a fast vq codebook search with initialization and search order vector quantization vq a lossy image compression is widely used for many applications due to its simple architecture fast decoding ability and high compression rate traditionally vq applies the full search algorithm to search for the codeword that best matches each image vector in the encoding procedure however matching in this manner consumes a lot of computation time and leads to a heavy burden for the vq method therefore torres and huguet proposed a double test algorithm to improve the matching efficiency however their scheme does not include an initiation strategy to choose an initially searched codeword for each image vector and as a result matching efficiency may be affected significantly to overcome this drawback we propose an improved double test scheme with a fine initialization as well as a suitable search order our experimental results indicate that the computation time of the double test algorithm can be significantly reduced by the proposed method in addition the proposed method is more flexible than existing schemes

constructing robust good lattice rules for computational finance the valuation of many financial derivatives leads to high dimensional integrals the constructions of robust or universal good lattice rules for financial applications are both important and challenging an important common feature of the integrands in computational finance is that they can often be well approximated by a sum of low dimensional functions i e functions that depend on only a small number of variables usually just two variables for numerical integration of such functions the quality of the low order i e low dimensional projections of the node set is crucial in this paper we propose methods to construct good lattice points with optimal low order projections the quality of a point set is measured by a new measure called elementary order ell discrepancy which measures the quality of all order ell projections and is more informative than usual measures two constructions namely the korobov and the component by component constructions are studied such that the low order projections are optimized numerical experiments demonstrate that even in high dimensions it is possible to construct new good lattice points with order 2 projections that are better than those of the sobol points and random points and with higher order projections that are no worse while the sobol points lost the advantage over random points in order 2 projections on the average the new lattice rules have the potential to improve upon the accuracy for favorable functions while doing no harm for unfavorable ones their applications for pricing path dependent options and american options based on the least square monte carlo method are studied and their high efficiency is demonstrated a nice surprise revealed is the robustness property of such lattice rules the good projection property and the suitability for a large range of problems the potential possibility and limitations of good lattice points in achieving good quality of moderate and high order projections is investigated the reason why classical lattice rules may not be efficient for high dimensional finance problems is also discussed

pricing derivative securities using integrated quasi monte carlo methods with dimension reduction and discontinuity realignment the dimension and the smoothness of the integrands are the two key factors that affect the efficiency of the quasi monte carlo qmc method the first factor implies that the qmc method can have high performance on a problem with low effective dimension even though its nominal dimension can be very large the second factor suggests that the qmc method can be very effective on a problem that is smooth or even on a problem that is discontinuous as long as its discontinuities are parallel to the axes motivated by these findings methods such as the linear transformation lt method of j imai and k s tan j comput finance 10 2006 pp 129 155 and the orthogonal transformation ot method of x wang and k s tan management sci 59 2013 pp 376 389 have been proposed to increase the efficiency of qmc however both of these methods are unsatisfactory in that they only achieve optimality single sidedly in the sense that the lt method enhances qmc by effective dimension reduction while the o

a hybrid monte carlo acceleration method of pricing basket options based on splitting abstract pricing basket options has always been one of the key problems in financial engineering because of high dimensionality and low convergence rate this paper proposes a hybrid monte carlo variance reduction method for pricing basket options first by splitting the payoff of the basket option into two parts we can price basket options by value the two parts respectively the first part has a closed form expectation formula the second part can be considered as a small probability event to reduce variance for simulating the second part the conditional monte carlo cmc method combined with the importance sampling is method is adapted because these two methods are all effective to deal with small probability events for is method it is a challenge to compute the optimal parameters with as little computational cost as possible therefore an efficient prediction correction pc iteration algorithm based on moment estimation is proposed to determine the optimal parameters in the importance sampling method some theoretical analyses for the existence and uniqueness of the optimal parameters in the is method and the convergence of the pc method are also given numerical results show that the hybrid variance reduction method has great variance reduction effect and pc iteration algorithm can save a lot of computing costs comparing with the traditional newton s iteration method

the hexanomial lattice for pricing multi asset options abstract multi asset options are important financial derivatives because closed form solutions do not exist for most of them numerical alternatives such as lattice are mandatory but lattices that require the correlation between assets to be confined to a narrow range will have limited uses let ij denote the correlation between assets i and j this paper defines a correlation optimal lattice as one that guarantees validity as long as 1 o t ij 1 o t for all pairs of assets i and j where t is the duration of a time period this paper then proposes the first optimal bivariate lattice generalizable to higher dimensions called the hexanomial lattice this lattice furthermore has the flexibility to handle a barrier on each asset experiments confirm its excellent numerical performance compared with alternative lattices

cp methods and the evaluation of negative energy coulomb whittaker functions the cp constant reference potential perturbation methods are specialized methods for the solution of schrodinger equations they allow big step sizes and are well suited for oscillatory problems as an illustration of the power of these methods oscillating whittaker functions of the second kind are obtained with high accuracy using a high order cp method

pricing and hedging with discontinuous functions quasi monte carlo methods and dimension reduction quasi monte carlo qmc methods are important numerical tools in the pricing and hedging of complex financial instruments the effectiveness of qmc methods crucially depends on the discontinuity and the dimension of the problem this paper shows how the two fundamental limitations can be overcome in some cases we first study how path generation methods pgms affect the structure of the discontinuities and what the effect of discontinuities is on the accuracy of qmc methods the insight is that the discontinuities can be qmc friendly i e aligned with the coordinate axes or not depending on the pgm the pgms that offer the best performance in qmc methods are those that make the discontinuities qmc friendly the structure of discontinuities can affect the accuracy of qmc methods more significantly than the effective dimension this insight motivates us to propose a novel way of handling the discontinuities the basic idea is to align the discontinuities with the coordinate axes by a judicious design of a method for simulating the underlying processes numerical experiments demonstrate that the proposed method leads to dramatic variance reduction in qmc methods for pricing options and for estimating greeks it also reduces the effective dimension of the problem this paper was accepted by assaf zeevi stochastic models and simulation

