve we consider this problem for constant negative gauss curvature surfaces pseudospherical surfaces in euclidean 3 space and for timelike constant non zero mean curvature cmc surfaces in the lorentz minkowski 3 space we prove that there is a unique solution if the prescribed curve is non characteristic and for characteristic initial curves asymptotic curves for pseudospherical surfaces and null curves for timelike cmc it is nec essary and sufficient for similar data to be prescribed along an additional characteristic curve that intersects the first the proofs also give a means of constructing all solutions using loop group techniques the method used is the infinite dimensional d alembert type representation for surfaces as sociated with lorentzian harmonic maps 1 1 wave maps into symmetric spaces developed since the 1990 s explicit formulae for the potentials in terms of the prescribed data are given and some applications are consid ered

second order lattice boltzmann methods for pdes of asian option pricing with regime switching this paper establishes lattice boltzmann models with five amending functions for solving system of partial differential equations pdes arising in asian options pricing with regime switching with the chapman enskog multi scale expansion the pdes are recovered correctly from the continuous boltzmann equation and then the lattice boltzmann method lbm is proposed in the lbm the coefficients of equilibrium distribution and amending functions are taken as polynomials instead of constants in the traditional lbms the lbm has second order convergence rate in space and first order convergence rate in time the stability convergence rates and computational cost of lbms are studied and verified by numerical examples