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

long time behavior for a class of feller processes in this paper as a main result we derive a chung fuchs type condition for the recurrence of feller processes associated with pseudo dierentia l operators in the l evy process case this condition reduces to the classical and well known chung fuchs condition further we also discuss the recurrence and transience of feller processes with respect to the dimension of the state space and pruitt indices and the recurrence and transience of feller dynkin diusions and stable like processes finally in the one dimensional symmetric case we study perturbations of feller processes which do not aect their recurrence and transience properties and we derive sucient conditions for their recurrence and transience in terms of the corresponding l evy measure in addition some comparison conditions for recurrence and transience also in terms of the l evy measures are obtained ams 2010 mathematics subject classication 60j75 60j25 60g17

combination of general antithetic transformations and control variables several methods for reducing the variance in the context of monte carlo simulation are based on correlation induction this includes antithetic variates latin hypercube sampling and randomized version of quasi monte carlo methods such as lattice rules and digital nets where the resulting estimators are usually weighted averages of several dependent random variables that can be seen as function evaluations at a finite set of random points in the unit hypercube in this paper we consider a setting where these methods can be combined with the use of control variates and we provide conditions under which we can formally prove that the variance is minimized by choosing equal weights and equal control variate coefficients across the different points of evaluation regardless of the function integrand that is evaluated

the k dimensional distribution of combined gfsr sequences we develop an efficient method for analysis of the fc dimensional distribution of combinations of several gfsr sequences by bitwise exclusive or operations first we introduce the notion of a resolution wise lattice structure for gfsr sequences and show that by applying a theorem of couture to this type of lattice we obtain a precise description of a dimensional distribution of combined gfsr sequences in the same way as for combined tausworthe sequences finally we apply this method to the combination of two different twisted gfsr generators which were recently proposed by matsumoto and kurita and investigate the order of equidistribution of the combined sequence

quasi random methods for estimating integrals using relatively small samples much of the recent work dealing with quasi random methods has been aimed at establishing the best possible asymptotic rates of convergence to zero of the error resulting when a finite dimensional integral is replaced by a finite sum of integrand values in contrast with this perspective to concentrate on asymptotic convergence rates this paper emphasizes quasi random methods that are effective for all sample sizes throughout the paper the problem of estimating finite dimensional integrals is used to illustrate the major ideas although much of what is done applies equally to the problem of solving certain fredholm integral equations some new techniques based on error reducing transformations of the integrand are described that have been shown to be useful both in estimating high dimensional integrals and in solving integral equations these techniques illustrate the utility of carrying over to the quasi monte carlo method certain devices that have proven to be very valuable in statistical pseudoran

homogenization of periodic diffusion with small jumps abstract in this paper we study the homogenization of a diffusion process with jumps that is feller process generated by an integro differential operator this problem is closely related to the problem of homogenization of boundary value problems arising in studying the behavior of heterogeneous media under the assumptions that the corresponding generator has vanishing drift coefficient rapidly periodically oscillating diffusion and jump coefficients that it admits only small jumps that is the jump kernel has finite second moment and under certain additional regularity conditions we prove that the homogenized process is a brownian motion the presented results generalize the classical and well known results related to the homogenization of a diffusion process

construction algorithms for polynomial lattice rules for multivariate integration we introduce a new construction algorithm for digital nets for integration in certain weighted tensor product hilbert spaces the first weighted hilbert space we consider is based on walsh functions dick and pillichshammer calculated the worst case error for integration using digital nets for this space here we extend this result to a special construction method for digital nets based on polynomials over finite fields this result allows us to find polynomials which yield a small worst case error by computer search we prove an upper bound on the worst case error for digital nets obtained by such a search algorithm which shows that the convergence rate is best possible and that strong tractability holds under some condition on the weights we extend the results for the weighted hilbert space based on walsh functions to weighted sobolev spaces in this case we use randomly digitally shifted digital nets the construction principle is the same as before only the worst case error is slightly different again digital nets obtained from our search algorithm yield a worst case error achieving the optimal rate of convergence and as before strong tractability holds under some condition on the weights these results show that such a construction of digital nets yields the until now best known results of this kind and that our construction methods are comparable to the construction methods known for lattice rules we conclude the article with numerical results comparing the expected worst case error for randomly digitally shifted digital nets with those for randomly shifted lattice rules

scrambled net variance for integrals of smooth functions hybrids of equidistribution and monte carlo methods of integration can achieve the superior accuracy of the former while allowing the simple error estimation methods of the latter one version randomized t m s nets has the property that the integral estimates are unbiased and that the variance is o 1 n for any square integrable integrand stronger assumptions on the integrand allow one to find rates of convergence this paper shows that for smooth integrands over s dimensions the variance is of order n 3 log n 3 1 compared to n 1 for ordinary monte carlo thus the integration errors are of order n 3 2 log n s 1 2 in probability this compares favorably with the rate n 1 log n s 1 for unrandomized t m s nets