an exponentially convergent functional discrete method for solving sturm liouville problems with a potential including the dirac δ function abstract in this paper we present a functional discrete method for solving sturm liouville problems with a potential that includes a function from l 1 0 1 and the dirac function for both the linear and the nonlinear case sufficient conditions for an exponential rate of convergence of the method are obtained the question of a possible software implementation of the method is discussed in detail the theoretical results are successfully confirmed by a numerical example

numerical methods for the eigenvalue determination of second order ordinary differential equations an accurate method for the numerical solution of the eigenvalue problem of second order ordinary differential equation using the shooting method is presented the method has three steps firstly initial values for the eigenvalue and eigenfunction at both ends are obtained by using the discretized matrix eigenvalue method secondly the initial value problem is solved using new highly accurate formulas of the linear multistep method thirdly the eigenvalue is properly corrected at the matching point the efficiency of the proposed methods is demonstrated by their applications to bound states for the one dimensional harmonic oscillator anharmonic oscillators the morse potential and the modified poschl teller potential in quantum mechanics

strong tractability of multivariate integration using quasi monte carlo algorithms we study quasi monte carlo algorithms based on low discrepancy sequences for multivariate integration we consider the problem of how the minimal number of function evaluations needed to reduce the worst case error from its initial error by a factor of e depends on e 1 and the dimension s strong tractability means that it does not depend on s and is bounded by a polynomial in e 1 the least possible value of the power of e 1 is called the e exponent of strong tractability sloan and wozniakowski established a necessary and sufficient condition of strong tractability in weighted sobolev spaces and showed that the e exponent of strong tractability is between 1 and 2 however their proof is not constructive in this paper we prove in a constructive way that multivariate integration in some weighted sobolev spaces is strongly tractable with e exponent equal to 1 which is the best possible value under a stronger assumption than sloan and wozniakowski s assumption we show that quasi monte carlo algorithms using niederreiter s t s sequences and sobol sequences achieve the optimal convergence order o n 1 for any 0 independent of the dimension with a worst case deterministic guarantee where n is the number of function evaluations this implies that strong tractability with the best e exponent can be achieved in appropriate weighted sobolev spaces by using niederreiter s t s sequences and sobol sequences

a generalized discrepancy and quadrature error bound an error bound for multidimensional quadrature is derived that includes the koksma hlawka inequality as a special case this error bound takes the form of a product of two terms one term which depends only on the integrand is defined as a generalized variation the other term which depends only on the quadrature rule is defined as a generalized discrepancy the generalized discrepancy is a figure of merit for quadrature rules and includes as special cases the l p star discrepancy and p that arises in the study of lattice rules

a general control variate method for option pricing under levy processes abstract we present a general control variate method for simulating path dependent options under levy processes it is based on fast numerical inversion of the cumulative distribution functions and exploits the strong correlation of the payoff of the original option and the payoff of a similar option under geometric brownian motion the method is applicable for all types of levy processes for which the probability density function of the increments is available in closed form numerical experiments confirm that our method achieves considerable variance reduction for different options and levy processes we present the applications of our general approach for asian lookback and barrier options under variance gamma normal inverse gaussian generalized hyperbolic and meixner processes

pitfalls in parameter estimation for delay differential equations given a set of data u gamma i approx u gamma i sfp star corresponding to the em delay differential equation vspace 3pt begin array rcll u t sfp f t u t sfp u alpha t sfp sfp sfp hbox for t ge t 0 sfp u t sfp psi t sfp hbox for t le t 0 sfp end array the basic problem addressed here is that of calculating the vector sfp star in blackboardd r n we also consider neutral differential equations most approaches to parameter estimation calculate sfp star by minimizing a suitable objective function that is assumed by the minimization algorithm to be sufficiently smooth in this paper we use the derivative discontinuity tracking theory for delay differential equations to analyze how jumps can arise in the derivatives of a natural objective function these jumps can occur when estimating parameters in lag functions and estimating the position of the initial point and as such are not expected to occur in parameter estimation problems for ordinary differential equations

parameter estimation for differential equations a generalized smoothing approach summary we propose a new method for estimating parameters in models that are defined by a system of non linear differential equations such equations represent changes in system outputs by linking the behaviour of derivatives of a process to the behaviour of the process itself current methods for estimating parameters in differential equations from noisy data are computationally intensive and often poorly suited to the realization of statistical objectives such as inference and interval estimation the paper describes a new method that uses noisy measurements on a subset of variables to estimate the parameters defining a system of non linear differential equations the approach is based on a modification of data smoothing methods along with a generalization of profiled estimation we derive estimates and confidence intervals and show that these have low bias and good coverage properties respectively for data that are simulated from models in chemical engineering and neurobiology the performance of the method is demonstrated by using real world data from chemistry and from the progress of the autoimmune disease lupus

on asian option pricing for nig levy processes in this paper we derive approximations and bounds for the esscher price of european style arithmetic and geometric average options the asset price process is assumed to be of exponential levy type with normal inverse gaussian nig distributed log returns numerical illustrations of the accuracy of these bounds as well as approximations and comparisons of the nig average option prices with the corresponding black scholes prices are given

