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Extensible Lattice Sequences For QuasiMonte Carlo Quadrature
 SIAM Journal on Scientific Computing
, 1999
"... Integration lattices are one of the main types of low discrepancy sets used in quasiMonte 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 b m of which form a lattice for any nonnegative ..."
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Cited by 35 (11 self)
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Integration lattices are one of the main types of low discrepancy sets used in quasiMonte 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 b m of which form 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.
Constructing embedded lattice rules for multivariate integration
 SIAM J. Sci. Comput
"... Abstract. Lattice rules are a family of equalweight cubature formulas for approximating highdimensional integrals. By now it is well established that good generating vectors for lattice rules having n points can be constructed componentbycomponent for integrands belonging to certain weighted fun ..."
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Cited by 30 (9 self)
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Abstract. Lattice rules are a family of equalweight cubature formulas for approximating highdimensional integrals. By now it is well established that good generating vectors for lattice rules having n points can be constructed componentbycomponent 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 componentbycomponent 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 orderdependent weight settings, at the cost of O(dn(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 lowdiscrepancy 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 O(n−1+δ), δ> 0, as those constructed for a fixed number of points, and that the implied constant only gains a factor of 1.30 to 1.55.
QuasiMonte Carlo Via Linear ShiftRegister Sequences
, 1999
"... Linear recurrences modulo 2 with long periods have been widely used for contructing (pseudo)random number generators. Here, we use them for quasiMonte Carlo integration over the unit hypercube. Any stochastic simulation fits this framework. The idea is to choose a recurrence with a short period leng ..."
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Cited by 13 (4 self)
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Linear recurrences modulo 2 with long periods have been widely used for contructing (pseudo)random number generators. Here, we use them for quasiMonte Carlo integration over the unit hypercube. Any stochastic simulation fits this framework. The idea is to choose a recurrence with a short period length and to estimate the integral by the average value of the integrand over all vectors of successive output values produced by the small generator. We examine randomizations of this scheme, discuss criteria for selecting the parameters, and provide examples. This approach can be viewed as a polynomial version of lattice rules.
Conditioning on onestep survival for barrier option simulations
 Operations Research
, 2001
"... Pricing financial options often requires Monte Carlo methods. One particular case is that of barrier options, whose payoff may be zero depending on whether or not an underlying asset crosses a barrier during the life of the option. This paper develops variance reduction techniques that take advantag ..."
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Cited by 12 (1 self)
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Pricing financial options often requires Monte Carlo methods. One particular case is that of barrier options, whose payoff may be zero depending on whether or not an underlying asset crosses a barrier during the life of the option. This paper develops variance reduction techniques that take advantage of the special structure of barrier options, and are appropriate for general simulation problems with similar structure. We use a change of measure at each step of the simulation to reduce the variance due to the possibility of a barrier crossing at each monitoring date. The paper details the theoretical underpinnings of this method, and evaluates alternative implementations when exact distributions conditional on onestep survival are available and when not available. When these onestep conditional distributions are unavailable, we introduce algorithms that combine change of measure and estimation of conditional probabilities simultaneously. The methods proposed are more generally applicable to terminal reward problems on Markov processes with absorbing states.
QuasiMonte Carlo sampling to improve the efficiency of Monte Carlo EM
 Computational Statistics and Data Analysis
, 2005
"... In this paper we investigate an efficient implementation of the Monte Carlo EM algorithm based on QuasiMonte Carlo sampling. The Monte Carlo EM algorithm is a stochastic version of the deterministic EM (ExpectationMaximization) algorithm in which an intractable Estep is replaced by a Monte Carlo ..."
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Cited by 11 (5 self)
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In this paper we investigate an efficient implementation of the Monte Carlo EM algorithm based on QuasiMonte Carlo sampling. The Monte Carlo EM algorithm is a stochastic version of the deterministic EM (ExpectationMaximization) algorithm in which an intractable Estep is replaced by a Monte Carlo approximation. QuasiMonte Carlo methods produce deterministic sequences of points that can significantly improve the accuracy of Monte Carlo approximations over purely random sampling. One drawback to deterministic QuasiMonte Carlo methods is that it is generally difficult to determine the magnitude of the approximation error. However, in order to implement the Monte Carlo EM algorithm in an automated way, the ability to measure this error is fundamental. Recent developments of randomized QuasiMonte Carlo methods can overcome this drawback. We investigate the implementation of an automated, datadriven Monte Carlo EM algorithm based on randomized QuasiMonte Carlo methods. We apply this algorithm to a geostatistical model of online purchases and find that it can significantly decrease the total simulation effort, thus showing great potential for improving upon the efficiency of the classical Monte Carlo EM algorithm. Key words and phrases: Monte Carlo error; lowdiscrepancy sequence; Halton sequence; EM algorithm; geostatistical model.