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

on the problem of minimum asymptotic exit rate for stochastically perturbed multi channel dynamical systems we consider the problem of minimizing the asymptotic exit rate with which the controlled diffusion process of a stochastically perturbed multi channel dynamical system exits from a given bounded open domain in particular for a class of admissible bounded linear feedback operators we establish a connection between the asymptotic exit rate with which such a controlled diffusion process exits from the given domain and the asymptotic behavior i e a probabilistic characterization of the principal eigenvalue of the infinitesimal generator which corresponds to the stochastically perturbed dynamical system with zero boundary conditions on the given domain finally we briefly remark on the implication of our result for evaluating the performance of the associated deterministic multi channel dynamical system when such a dynamical system is composed with a set of sub optimal admissible linear feedback operators

multiple integration over bounded and unbounded regions abstract a method is proposed for integrating over r n functions that are reasonably smooth and rapidly decaying at infinity the method makes use of an infinite lattice of quadrature points truncated at some suitable radius in certain circumstances it is shown that the best lattice among all those with a given determinant is the one whose dual lattice corresponds to the densest sphere packing the method of sag and szekeres for integration over n dimensional spheres and cubes is shown to be a special case with a lattice that is not necessarily optimal

an efficient lattice reduction method for f 2 linear pseudorandom number generators using mulders and storjohann algorithm abstract recent simulations often use highly parallel machines with many processors and they need many pseudorandom number generators with distinct parameter sets and hence we need an effective fast assessment of the generator with a given parameter set linear generators over the two element field are good candidates because of the powerful assessment via their dimensions of equidistribution some efficient algorithms to compute these dimensions use reduced bases of lattices associated with the generator in this article we use a fast lattice reduction algorithm by mulders and storjohann instead of schmidt s algorithm and show that the order of computational complexity is lessened experiments show an improvement in the speed by a factor of three we also report that just using a sparsest initial state i e consisting of all 0 bits except one significantly accelerates the lattice computation in the case of mersenne twister generators

the jackknife estimate of variance tukey s jackknife estimate of variance for a statistic s x 1 x 2 cdots x n which is a symmetric function of i i d random variables x i is investigated using an anova like decomposition of s it is shown that the jackknife variance estimate tends always to be biased upwards a theorem to this effect being proved for the natural jackknife estimate of operatorname var s x 1 x 2 cdots x n 1 based on x 1 x 2 cdots x n

on the use of reducible polynomials as random number generators the randomness properties and the hierarchy of correlation coefficients are studied of approximate maximum length sequences for which the characteristic polynomial is a product of several primitive polynomials the randomness properties are almost the same as for maximum length sequences characterized by a primitive polynomial with many terms and of the same degree reducible characteristic polynomials have acceptable figures of merit and can be of extremely high degree since they are also easily constructed and implemented reducible polynomials are strong candidates for reliable random number generation especially at the bit rates needed in large scale monte carlo simulations

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 construction of polynomial lattice rules with small gain coefficients in this paper we construct polynomial lattice rules which have in some sense small gain coefficients using a component by component approach the gain coefficients as introduced by owen indicate to what degree the method improves upon monte carlo we show that the variance of an estimator based on a scrambled polynomial lattice rule constructed component by component decays at a rate of n 2 1 for all 0 assuming that the function under consideration has bounded variation of order for some 0 1 and where n denotes the number of quadrature points an analogous result is obtained for korobov polynomial lattice rules it is also established that these rules are almost optimal for the function space considered in this paper furthermore we discuss the implementation of the component by component approach and show how to reduce the computational cost associated with it finally we present numerical results comparing scrambled polynomial lattice rules and scrambled digital nets

the asymptotic efficiency of randomized nets for quadrature an l 2 type discrepancy arises in the average and worst case error analyses for multidimensional quadrature rules this discrepancy is uniquely defined by k x y which serves as the covariance kernel for the space of random functions in the average case analysis and a reproducing kernel for the space of functions in the worst case analysis this article investigates the asymptotic order of the root mean square discrepancy for randomized 0 m s nets in base b for moderately smooth k x y the discrepancy is o n 1 log n s 1 2 and for k x y with greater smoothness the discrepancy is o n 3 2 log n s 1 2 where n b m is the number of points in the net numerical experiments indicate that the t m s nets of faure niederreiter and sobol do not necessarily attain the higher order of decay for sufficiently smooth kernels however niederreiter nets may attain the higher order for kernels corresponding to spaces of periodic functions

figures of merit for digital multistep pseudorandom numbers the statistical independence properties of s successive digital multistep pseudorandom numbers are governed by the figure of merit p s i which depends on s and the characteristic polynomial f of the recursion used in the generation procedure we extend previous work for s 2 and describe how to obtain large figures of merit for s 2 thus arriving at digital multistep pseudorandom numbers with attractive statistical independence properties tables of figures of merit for s 3 4 5 and degrees 32 are included