constructing embedded lattice rules for multivariate integration lattice rules are a family of equal weight cubature formulae for approximating high dimensional integrals by now it is well established that good generating vectors for lattice rules having n points can be constructed component by component for integrands belonging to certain weighted function spaces and that they can achieve the optimal rate of convergence although the lattice rules constructed this way are extensible in dimension they are not extensible in n thus when n is changed the generating vector needs to be constructed anew in this paper we introduce a new algorithm for constructing good generating vectors for embedded lattice rules which can be used for a range of n while still being extensible in dimension by using an adaptation of the fast component by component construction algorithm which makes use of fast fourier transforms we are able to obtain good generating vectors for thousands of dimensions and millions of points under both product weight and order dependent weight settings at the cost of calo d n log n 2 operations with a sufficiently large number of points and good overall quality these embedded lattice rules can be used for practical purposes in the same way as a low discrepancy sequence we show for a range of weight settings in an unanchored sobolev space that our embedded lattice rules achieve the same optimal rate of convergence calo n 1 delta delta 0 as those constructed for a fixed number of points and that the implied constant gains only a factor of 1 30 to 1 55

conditional quasi monte carlo methods and dimension reduction for option pricing and hedging with discontinuous functions abstract quasi monte carlo qmc methods are efficient simulation tools for multidimensional integrations in financial engineering discontinuous functions occur commonly in pricing or hedging financial options high dimensionality and the lack of smoothness may cause an adverse impact on the performance of qmc in this paper we develop a two step procedure to handle the discontinuities and to reduce the effective dimensions successively the first step is to smooth the discontinuous functions by using conditional qmc methods which are deterministic versions of conditional monte carlo methods we show how the discontinuities can be removed completely and how the required conditional expectations can be calculated analytically for some typical pricing and hedging problems the second step is to reduce the effective dimensions of the resulting functions by using a new dimension reduction method based on principal component analysis pca rather than aiming at decomposing the covariance matrix of the brownian motion as in the traditional pca the new dimension reduction method implements pca on the gradients of the smoothed functions at a number of points in order to learn the inherent structure information a suitable orthogonal transformation is obtained to reduce the effective dimensions numerical experiments show that the two step procedure significantly improves qmc methods in pricing or hedging options with discontinuous functions

quasi monte carlo methods in financial engineering an equivalence principle and dimension reduction quasi monte carlo qmc methods are playing an increasingly important role in the pricing of complex financial derivatives for models in which the prices of the underlying assets are driven by brownian motions the performance of qmc methods is known to depend crucially on the construction of brownian motions this paper focuses on the impact of various constructions although the brownian bridge bb construction often yields very good results as papageorgiou pointed out there are financial derivatives for which the bb construction performs badly papageorgiou a 2002 the brownian bridge does not offer a consistent advantage in quasi monte carlo integration j complexity18 1 171 186 in this paper we first extend papageorgiou s analysis to establish an equivalence principle if the bb construction or any other construction is the preferred construction for a particular financial derivative then for any other construction there is another financial derivative for which the latter construction is the preferred one in this sense all methods of construction are equivalent and no method is consistently superior to others it all depends on the particular financial derivative we then show how to find a good construction for a particular class of financial derivatives in practice our strategy is to find a good construction for an easy problem and then apply it to more complicated problems related to the easy one this strategy is applied to the arithmetic asian options including bermudan asian options based on the weighted average of the stock prices we do this by studying a simpler problem namely the geometric asian option for which the best construction is easily available and applying it to the arithmetic asian option numerical experiments confirm the success of this strategy whereas in qmc all the common methods the standard method bb and principal component analysis may lose their power in some situations the new method behaves very well in all cases further large variance reduction can be achieved in combination with a control variate the new method can be interpreted as a practical way of reducing the effective dimension for some class of functions

space time adaptive finite difference method for european multi asset options the multi dimensional black scholes equation is solved numerically for a european call basket option using a priori a posteriori error estimates the equation is discretized by a finite difference method on a cartesian grid the grid is adjusted dynamically in space and time to satisfy a bound on the global error the discretization errors in each time step are estimated and weighted by the solution of the adjoint problem bounds on the local errors and the adjoint solution are obtained by the maximum principle for parabolic equations comparisons are made with monte carlo and quasi monte carlo methods in one dimension and the performance of the method is illustrated by examples in one two and three dimensions

quasi monte carlo methods in numerical finance this paper introduces and illustrates a new version of the monte carlo method that has attractive properties for the numerical valuation of derivatives the traditional monte carlo method has proven to be a powerful and flexible tool for many types of derivatives calculations under the conventional approach pseudo random numbers are used to evaluate the expression of interest unfortunately the use of pseudo random numbers yields an error bound that is probabilistic which can be a disadvantage another drawback of the standard approach is that many simulations may be required to obtain a high level of accuracy there are several ways to improve the convergence of the standard method this paper suggests a new approach which promises to be very useful for applications in finance quasi monte carlo methods use sequences that are deterministic instead of random these sequences improve convergence and give rise to deterministic error bounds the method is explained and illustrated with several examples these examples include complex derivatives such as basket options asian options and energy swaps

an accelerating quasi monte carlo method for option pricing under the generalized hyperbolic levy process in this paper we develop a simple and yet practically efficient algorithm for simulating high dimensional exotic options our method is based on an extension of imai and tan s linear transformation method which is originally proposed in the context of simulating a gaussian process by generalizing this method to other stochastic processes and exploiting the numerical inversion method of hormann and leydold this method can be used to enhance quasi monte carlo method in a wide range of applications we demonstrate the relative efficiency of our proposed simulation technique using exotic option examples including asian lookback barrier and cliquet options for which the underlying asset price follows an exponential generalized hyperbolic levy process we also illustrate the impact of our proposed method on dimension reduction