The EM Algorithm, Its Stochastic Implementation and Global Optimization: Some Challenges and Opportunities for OR
, 2006
"... The EM algorithm is a very powerful optimization method and has reached popularity in many fields. Unfortunately, EM is only a local optimization method and can get stuck in suboptimal solutions. While more and more contemporary data/model combinations yield more than one optimum, there have been on ..."
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Cited by 11 (4 self)
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The EM algorithm is a very powerful optimization method and has reached popularity in many fields. Unfortunately, EM is only a local optimization method and can get stuck in suboptimal solutions. While more and more contemporary data/model combinations yield more than one optimum, there have been only very few attempts at making EM suitable for global optimization. In this paper we review the basic EM algorithm, its properties and challenges and we focus in particular on its stochastic implementation. The stochastic EM implementation promises relief to some of the contemporary data/model challenges and it is particularly wellsuited for a wedding with global optimization ideas since most global optimization paradigms are also based on the principles of stochasticity. We review some of the challenges of the stochastic EM implementation and propose a new algorithm that combines the principles of EM with that of the Genetic Algorithm. While this new algorithm shows some promising results for clustering of an online auction database of functional objects, the primary goal of this work is to bridge a gap between the field of statistics, which is home to extensive research on the EM algorithm, and the field of operations research, in which work on global optimization thrives, and to stir new ideas for joint research between the two.
A Dynamic Programming Procedure for Pricing AmericanStyle Asian Options
, 2002
"... Pricing Europeanstyle Asian options based on the arithmetic average, under the Black and Scholes model, involves estimating an integral (a mathematical expectation) for which no easily computable analytical solution is available. Pricing their Americanstyle counterparts, which provide early exerci ..."
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Cited by 8 (2 self)
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Pricing Europeanstyle Asian options based on the arithmetic average, under the Black and Scholes model, involves estimating an integral (a mathematical expectation) for which no easily computable analytical solution is available. Pricing their Americanstyle counterparts, which provide early exercise opportunities, poses the additional difficulty of solving a dynamic optimization problem to determine the optimal exercise strategy. A procedure for pricing Americanstyle Asian options of the Bermudan flavor, based on dynamic programming combined with finiteelement piecewisepolynomial approximation of the value function, is developed here. A convergence proof is provided. Numerical experiments illustrate the consistency and efficiency of the procedure. Theoretical properties of the value function and of the optimal exercise strategy are also established.
Using Simulation for Option Pricing
, 2000
"... Monte Carlo simulation is a popular method for pricing financial options and other derivative securities because of the availability of powerful workstations and recent advances in applying the tool. The existence of easytouse software makes simulation accessible to many users who would otherwise ..."
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Cited by 8 (2 self)
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Monte Carlo simulation is a popular method for pricing financial options and other derivative securities because of the availability of powerful workstations and recent advances in applying the tool. The existence of easytouse software makes simulation accessible to many users who would otherwise avoid programming the algorithms necessary to value derivative securities. This paper presents examples of option pricing and variance reduction, and demonstrates their implementation with Crystal Ball 2000, a spreadsheet simulation addin program.
On Selection Criteria for Lattice Rules and Other QuasiMonte Carlo Point Sets
, 2001
"... We define new selection criteria for lattice rules for quasiMonte Carlo integration. The criteria examine the projections of the lattice over subspaces of small or successive dimensions. Their computation exploits the dimensionstationarity of certain lattice rules, and of other lowdiscrepancy poi ..."
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Cited by 7 (0 self)
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We define new selection criteria for lattice rules for quasiMonte Carlo integration. The criteria examine the projections of the lattice over subspaces of small or successive dimensions. Their computation exploits the dimensionstationarity of certain lattice rules, and of other lowdiscrepancy point sets sharing this property. Numerical results illustrate the usefulness of these new figures of merit.