maximally equidistributed combined tausworthe generators tausworthe random number generators based on a primitive trinomial allow an easy and fast implementation when their parameters obey certain restrictions however such generators with those restrictions have bad statistical properties unless we combine them a generator is called maximally equidistributed if its vectors of successive values have the best possible equidistribution in all dimensions this paper shows how to find maximally equidistributed combinations in an efficient manner and gives a list of generators with that property such generators have a strong theoretical support and lend themselves to very fast software implementations

quasi monte carlo methods for two stage stochastic mixed integer programs we consider randomized qmc methods for approximating the expected recourse in two stage stochastic optimization problems containing mixed integer decisions in the second stage it is known that the second stage optimal value function is piecewise linear quadratic with possible kinks and discontinuities at the boundaries of certain convex polyhedral sets this structure is exploited to provide conditions implying that first and higher order terms of the integrand s anova decomposition math comp 79 2010 953 966 have mixed weak first order partial derivatives this leads to a good smooth approximation of the integrand and hence to good convergence rates of randomized qmc methods if the effective superposition dimension is low

variance bounds and existence results for randomly shifted lattice rules we study the convergence of the variance for randomly shifted lattice rules for numerical multiple integration over the unit hypercube in an arbitrary number of dimensions we consider integrands that are square integrable but whose fourier series are not necessarily absolutely convergent for such integrands a bound on the variance is expressed through a certain type of weighted discrepancy we prove existence and construction results for randomly shifted lattice rules such that the variance bounds are almost o n a where n is the number of function evaluations and a 1 depends on our assumptions on the convergence speed of the fourier coefficients these results hold for general weights arbitrary n and any dimension with additional conditions on the weights we obtain a convergence that holds uniformly in the dimension and this provides sufficient conditions for strong tractability of the integration problem we also show that lattice rules that satisfy these bounds are not difficult to construct explicitly and we provide numerical illustrations of the behaviour of construction algorithms

some properties of rank 2 lattice rules a rank 2 lattice rule is a quadrature rule for the unit ital s dimensional hypercube of the form ital qf 1 ital n sub 1 ital n sub 2 summation sub ital j sub 1 1 ital j sub 2 1 sup ital n sub 1 summation sub ital j sub 2 1 sup ital n sub 2 ital bar f ital j sub 1 ital z1 ital n sub 1 ital j sub 2 ital z2 ital n sub 2 which cannot be re expressed in an analogous form with a single sum here ital bar f is a periodic extension of ital f and ital z sub 1 ital z sub 2 are integer vectors in this paper we discuss these rules in detail in particular we categorize a special subclass whose leading one and two dimensional projections contain the maximum feasible number of abscissas we show that rules of this subclass can be expressed uniquely in a simple tricycle form

short time behavior of hermite functions on compact lie groups abstract let p t x be the gaussian heat kernel on r n at time t the classical hermite polynomials at time t may be defined by a rodriguez formula given by h x t p t x p t x where is a constant coefficient differential operator on r n recent work of gross 1993 and hijab 1994 has led to the study of a new class of functions on a general compact lie group g in analogy with the r n case these hermite functions on g are obtained by the same formula wherein p t x is now the heat kernel on the group x is replaced by x 1 and is a right invariant differential operator let g be the lie algebra of g composing a hermite function on g with the exponential map produces a family of functions on g we prove that these functions scaled appropriately in t approach the classical hermite polynomials at time 1 as t tends to 0 both uniformly on compact subsets of g and in l p g where 1 p is a gaussian measure on g similar theorems are established when g is replaced by g k where k is some closed connected subgroup of g

the hierarchy of correlations in random binary sequences the meaning of randomness is studied for the simple case of binary sequences ensemble theory is used together with correlation coefficients of any order conservation laws for the total amount of correlation are obtained they imply that true randomness is an ensemble property and can never be achieved in a single sequence the relation with entropy is discussed for different ensembles well tempered pseudorandom sequences turn out to be suitable sources of random numbers and practical recipes to generate them for use in large scale monte carlo simulations are found

efficient lattice assessment for lcg and glp parameter searches in the present paper we show how to speed up lattice parameter searches for monte carlo and quasi monte carlo node sets the classical measure for such parameter searches is the spectral test which is based on a calculation of the shortest nonzero vector in a lattice instead of the shortest vector we apply an approximation given by the lll algorithm for lattice basis reduction we empirically demonstrate the speed up and the quality loss obtained by the lll reduction and we present important applications for parameter selections