bounds for the price of discrete arithmetic asian options in this paper the pricing of european style discrete arithmetic asian options with fixed and floating strike is studied by deriving analytical lower and upper bounds in our approach we use a general technique for deriving upper and lower bounds for stop loss premiums of sums of dependent random variables as explained in kaas et al ins math econom 27 2000 151 168 and additionally the ideas of rogers and shi j appl probab 32 1995 1077 1088 and of nielsen and sandmann j financial quant anal 38 2 2003 449 473 we are able to create a unifying framework for european style discrete arithmetic asian options through these bounds that generalizes several approaches in the literature as well as improves the existing results we obtain analytical and easily computable bounds the aim of the paper is to formulate an advice of the appropriate choice of the bounds given the parameters investigate the effect of different conditioning variables and compare their efficiency numerically several sets of numerical results are included we also discuss hedging using these bounds moreover our methods are applicable to a wide range of pricing problems involving a sum of dependent random variables

a stochastic algorithm for high dimensional integrals over unbounded regions with gaussian weight abstract details are given for a fortran implementation of an algorithm that uses stochastic spherical radial rules for the numerical computation of multiple integrals over unbounded regions with gaussian weight the implemented rules are suitable for high dimensional problems a high dimensional example from a computational finance application is used to illustrate the use of the rules

path generation for quasi monte carlo simulation of mortgage backed securities monte carlo simulation is playing an increasingly important role in the pricing and hedging of complex path dependent financial instruments low discrepancy simulation methods offer the potential to provide faster rates of convergence than those of standard monte carlo methods however in high dimensional problems special methods are required to ensure that the faster convergence rates hold indeed ninomiya and tezuka 1996 have shown highdimensional examples in which low discrepancy methods perform worse than monte carlo methods the principal component construction introduced by acworth et al 1998 provides one solution to this problem however the computational effort required to generate each path grows quadratically with the dimension of the problem this article presents two new methods that offer accuracy equivalent in terms of explained variability to the principal components construction with computational requirements that are linearly related to the problem dimension one method is to take into account knowledge about the payoff function which makes it more flexible than the brownian bridge construction numerical results are presented that show the benefits of such adjustments the different methods are compared for the case of pricing mortgage backed securities using the hull white term structure model

on the explicit evaluation of the geometric asian options in stochastic volatility models with jumps in the present paper we provide a semiexplicit valuation formula for geometric asian options with fixed and floating strike under continuous monitoring when the underlying stock price process exhibits both stochastic volatility and jumps more precisely we shall work in the barndorff nielsen and shephard bns model framework we shall provide some numerical illustrations of the results obtained

low discrepancy sequences in high dimensions how well are their projections distributed quasi monte carlo qmc methods have been successfully used to compute high dimensional integrals arising in many applications especially in finance to understand the success and the potential limitation of qmc this paper focuses on quality measures of point sets in high dimensions we introduce the email protected superposition and truncation discrepancies which measure the quality of selected projections of a point set on lower dimensional spaces these measures are more informative than the classical ones we study their relationships with the integration errors and study the tractability issues we present efficient algorithms to compute these discrepancies and perform computational investigations to compare the performance of the sobol nets with that of the sets of latin hypercube sampling and random points numerical results show that in high dimensions the superiority of the sobol nets mainly derives from the one dimensional projections and the projections associated with the earlier dimensions for order 2 and higher order projections all these point sets have similar behavior on the average in weighted cases with fast decaying weights the sobol nets have a better performance than the other two point sets the investigation enables us to better understand the properties of qmc and throws new light on when and why qmc can have a better or no better performance than monte carlo for multivariate integration in high dimensions

rapid and brief communication initialization insensitive lvq algorithm based on cost function adaptation a learning vector quantization lvq algorithm called harmonic to minimum lvq algorithm h2m lvq is presented to tackle the initialization sensitiveness problem associated with the original generalized lvq glvq algorithm experimental results show superior performance of the h2m lvq algorithm over the glvq and one of its variants on several datasets datasets

numerical solution of sturm liouville problems via fer streamers we address the numerical challenge of solving regular sturm liouville problems in liouville s normal form with a continuous and piecewise analytic potential and self adjoint separated boundary conditions the novelty of our approach which is based on a non standard truncation of fer expansions which we call fer streamers lies in the construction of a new numerical method which 1 does not impose any restriction on the step size for eigenvalues which are greater than or equal to the minimum of the potential 2 requires only a mild restriction on the step size for the remaining finite number of eigenvalues 3 can attain any convergence rate which grows exponentially with the number of terms and is uniform for every eigenvalue and 4 lends itself to a clear understanding of the manner in which the potential affects the local and global errors we provide our numerical method with its analytical underpinning but emphasize that it is at an early stage of development and that much remains to be done in particular we comment on our investigation of efficient discretization schemes for the integrals which arise in fer streamers

a comparison of three methods for selecting values of input variables in the analysis of output from a computer code two types of sampling plans are examined as alternatives to simple random sampling in monte carlo studies these plans are shown to be improvements over simple random sampling with respect to variance for a class of estimators which includes the sample mean and the empirical distribution function

average case complexity of multivariate integration we study the average case complexity of multivariate integration for the class of continuous functions of d variables equipped with the classical wiener sheet measure to derive the average case complexity one needs to obtain optimal sample points we prove that optimal design is closely related to discrepancy theory which has been extensively studied for many years this relation enables us to show that optimal sample points can be derived from hammersley points extending the result of roth and using the recent result of wasilkowski we conclude that the average case complexity is e 1 lne 1 d 1 2

on the effects of dimension reduction techniques on some high dimensional problems in finance many problems in finance can be formulated as high dimensional integrals which are often attacked by quasi monte carlo qmc algorithms to enhance qmc algorithms dimension reduction techniques such as the brownian bridge bb and principal component analysis pca are used to reduce the effective dimension this paper explores in depth the effects of these techniques on the dimension structure of some typical high dimensional problems from finance the pricing of path dependent options and bond valuation according to term structure models by deriving explicit expressions for the underlying integrands and the associated weights that control the relative importance of different variables and by investigating the variance ratios the effective dimensions the mean dimension and their limiting behavior as the nominal dimension tends to infinity we show theoretically and empirically how and to what extent the bb and pca algorithms change the dimension structure including the degree of additivity of the underlying functions they change the functions to be strongly weighted and substantially reduce the effective dimensions and the mean dimension and they enhance the degree of additivity which is particularly important for qmc moreover the resulting functions are of low effective dimension not only in the superposition sense but also in the truncation sense the variance ratios the effective dimensions and the mean dimension associated with these techniques are very insensitive to the nominal dimension they are essentially constant which highlights the possibility of removing the curse of dimensionality when dimension reduction techniques are used in combination with qmc a counterexample is also shown for which the bb and pca may increase the effective dimension the investigation provides further insight into the effects of dimension reduction techniques

pricing bounds for discrete arithmetic asian options under levy models abstract analytical bounds for asian options are almost exclusively available in the black scholes framework in this paper we derive bounds for the price of a discretely monitored arithmetic asian option when the underlying asset follows an arbitrary levy process explicit formulas are given for kou s model merton s model the normal inverse gaussian model the cgmy model and the variance gamma model the results are compared with the comonotonic upper bound existing numerical results monte carlo simulations and in the case of the variance gamma model with an existing lower bound the method outlined here provides lower and upper bounds that are quick to evaluate and more accurate than existing bounds

reclaiming quasimonte carlo efficiency in portfolio value at risk simulation through fourier transform quasimonte carlo methods overcome the problem of sample clustering in regular monte carlo simulation and have been shown to improve simulation efficiency in the derivatives pricing literature when the price is expressed as a multidimensional integration and the integrand is suitably smooth for portfolio value at risk var problems the distribution of portfolio value change is based on the expectation of an indicator function hence the integrand is discontinuous the purpose of this paper is to smooth the expectation estimation of an indicator function via fourier transform so that the faster convergence rate of quasimonte carlo methods can be reclaimed theoretically under fairly mild assumptions the simulation of portfolio value at risk is fast and accurate numerical examples elucidate the advantage of the proposed approach over regular monte carlo and quasimonte carlo methods

application of the cross entropy method to the glvq algorithm this paper discusses an alternative approach to parameter optimization of well known prototype based learning algorithms minimizing an objective function via gradient search the proposed approach considers a stochastic optimization called the cross entropy method ce method the ce method is used to tackle efficiently the initialization sensitiveness problem associated with the original generalized learning vector quantization glvq algorithm and its variants results presented in this paper indicate that the ce method can be successfully applied to this kind of problem on real world data sets as far as known by the authors it is the first use of the ce method in prototype based learning

a posteriori error bounds for the approximate solution of second order odes by piecewise coefficients perturbation methods this paper presents a constructive approach to compute upper and lower bounds for the error committed when some second order ordinary differential equations are solved by means of piecewise coefficients perturbation methods ppm the obtained error bounds are expressed in terms of perturbations introduced in the differential operator and in the prescribed initial conditions the results are applied to collocation tau and finite difference methods numerical examples demonstrating the sharpness of the constructed bounds are given

an improved tau method for a class of sturm liouville problems we consider the numerical solution of sturm liouville eigenvalue problems by legendre gauss tau method the latter approximates the solution of differential equations as a finite sum of legendre polynomials l k x k n we propose an improved approach which seeks approximants in terms of a finite sum of exponentially weighted legendre polynomials e w k xl k x k n for some real or complex frequencies w k with the introduction of such exponentials legendre gauss tau method can detect the sharp variations exhibited by the highly indexed sturm liouville eigenfunctions the efficiency of our results is illustrated through numerical examples

cp methods for the schro dinger equation revisited abstract on constructing cpm propagators with an abundant number of terms by mathematica we have shown that the cpm n q where n is the number of polynomial terms by which the potential is approximated in each interval and q the number of corrections introduced is a method of order 2n 2 at low energies if q 2 3 n 1 and of order n at high energies if q 1 we have also proved that in the last case the error damps out as 1 e for both initial and boundary value problems we have written a program for boundary value problems which is of order 12 10 at low and high energies respectively and have found out that it is far more efficient than the well established codes sl02f sledge and sleign

handling discontinuities in financial engineering good path simulation and smoothing discontinuities are common in the pricing and hedging of complex financial derivatives quasi monte carlo qmc methods for high dimensional finance problems with discontinuities can be inefficient because of the lack of good smoothness and high dimensionality interestingly path simulation method psm may affect both factors implying its significance in qmc methods what defines a good psm for problems with discontinuities the ability to align the discontinuities with the coordinate axes is a desirable property for a psm we show that for an arbitrary psm there exists a class of options with discontinuous payoff functions such that the transformed functions have only axis parallel discontinuities for which good qmc performance can be expected in this sense any psm can be good in qmc methods for a specific class of problems we analyze the structure of discontinuities for digital options using the new approach and show the superiority and the uniqueness up to a permutation of the standard co