growth and holder conditions for the sample paths of feller processes let a d a be the infinitesimal generator of a feller semigroup such that cc n d a and a cc n is a pseudo differential operator with symbol p x satisfying p c 1 2 and imp x c0rep x we show that the associated feller process xt t 0 on n is a semimartingale even a homogeneous diffusion with jumps in the sense of 21 and characterize the limiting behaviour of its trajectories as t 0 and to this end we introduce various indices e g x 0 lim x y 2 p y 0 or x 0 liminf x y 2 e 1 p y e 0 and obtain a s x that lim t 0t 1 s t xs x 0 or according to x or x similar statements hold for the limit inferior and superior and also for t our results extend the constant coefficient i e levy case considered by w pruitt 27

extensible lattice sequences for quasi monte carlo quadrature integration lattices are one of the main types of low discrepancy sets used in quasi monte carlo methods however they have the disadvantage of being of fixed size this article describes the construction of an infinite sequence of points the first bm of which forms a lattice for any nonnegative integer m thus if the quadrature error using an initial lattice is too large the lattice can be extended without discarding the original points generating vectors for extensible lattices are found by minimizing a loss function based on some measure of discrepancy or nonuniformity of the lattice the spectral test used for finding pseudorandom number generators is one important example of such a discrepancy the performance of the extensible lattices proposed here is compared to that of other methods for some practical quadrature problems

hardware implementation trade offs of polynomial approximations and interpolations this paper examines the hardware implementation trade offs when evaluating functions via piecewise polynomial approximations and interpolations for precisions of up to 24 bits in polynomial approximations polynomials are evaluated using stored coefficients polynomial interpolations however require the coefficients to be computed on the fly by using stored function values although it is known that interpolations require less memory than approximations but at the expense of additional computations the trade offs in memory area delay and power consumption between the two approaches have not been examined in detail this work quantitatively analyzes these trade offs for optimized approximations and interpolations across different functions and target precisions hardware architectures for degree 1 and degree 2 approximations and interpolations are described the results show that the extent of memory savings realized by using interpolation is significantly lower than what is commonly believed furthermore experimental results on a field programmable gate array fpga show that for high output precision degree 1 interpolations offer considerable area and power savings over degree 1 approximations but similar savings are not realized when degree 2 interpolations and approximations are compared the availability of both interpolation based and approximation based designs offers a richer set of design trade offs than what is available using either interpolation or approximation alone

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

on the lattice basis reduction multisequence synthesis algorithm a lattice basis reduction multisequence synthesis lbrms algorithm was presented with a new intuitive and vector form model in this paper a refined version of the lbrms algorithm is deduced from which massey s conjectured algorithm can be derived moreover if we modify the lattice basis reduction algorithm used in the lbrms algorithm a new synthesis algorithm is deduced which is equivalent to the generalization of the berlekamp massey algorithm proposed by feng and tzeng 1989 therefore the lbrms algorithm provides a unified approach to the multisequence shift register synthesis problem

on the distribution of k dimensional vectors for simple and combined tausworthe sequences the lattice structure of conventional linear congruential random number generators lcgs over integers is well known in this paper we study lcgs in the field of formal laurent series with coefficients in the galois field f2 the state of the generator a laurent series evolves according to a linear recursion and can be mapped to a number between 0 and 1 producing what we call a ls2 sequence in particular the sequences produced by simple or combined tausworthe generators are special cases of ls2 sequences by analyzing the lattice structure of the lcg we obtain a precise description of how all the k dimensional vectors formed by successive values in the ls2 sequence are distributed in the unit hypercube more specifically for any partition of the k dimensional hypercube into 2kl identical subcubes we can quickly compute a table giving the exact number of subcubes that contain exactly n points for each integer n we give numerical examples and discuss the practical implications of our results

the log sobolev inequality on loop space over a compact riemannian manifold abstract we obtain a log sobolev inequality with a neat and explicit potential for the gradient on a based loop space over a compact riemannian manifold the potential term relies only on the curvature of the manifold and the hessian of the heat kernel and is l p integrable for all p 1 the log sobolev inequality is derived by a martingale representation theorem for the differentiable functions on loop space which is a variation of the clark ocone haussmann formula

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

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

periodic orbits of the ensemble of sinai arnold cat maps and pseudorandom number generation we propose methods for constructing high quality pseudorandom number generators rngs based on an ensemble of hyperbolic automorphisms of the unit two dimensional torus sinai arnold map or cat map while keeping a part of the information hidden the single cat map provides the random properties expected from a good rng and is hence an appropriate building block for an rng although unnecessary correlations are always present in practice we show that introducing hidden variables and introducing rotation in the rng output accompanied with the proper initialization dramatically suppress these correlations we analyze the mechanisms of the single cat map correlations analytically and show how to diminish them we generalize the percival vivaldi theory in the case of the ensemble of maps find the period of the proposed rng analytically and also analyze its properties we present efficient practical realizations for the rngs and check our predictions numerically we also test our rngs using the known stringent batteries of statistical tests and find that the statistical properties of our best generators are not worse than those of other best modern generators