the error bounds and tractability of quasi monte carlo algorithms in infinite dimension dimensionally unbounded problems are frequently encountered in practice such as in simulations of stochastic processes in particle and light transport problems and in the problems of mathematical finance this paper considers quasi monte carlo integration algorithms for weighted classes of functions of infinitely many variables in which the dependence of functions on successive variables is increasingly limited the dependence is modeled by a sequence of weights the integrands belong to rather general reproducing kernel hilbert spaces that can be decomposed as the direct sum of a series of their subspaces each subspace containing functions of only a finite number of variables the theory of reproducing kernels is used to derive a quadrature error bound which is the product of two terms the generalized discrepancy and the generalized variation tractability means that the minimal number of function evaluations needed to reduce the initial integration error by a factor s is bounded by ce p for some exponent p and some positive constant c the e exponent of tractability is defined as the smallest power of e 1 in these bounds it is shown by using monte carlo quadrature that the e exponent is no greater than 2 for these weighted classes of integrands under a somewhat stronger assumption on the weights and for a popular choice of the reproducing kernel it is shown constructively using the halton sequence that the e exponent of tractability is 1 which implies that infinite dimensional integration is no harder than one dimensional integration

parameter estimation for odes using a cross entropy approach parameter estimation for odes is an important topic in numerical analysis in this paper we present a novel approach to address this inverse problem that can be applied to differential equations that may include delay terms cross entropy algorithms are general algorithms which can be applied to solve global optimization problems the main steps of cross entropy methods are first to generate a set of trial samples from a certain distribution and then to update the distribution based on these generated sample trials to overcome the prohibitive computation of standard cross entropy algorithms we develop a modification combining local search techniques the modified cross entropy algorithm can improve the convergence rate and reduce the chances of converging to a local optimum two different coding schemes continuous coding and discrete coding are introduced to represent the search space that we are optimizing over continuous coding uses a truncated multivariate gaussian to generate trial samples wh

solution of sturm liouville problems using modified neumann schemes the main purpose of this paper is to describe the extension of the successful modified integral series methods for schrodinger problems to more general sturm liouville eigenvalue problems we present a robust and reliable modified neumann method which can handle a wide variety of problems this modified neumann method is closely related to the second order pruess method but provides for higher order approximations we show that the method can be successfully implemented in a competitive automatic general purpose software package

why are high dimensional finance problems often of low effective dimension many problems in mathematical finance can be formulated as high dimensional integrals where the large number of dimensions arises from small time steps in time discretization and or a large number of state variables quasi monte carlo qmc methods have been successfully used for approximating such integrals to understand this success this paper focuses on investigating the special features of some typical high dimensional finance problems namely option pricing and bond valuation we provide new insight into the connection between the effective dimension and the efficiency of qmc and we present methods to analyze the dimension structure of a function we confirm the observation of caflisch morokoff and owen that functions from finance are often of low effective dimension in the sense that they can be well approximated by their low order anova analysis of variance terms usually just the order 1 and order 2 terms we explore why the effective dimension is small for many integrals from finance by deriving explicit forms of the anova terms in simple cases we find that the importance of each dimension is naturally weighted by certain hidden weights these weights characterize the relative importance of different variables or groups of variables and limit the importance of the higher order anova terms we study the variance ratios captured by low order anova terms and their asymptotic properties as the dimension tends to infinity and we show that with the increase of dimension the lower order terms continue to play a significant role and the higher order terms tend to be negligible this provides some insight into high dimensional problems from finance and explains why qmc algorithms are efficient for problems of this kind

an auto realignment method in quasi monte carlo for pricing financial derivatives with jump structures abstract discontinuities are common in the pricing of financial derivatives and have a tremendous impact on the accuracy of quasi monte carlo qmc method while if the discontinuities are parallel to the axes good efficiency of the qmc method can still be expected by realigning the discontinuities to be axes parallel wang tan 2013 succeeded in recovering the high efficiency of the qmc method for a special class of functions motivated by this work we propose an auto realignment method to deal with more general discontinuous functions the k means clustering algorithm a classical algorithm of machine learning is used to select the most representative normal vectors of the discontinuity surface by applying this new method the discontinuities of the resulting function are realigned to be friendly for the qmc method numerical experiments demonstrate that the proposed method significantly improves the performance of the qmc method

regularized margin based conditional log likelihood loss for prototype learning the classification performance of nearest prototype classifiers largely relies on the prototype learning algorithm the minimum classification error mce method and the soft nearest prototype classifier snpc method are two important algorithms using misclassification loss this paper proposes a new prototype learning algorithm based on the conditional log likelihood loss cll which is based on the discriminative model called log likelihood of margin logm a regularization term is added to avoid over fitting in training as well as to maximize the hypothesis margin the cll in the logm algorithm is a convex function of margin and so shows better convergence than the mce in addition we show the effects of distance metric learning with both prototype dependent weighting and prototype independent weighting our empirical study on the benchmark datasets demonstrates that the logm algorithm yields higher classification accuracies than the mce generalized learning vector quantization glvq soft nearest prototype classifier snpc and the robust soft learning vector quantization rslvq and moreover the logm with prototype dependent weighting achieves comparable accuracies to the support vector machine svm classifier

estimation of time and state dependent delays and other parameters in functional differential equations a parameter estimation algorithm is developed that can be used to estimate unknown time or state dependent delays and other parameters e g initial condition appearing within a nonlinear nonautonomous functional differential equation the original infinite dimensional differential equation is approximated using linear splines which are allowed to move with the variable delay the variable delays are approximated using linear splines as well the approximation scheme produces a system of ordinary differential equations with nice computational properties the unknown parameters are estimated within the approximating systems by minimizing a least squares fit to data criterion convergence theorems are proved for time dependent delays and state dependent delays within two classes which say essentially that fitting the data by using approximations will in the limit provide a fit to the data using the original system numerical test examples are presented that illustrate the method for all types of delay

variance reduction via lattice rules this is a review article on lattice methods for multiple integration over the unit hypercube with a variance reduction viewpoint it also contains some new results and ideas the aim is to examine the basic principles supporting these methods and how they can be used effectively for the simulation models that are typically encountered in the area of management science these models can usually be reformulated as integration problems over the unit hypercube with a large sometimes infinite number of dimensions we examine selection criteria for the lattice rules and suggest criteria which take into account the quality of the projections of the lattices over selected low dimensional subspaces the criteria are strongly related to those used for selecting linear congruential and multiple recursive random number generators numerical examples illustrate the effectiveness of the approach

default prediction with the merton type structural model based on the nig levy process merton s model merton 1974 has long been a standard for estimating company s probability of default pd for listed companies the major advantage of merton s model is the use of current market prices to determine the probability of default the logic behind the model is simple the market prices best reflect all the relevant information being forward looking estimates of company s prospect and should be and are superior to the balance sheet disclosures which at best are ex post realisations of company s performance it is thus a pity that the benefits strengths of merton s model are hindered by a significant shortcoming of the model namely the assumption of normally distributed returns as numerous authors point out barndorff nielsen 1997 5 6 prause 1999 eberlein 2001 brambilla et al 2015 stock returns are not normally distributed which significantly limits the use of model in practice moreover the estimates of pds can be biased downwards exposing the banks to the possibility of undercapitalisation and systematic shocks it is the purpose of this paper to remedy this situation firstly we extend the merton model by allowing for normal inverse gaussian nig distributed returns as several authors point out using the examples of options schoutens 2009 nig in most cases provides a robust statistical platform for estimating stock returns we further extend our approach by constructing a robust em algorithm for estimating pds within the merton nig framework we also test the reliability of the nig improved merton model against classical merton s model for estimating pds applying our results to ljubljana stock exchange we find that the pd estimates using classical merton s model are biased whereas the estimates from nig merton s model are robust

quasi random sequences and their discrepancies quasi random also called low discrepancy sequences are a deterministic alternative to random sequences for use in monte carlo methods such as integration and particle simulations of transport processes the error in uniformity for such a sequence of n points in the s dimensional unit cube is measured by its discrepancy which is of size log n s n 1 for large n as opposed to discrepancy of size log log n 1 2 n 1 2 for a random sequence i e for almost any randomly chosen sequence several types of discrepancies one of which is new are defined and analyzed a critical discussion of the theoretical bounds on these discrepancies is presented computations of discrepancies are presented for a wide choice of dimension s number of points n and different quasi random sequences in particular for moderate or large s there is an intermediate regime in which the discrepancy of a quasi random sequence is almost exactly the same as that of a randomly chosen sequence a simplified p

automatic solution of sturm liouville problems using the pruess method abstract we develop algorithms based on coefficient approximation for the automatic solution of regular and singular sturm liouville problems

least squares polynomial chaos expansion a review of sampling strategies abstract as non intrusive polynomial chaos expansion pce techniques have gained growing popularity among researchers we here provide a comprehensive review of major sampling strategies for the least squares based pce traditional sampling methods such as monte carlo latin hypercube quasi monte carlo optimal design of experiments ode gaussian quadratures as well as more recent techniques such as coherence optimal and randomized quadratures are discussed we also propose a hybrid sampling method dubbed alphabetic coherence optimal that employs the so called alphabetic optimality criteria used in the context of ode in conjunction with coherence optimal samples a comparison between the empirical performance of the selected sampling methods applied to three numerical examples including high order pce s high dimensional problems and low oversampling ratios is presented to provide a road map for practitioners seeking the most suitable sampling technique for a problem at hand we observed that the alphabetic coherence optimal technique outperforms other sampling methods specially when high order ode are employed and or the oversampling ratio is low

constructing sobol sequences with better two dimensional projections direction numbers for generating sobol sequences that satisfy the so called property a in up to 1111 dimensions have previously been given in joe and kuo acm trans math software 29 2003 pp 49 57 however these sobol sequences may have poor two dimensional projections here we provide a new set of direction numbers alleviating this problem these are obtained by treating sobol sequences in d dimensions as t d sequences and then optimizing the t values of the two dimensional projections our target dimension is 21201

quasi monte carlo method for infinitely divisible random vectors via series representations an infinitely divisible random vector without gaussian component admits representations of shot noise series due to possible slow convergence of the series they have not been investigated as a device for monte carlo simulation in this paper we investigate the structure of shot noise series representations from a simulation point of view and discuss the effectiveness of quasi monte carlo methods applied to series representations the structure of series representations in nature tends to decrease their effective dimension and thus increase the efficiency of quasi monte carlo methods thanks to the greater uniformity of low discrepancy sequence in the lower dimension we illustrate the effectiveness of our approach through numerical results of moment and tail probability estimations for stable and gamma random variables

acceleration of quasi monte carlo approximations with applications in mathematical finance a procedure to accelerate the convergence of quasi monte carlo approximations is described and investigated the procedure applies a weighted least squares smoothing method to improve the approximations that were obtained using low discrepancy sequences the acceleration procedure is tested computationally on several numerical integration problems including two problems of interest in mathematical finance the results indicate that when halton low discrepancy points are used in the quasi monte carlo method the accelerated sequence often converges a factor of five or more faster than the original quasi monte carlo approximations