epi convergent discretizations of multistage stochastic programs via integration quadratures this paper presents procedures for constructing numerically solvable discretizations of multistage stochastic programs that epi converge to the original problem as the discretizations are made finer epi convergence implies in particular that the cluster points of the first stage solutions of the discretized problems are optimal first stage solutions of the original problem the discretization procedures apply to a general class of nonlinear stochastic programs where the uncertain factors are driven by time series models using existing routines for numerical integration allows for an easy and efficient implementation of the procedures

a randomized quasi monte carlo simulation method for markov chains we introduce and study a randomized quasi monte carlo method for the simulation of markov chains up to a random and possibly unbounded stopping time the method simulates n copies of the chain in parallel using a d 1 dimensional highly uniform point set of cardinality n randomized independently at each step where d is the number of uniform random numbers required at each transition of the markov chain the general idea is to obtain a better approximation of the state distribution at each step of the chain than with standard monte carlo the technique can be used in particular to obtain a low variance unbiased estimator of the expected total cost when state dependent costs are paid at each step it is generally more effective when the state space has a natural order related to the cost function we provide numerical illustrations where the variance reduction with respect to standard monte carlo is substantial the variance can be reduced by factors of several thousands in some cases we prove bounds on the convergence rate of the worst case error and of the variance for special situations where the state space of the chain is a subset of the real numbers in line with what is typically observed in randomized quasi monte carlo contexts our empirical results indicate much better convergence than what these bounds guarantee

generating quasi random paths for stochastic processes the need to simulate stochastic processes numerically arises in many fields frequently this is done by discretizing the process into small time steps and applying pseudorandom sequences to simulate the randomness this paper addresses the question of how to use quasi monte carlo methods to improve this simulation special techniques must be applied to avoid the problem of high dimensionality which arises when a large number of time steps is required two such techniques the generalized brownian bridge and particle reordering are described here these methods are applied to a problem from finance the valuation of a 30 year bond with monthly coupon payments assuming a mean reverting stochastic interest rate when expressed as an integral this problem is nominally 360 dimensional the analysis of the integrand presented here explains the effectiveness of the quasi random sequences on this high dimensional problem and suggests methods of variance reduction which can be used in conjunction with the quasi random sequences

integrated variance reduction strategies for simulation we develop strategies for integrated use of certain well known variance reduction techniques to estimate a mean response in a finite horizon simulation experiment the building blocks for these integrated variance reduction strategies are the techniques of conditional expectation correlation induction including antithetic variates and latin hypercube sampling and control variates all pairings of these techniques are examined for each integrated strategy we establish sufficient conditions under which that strategy will yield a smaller response variance than its constituent variance reduction techniques will yield individually we also provide asymptotic variance comparisons between many of the methods discussed with emphasis on integrated strategies that incorporate latin hypercube sampling an experimental performance evaluation reveals that in the simulation of stochastic activity networks substantial variance reductions can be achieved with these integrated strategies both the theoretical and experimental results indicate that superior performance is obtained via joint application of the techniques of conditional expectation and latin hypercube sampling

hurst s rescaled range statistical analysis for pseudorandom number generators used in physical simulations the rescaled range statistical analysis r s is proposed as a method to detect correlations in pseudorandom number generators used in monte carlo simulations in an extensive test it is demonstrated that the r s analysis provides a very sensitive method to reveal hidden long run and short run correlations several widely used and also some recently proposed pseudorandom number generators are subjected to this test in many generators correlations are detected and quantified

efficient weighted lattice rules with applications to finance good lattice rules are an important type of quasi monte carlo algorithm they are known to have good theoretical properties in the sense that they can achieve an error bound or optimal error bound that is independent of the dimension for weighted function spaces with suitably decaying weights to use the theory of weighted function spaces for practical applications one has to determine what weights should be used after explaining that lattice rules based on figures of merit with classical weights classical lattice rules may not give good results for high dimensional problems we propose a matching strategy which chooses the weights in such a way that the typical functions of a weighted korobov space are similar in the sense of similar relative sensitivity indices to those of the given function the matching strategy relates the weights of the spaces to the sensitivity indices of the given function we apply the korobov construction and the component by component construction of lattice rules with the suitably chosen weights to the valuation of high dimensional financial derivative securities and find that such weighted lattice rules improve dramatically on the results for the classical lattice rules moreover they are competitive with other types of low discrepancy sequences we also demonstrate that lattice rules combined with variance reduction techniques and dimension reduction techniques increase the efficiency significantly we show that an acceptance rejection approach to the construction of lattice rules can reduce the construction cost with almost no loss of accuracy