efficient algorithms for computing the l 2 discrepancy the l 2 discrepancy is a quantitative measure of precision for multivariate quadrature rules it can be computed explicitly previously known algorithms needed o m 2 operations where m is the number of nodes in this paper we present algorithms which require o m log m d operations

on the step by step construction of quasi monte carlo integration rules that achieve strong tractability error bounds in weighted sobolev spaces we develop and justify an algorithm for the construction of quasi monte carlo qmc rules for integration in weighted sobolev spaces the rules so constructed are shifted rank 1 lattice rules the parameters characterising the shifted lattice rule are found component by component the d 1 th component of the generator vector and the shift are obtained by successive 1 dimensional searches with the previous d components kept unchanged the rules constructed in this way are shown to achieve a strong tractability error bound in weighted sobolev spaces a search for n point rules with n prime and all dimensions 1 to d requires a total cost of o n3d2 operations this may be reduced to o n3d operations at the expense of o n2 storage numerical values of parameters and worst ease errors are given for dimensions up to 40 and n up to a few thousand the worst case errors for these rules are found to be much smaller than the theoretical bounds

good lattice rules in weighted korobov spaces with general weights we study the problem of multivariate integration and the construction of good lattice rules in weighted korobov spaces with general weights these spaces are not necessarily tensor products of spaces of univariate functions sufficient conditions for tractability and strong tractability of multivariate integration in such weighted function spaces are found these conditions are also necessary if the weights are such that the reproducing kernel of the weighted korobov space is pointwise non negative the existence of a lattice rule which achieves the nearly optimal convergence order is proven a component by component cbc algorithm that constructs good lattice rules is presented the resulting lattice rules achieve tractability or strong tractability error bounds and achieve nearly optimal convergence order for suitably decaying weights we also study special weights such as finite order and order dependent weights for these special weights the cost of the cbc algorithm is polynomial numerical computations show that the lattice rules constructed by the cbc algorithm give much smaller worst case errors than the mean worst case errors over all quasi monte carlo rules or over all lattice rules and generally smaller worst case errors than the best korobov lattice rules in dimensions up to hundreds numerical results are provided to illustrate the efficiency of cbc lattice rules and korobov lattice rules with suitably chosen weights in particular for high dimensional finance problems

k harmonic means data clustering with simulated annealing heuristic abstract clustering procedures partition a set of objects into clusters such that objects in the same cluster are more similar to each other than objects in different clusters according to some predefined criteria clustering is a popular data analysis and data mining technique since clustering problem have np complete nature the larger the size of the problem the harder to find the optimal solution and furthermore the longer to reach a reasonable results one of the most used techniques for clustering is based on k means such that the data is partitioned into k clusters although k means algorithm is easy to implement and works fast in most situations it suffers from two major drawbacks one is sensitivity to initialization and the other is convergence to local optima it is seen from the studies k harmonic means clustering solves the problem of initialization but since its greedy search nature the second problem convergence to local optima still remains in this paper we develop a new algorithm for solving this problem based on a simulated annealing technique simulated annealing k harmonic means clustering sakhmc the experiment results on the iris and the other well known data illustrate the robustness of the sakhmc clustering algorithm

fast convergence of quasi monte carlo for a class of isotropic integrals we consider the approximation ofd dimensional weighted integrals of certain isotropic functions we are mainly interested in cases where d is large we show that the convergence rate of quasi monte carlo for the approximation of these integrals iso p logn n since this is a worst case result compared to the expected convergence rate o n 1 2 of monte carlo it shows the superiority of quasi monte carlo for this type of integral this is much faster than the worst case convergence o log d n n of quasi monte carlo

cp methods for the schrodinger equation abstract after a short survey over the efforts in the direction of solving the schrodinger equation by using piecewise approximations of the potential function the paper focuses on the piecewise perturbation methods in their cp implementation the presentation includes a short list of problems for which cp versions are available a sketch of the derivation of the cpm formulae a description of various ways to construct or identify a certain version and also the main results of the error analysis one of the most relevant results of the latter is that the energy dependence of the error is bounded a fact which places these methods on a special position among the numerical methods for differential equations a numerical illustration is also included in which a cpm based code for the regular sturm liouville problem is compared with some other well established codes

accurate pricing formulas for asian options asian options have payoffs that depend on the average price of the underlying asset such as stocks commodities or financial indices as exact closed form formulas do not exist for these popular options how to price them numerically in an efficient and accurate manner has been extensively investigated there are two types of asian options fixed strike and floating strike asian options excellent lower bound formulas for both types of options have been derived by rogers and shi these formulas are extremely easy to calculate but they restrict the option s maturity to exactly 1 year this paper extends the rogers shi formulas to general maturities numerical experiments are performed to compare the formulas with many other numerical methods in the literature and under a wide variety of situations they confirm the extreme accuracy of the formulas

pricing asian options via compound gamma and orthogonal polynomials we develop two approximations cg3 and cgn for pricing arithmetic asian options approximation cg3 utilizes a compound gamma distribution of the price average it is calibrated by analytically matching the first three moments approximation cg3 outperforms many other existing approximations approximation cgn expands cg3 by utilizing the concept of orthogonal polynomials and matches n 3 moments it produces near exact results within 1 s when volatility and maturity are not too high two useful insights of our work are i demonstrating that orthogonal polynomials can be used effectively to improve accuracy and ii showing that matching higher moments is beneficial