comment on superposition of elliptic functions as solutions for a large number of nonlinear equations j math phys 56 032104 2015 by using the method of planar dynamical systems we solve exactly a nonlinear schrodinger nls equation discussed by khare and saxena j math phys 56 032104 2015 and give the exact explicit parametric representations of all the traveling wave solutions

periodic and hyperbolic soliton solutions of a number of nonlocal nonlinear equations for a number of nonlocal nonlinear equations such as nonlocal nonlinear schrodinger equation nlse nonlocal ablowitz ladik al nonlocal saturable discrete nlse dnlse coupled nonlocal nlse coupled nonlocal al and coupled nonlocal saturable dnlse we obtain periodic solutions in terms of jacobi elliptic functions as well as the corresponding hyperbolic soliton solutions remarkably in all the six cases we find that unlike the corresponding local cases all the nonlocal models simultaneously admit both the bright and the dark soliton solutions further in all the six cases not only the elliptic functions dn x m and cn x m with modulus m but also their linear superposition is shown to be an exact solution finally we show that the coupled nonlocal nlse not only admits solutions in terms of lame polynomials of order 1 but also admits solutions in terms of lame polynomials of order 2 even though they are not the solution of the uncoupled nonlocal problem we also remark on the possible int

good parameters and implementations for combined multiple recursive random number generators combining parallel multiple recursive sequences provides an efficient way of implementing random number generators with long periods and good structural properties such generators are statistically more robust than simple linear congruential generators that fit into a computer word we made extensive computer searches for good parameter sets with respect to the spectral test for combined multiple recursive generators of different sizes we also compare different implementations and give a specific code in c that is faster than previous implementations of similar generators

correlation induction techniques for estimating quantiles in simulation experiments a simulation based quantile estimator measures the level of system performance that can be delivered with a prespecified probability to estimate selected quantiles of the response of a finite horizon simulation we develop procedures based on correlation induction techniques for variance reduction with emphasis on antithetic variates and latin hypercube sampling these procedures achieve improved precision by controlling the simulation s random number inputs as an integral part of the experimental design the proposed multiple sample quantile estimator is the average of negatively correlated quantile estimators computed from disjoint samples of the simulation response where negative correlation is induced between corresponding responses in different samples while mutual independence of responses is maintained within each sample the proposed single sample quantile estimator is computed from negatively correlated simulation responses within one all inclusive sample the single sample estimator based on latin hypercube sampling is shown to be asymptotically normal and unbiased with smaller variance than the comparable directsimulation estimator based on independent replications similar asymptotic comparisons of the multiple sample and directsimulation estimators focus on bias and mean square error monte carlo results suggest that the proposed procedures can yield significant reductions in bias variance and mean square error when estimating quantiles of the completion time of a stochastic activity network

comment on comment on superposition of elliptic functions as solutions for a large number of nonlinear equations j math phys 56 084101 2015 the assertion in a theorem that the superposition of two particular jacobian elliptic functions is not a solution of the nonlinearschrodinger equation is demonstrated to be invalid

variance reduction in sample approximations of stochastic programs this paper studies the use of randomized quasi monte carlo methods rqmc in sample approximations of stochastic programs in numerical integration rqmc methods often substantially reduce the variance of sample approximations compared to monte carlo mc it seems thus natural to use rqmc methods in sample approximations of stochastic programs it is shown that rqmc methods produce epi convergent approximations of the original problem rqmc and mc methods are compared numerically in five different portfolio management models in the tests rqmc methods outperform mc sampling substantially reducing the sample variance and bias of optimal values in all the considered problems

randomized dimension reduction for monte carlo simulations we present a new unbiased algorithm that estimates the expected value of f u via monte carlo simulation where u is a vector of d independent random variables and f is a function of d variables

on the asymptotic estimates for exit probabilities and minimum exit rates of diffusion processes pertaining to a chain of distributed control systems in this paper we consider a diffusion process pertaining to a distributed control system formed by a chain of n subsystems with n ge 2 where a random perturbation enters only in the first subsystem and is then subsequently transmitted to the other subsystems we also assume that for any ell in 2 ldots n the distributed control system formed by the first ell subsystems i e from the first subsystem up to the ell th subsystem satisfies an appropriate hormander condition as a result the diffusion process is degenerate in the sense that the backward operator associated with it is a degenerate parabolic equation in particular we consider the following two problems i we provide an asymptotic estimate for the exit probability with which the diffusion process corresponding to a particular subsystem exits from a given bounded open domain during a certain time interval the approach for such an asymptotic estimate basically relies on the interpretation of the exit probability

superposition of elliptic functions as solutions for a large number of nonlinear equations for a large number of nonlinear equations both discrete and continuum we demonstrate a kind of linear superposition we show that whenever a nonlinear equation admits solutions in terms of both jacobi elliptic functions cn x m and dn x m with modulus m then it also admits solutions in terms of their sum as well as difference we have checked this in the case of several nonlinear equations such as the nonlinear schrodinger equation mkdv a mixed kdv mkdv system a mixed quadratic cubic nonlinear schrodinger equation the ablowitz ladik equation the saturable nonlinear schrodinger equation 4 the discrete mkdv as well as for several coupled field equations further for a large number of nonlinear equations we show that whenever a nonlinear equation admits a periodic solution in terms of dn2 x m it also admits solutions in terms of dn 2 x m m cn x m dn x m even though cn x m dn x m is not a solution of these nonlinear equations finally we also obtain superposed solutions of variou

uniqueness in law for pure jump markov processes let a be the operator defined on c2 functions by af left x right smallint left f left x h right f left x right f left x right h 1 1 1 left h right right v left x dh right sufficient conditions are given for existence and uniqueness for the martingale problem associated with a in the case of stable like processes where v x dh is equal to the levy measure for the stable symmetric process of index x for each x the conditions reduce to x continuous for existence and x dini continuous for uniqueness

yang mills fields and random holonomy along brownian bridges we characterize yang mills connections in vector bundles in terms of covariant derivatives of stochastic parallel transport along variations of brownian bridges on the base manifold in particular we prove that a connection in a vector bundle e is yang mills if and only if the covariant derivative of parallel transport along brownian bridges in the direction of their drift is a local martingale when transported back to the starting point we present a taylor expansion up to order 3 for stochastic parallel transport in e along small rescaled brownian bridges and prove that the connection in e is yang mills if and only if all drift terms in the expansion up to order 3 vanish or equivalently if and only if the average rotation of parallel transport along small bridges and loops is of order 4

sparse serial tests of uniformity for random number generators different versions of the serial test for testing the uniformity and independence of vectors of successive values produced by a pseudo random number generator are studied these tests partition the t dimensional unit hypercube into k cubic cells of equal volume generate n points vectors in this hypercube count how many points fall in each cell and compute a test statistic defined as the sum of values of some univariate function f applied to these k individual counters both overlapping and nonoverlapping vectors are considered for different families of generators such as linear congruential tausworthe nonlinear inversive etc different ways of choosing these functions and of choosing k are compared and formulas are obtained for the estimated sample size required to reject the null hypothesis of independent uniform random variables as a function of the period length of the generator for the classes of alternatives that correspond to linear generators the most efficient tests turn out to have k gg n in contrast to what is usually done or recommended in simulation books and to use overlapping vectors

close point spatial tests and their application to random number generators we study statistical tests of uniformity based on thel p distances between them nearest pairs of points forn points generated uniformly over thek dimensional unit hypercube or unit torus the number of distinct pairs at distance no more thant fort 0 is a stochastic process whose initial part after an appropriate transformation and asn 8 is asymptotically a poisson process with unit rate convergence to this asymptotic is slow in the hypercube as soon ask exceeds 2 or 3 due to edge effects but is reasonably fast in the torus we look at the quality of approximation of the exact distributions of the tests statistics by their asymptotic distributions discuss computational issues and apply the tests to random number generators linear congruential generators fail decisively certain variants of the tests as soon asn approaches the square root of the period length

lattice computations for random numbers we improve on a lattice algorithm of tezuka for the computation of the k distribution of a class of random number generators based on finite fields we show how this is applied to the problem of constructing for such generators an output mapping yielding optimal k distribution

asymptotic behavior of divergences and cameron martin theorem on loop spaces we first prove the lp convergence p 1 and a fernique type exponential integrability of divergence functionals for all cameron martin vector fields with respect to the pinned wiener measure on loop spaces over a compact riemannian manifold we then prove that the driver flow is a smooth transform on path spaces in the sense of the malliavin calculus and has an quasi continuous modification which can be quasi surely well defined on path spaces this leads us to construct the driver flow on loop spaces through the corresponding flow on path spaces combining these two results with the cruzeiro lemma j funct anal 54 1983 206 227 we give an alternative proof of the quasi invariance of the pinned wiener measure under driver s flow on loop spaces which was established earlier by driver trans amer math soc 342 1994 375 394 and enchev and stroock adv math 119 1996 127 154 by doob s h processes approach together with the short time estimates of the gradient and the hessian of the logarithmic heat kernel on compact riemannian manifolds we also establish the lp convergence p 1 and a fernique type exponential integrability theorem for the stochastic anti development of pinned brownian motions on compact riemannian manifold with an explicit exponential exponent our results generalize and sharpen some earlier results due to gross j funct anal 102 1991 268 313 and hsu math ann 309 1997 331 339 our method does not need any heat kernel estimate and is based on quasi sure analysis and sobolev estimates on path spaces

density estimates for a random noise propagating through a chain of differential equations we here provide two sided bounds for the density of the solution of a system of n differential equations of dimension d the first one being forced by a non degenerate random noise and the n 1 other ones being degenerate the system formed by the n equations satisfies a suitable hormander condition the second equation feels the noise plugged into the first equation the third equation feels the noise transmitted from the first to the second equation and so on so that the noise propagates one way through the system when the coefficients of the system are lipschitz continuous we show that the density of the solution satisfies gaussian bounds with non diffusive time scales the proof relies on the interpretation of the density of the solution as the value function of some optimal stochastic control problem

on the numerical integration of walsh series by number theoretic methods in analogy to the theory of good lattice points for the numerical integration of rapidly converging fourier series a theory for the fast numerical integration of walsh series is developed the basis for this theory is provided by a class of very well distributed point sets in the s dimensional unit cube the so called t m s nets

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

fast lattice reduction for linear pseudorandom number generators sequences generated by an f 2 linear recursion have wide applications in particular pseudorandom number generation the dimension of equidistribution with v bit accuracy is a most important criterion for the uniformity of the generated sequence the fastest known method for computing these dimensions is proposed by couture and l ecuyer based on lenstra s lattice basis reduction and the dual lattice to the lattice of vector valued generating functions with components in the formal power series f2 t 1 associated to the output f 2 vector sequence in this paper we propose a similar but faster algorithm where 1 the state space is used to represent vectors with components in the formal power series 2 the dual lattice is not necessary and 3 lenstra reduction is replaced with a simpler basis reduction the computational complexity of our method is smaller than for the couture l ecuyer method experiments show that our method improves the speed by a factor of 10 for mersenne twister mt19937 and for well generators with state sizes of 19937 bits and 44497 bits

epi convergent discretizations of stochastic programs via integration quadratures the simplest and the best known method for numerical approximation of high dimensional integrals is the monte carlo method mc i e random sampling mc has also become the most popular method for constructing numerically solvable approximations of stochastic programs however certain modern integration quadratures are often superior to crude mc in high dimensional integration so it seems natural to try to use them also in discretization of stochastic programs this paper derives conditions that guarantee the epi convergence of the resulting objectives to the original one our epi convergence result is closely related to some of the existing ones but it is easier to apply to discretizations and it allows the feasible set to depend on the probability measure as examples we prove epi convergence of quadrature based discretizations of three different models of portfolio management and we study their behavior numerically besides mc our discretizations are the only existing ones with guaranteed epi convergence for these problem classes in our tests modern quadratures seem to result in faster convergence of optimal values than mc

the growth of random walks and levy processes let x i be a sequence of independent identically distributed non degenerate random variables taking values in mathbb r d and s n sum n i 1 x i m n max 1 leqq i leqq n s i define for x 0 g x p x 1 x k x x 2 e x 1 2 1 x 1 leq x m x x 1 e x 1 1 x 1 leq x and h x g x k x m x then if beta sup alpha lim sup x alpha h x 0 delta sup alpha lim inf x alpha h x 0 it is proved that n 1 alpha m n rightarrow 0 for alpha delta while the lim inf is 0 and the lim sup is infty for beta alpha delta some alternative characterizations of the indices beta delta are obtained as well as the analogous results for levy processes

perelman s entropy formula for the witten laplacian on riemannian manifolds via bakry emery ricci curvature in this paper we study perelman s mathcal w entropy formula for the heat equation associated with the witten laplacian on complete riemannian manifolds via the bakry emery ricci curvature under the assumption that the m dimensional bakry emery ricci curvature is bounded from below we prove an analogue of perelman s and ni s entropy formula for the mathcal w entropy of the heat kernel of the witten laplacian on complete riemannian manifolds with some natural geometric conditions in particular we prove a monotonicity theorem and a rigidity theorem for the mathcal w entropy on complete riemannian manifolds with non negative m dimensional bakry emery ricci curvature moreover we give a probabilistic interpretation of the mathcal w entropy for the heat equation of the witten laplacian on complete riemannian manifolds and for the ricci flow on compact riemannian manifolds

randomized polynomial lattice rules for multivariate integration and simulation lattice rules are among the best methods to estimate integrals in a large number of dimensions they are part of the quasi monte carlo set of tools a theoretical framework for a class of lattice rules defined in a space of polynomials with coefficients in a finite field is developed in this paper a randomized version is studied implementations and criteria for selecting the parameters are discussed and examples of its use as a variance reduction tool in stochastic simulation are provided certain types of digital net constructions as well as point sets constructed by taking all vectors of successive output values produced by a tausworthe random number generator are special cases of this method

sharp explicit lower bounds of heat kernels by using logarithmic transformations an explicit lower bound estimate of heat kernels is obtained for diffusion processes on riemannian manifolds this estimate is sharp for both short and long times especially for heat kernels on a compact manifold and is extended to manifolds with unbounded curvature