fixed width multiple comparison procedures using common random numbers for steady state simulations abstract suppose that there are k 2 different systems i e stochastic processes where each system has an unknown steady state mean performance we consider the problem of running a two stage simulation using common random numbers to construct fixed width confidence intervals for two multiple comparison problems under the assumptions that the stochastic processes representing the simulation output of the different systems satisfy a functional central limit theorem and that the asymptotic covariance matrix satisfies a condition known as sphericity we prove that our confidence intervals are asymptotically valid as the desired half width of the confidence intervals tend to zero we develop both absolute and relative width confidence intervals empirical results are presented indicating the procedures robustness to violations of the sphericity assumption

asymptotic properties of some confidence interval estimators for simulation output the classical confidence interval estimator commonly used in simulation is compared with four new estimators based on standardization of a time series presented in a previous paper these new interval estimators are shown to have asymptotic properties that strictly dominate the classical estimator when used with data from independent simulation replications or means of batched observations from a single simulation run two of the new estimators also can be used with a single unbatched replication of a simulation program a situation where the classical estimator is not defined

unbiased estimation with square root convergence for sde models in many settings in which monte carlo methods are applied there may be no known algorithm for exactly generating the random object for which an expectation is to be computed frequently however one can generate arbitrarily close approximations to the random object we introduce a simple randomization idea for creating unbiased estimators in such a setting based on a sequence of approximations applying this idea to computing expectations of path functionals associated with stochastic differential equations sdes we construct finite variance unbiased estimators with a square root convergence rate for a general class of multidimensional sdes we then identify the optimal randomization distribution numerical experiments with various path functionals of continuous time processes that often arise in finance illustrate the effectiveness of our new approach

multiple comparison procedures for steady state simulations suppose that there are k 2 different systems i e stochastic processes where each system has an unknown steady state mean performance and unknown asymptotic variance we allow for the asymptotic variances to be unequal and for the distributions of the k systems to be different we consider the problem of running independent single stage simulations to make multiple comparisons of the steady state means of the different systems we derive asymptotically valid as the run lengths of the simulations of the systems tend to infinity simultaneous confidence intervals for each of the following problems all pairwise comparisons of means all contrasts multiple comparisons with a control and multiple comparisons with the best our confidence intervals are based on standardized time series methods and we establish the asymptotic validity of each under the sole assumption that the stochastic processes representing the simulation output of the different systems satisfy a functional central limit theorem although simulation is the context of this paper the results naturally apply to asymptotically stationary time series

random generation of stochastic area integrals the authors describe a method of random generation of the integrals a 1 2 t t h int t t h int t s dw 1 r dw 2 s int t t h int t s dw 2 r dw 1 s together with the increments w 1 t h w 1 t and w 2 t h w 2 t of a two dimensional brownian path w 1 t w 2 t the method chosen is based on marsaglia s rectangle wedge tail method generalised to higher dimensions the motivation is the need for a numerical scheme for simulation of strong solutions of general multidimensional stochastic differential equations with an order of convergence o h where h is the stepsize previously no method has obtained an order of convergence better than o sqrt h in the general case

simultaneous confidence intervals for all distances from the best in practice comparisons with the best are often the ones of primary interest in this paper parametric and nonparametric simultaneous upper confidence intervals for all distances from the best are derived under the location model their improvement upon the results of bechhofer 1954 gupta 1956 1965 fabian 1962 and desu 1970 in the parametric case is discussed in the nonparametric case no comparable confidence statements were available previously

batch size effects in the analysis of simulation output batching is a commonly used method for calculating confidence intervals on the mean of a sequence of correlated observations arising from a simulation experiment several recent papers have considered the effect of using batch sizes too small to satisfy assumptions of normality and or independence and the resulting incorrect probabilities of the confidence interval covering the mean this paper quantifies the effects of using batch sizes larger than necessary to satisfy normality and independence assumptions these effects include 1 correct probability of covering the mean 2 an increase in expected half length 3 an increase in the standard deviation and coefficient of variation of the half length and 4 an increase in the probability of covering points not equal to the mean for any sample size and independent and normal batch means the results are 1 the effects of less than 10 batches are large and the effects of more than 30 batches small and 2 additional batches have lesser effects on

exact simulation of the ornstein uhlenbeck driven stochastic volatility model abstract this paper proposes a novel exact simulation method for the ornstein uhlenbeck driven stochastic volatility model to accomplish this goal our task hinges on properly handling the ornstein uhlenbeck volatility process the major challenge involves conditionally sampling the integral of its square with respect to time given its marginal state as well as its integral with respect to time we thus derive a closed form laplace transform of this conditional distribution via the techniques of changing probability measure as well as analytical extension then we obtain the corresponding conditional cumulative distribution function via fourier transform inversion and finally sample the distribution via the inverse transform method we show that our method achieves a faster convergence rate of root mean square errors comparing with euler discretization method and apply it in the valuation of discretely monitored path dependent options

the use of subseries values for estimating the variance of a general statistic from a stationary sequence let z i infty i infty be a strictly stationary infty mixing sequence without specifying the dependence model giving rise to z i and without specifying the marginal distribution of z i we address the question of variance estimation for a general statistic t n t n z 1 z n for estimating var t n from just the available data z 1 z n we propose computing subseries values t m z i 1 z i 2 z i m 0 leq i i m leq n these subseries values are used as replicates to model the sampling variability of t n in particular we use adjacent nonoverlapping subseries of length m m n with m rightarrow infty and m n n rigtharrow 0 our variance estimator is just the usual sample variance computed amongst these subseries values after appropriate standardization this estimator is shown to be consistent under mild integrability conditions we present optimal i e minimum m s e choices of m n for the special case where t n overset z n and z i is a normal ar 1 sequence a simulation study is conducted showing that those same choices of m n are effective when t n is a robust estimator of location and z i is subject to contamination

exact simulation of stochastic volatility and other affine jump diffusion processes the stochastic differential equations for affine jump diffusion models do not yield exact solutions that can be directly simulated discretization methods can be used for simulating security prices under these models however discretization introduces bias into the simulation results and a large number of time steps may be needed to reduce the discretization bias to an acceptable level this paper suggests a method for the exact simulation of the stock price and variance under hestons stochastic volatility model and other affine jump diffusion processes the sample stock price and variance from the exact distribution can then be used to generate an unbiased estimator of the price of a derivative security we compare our method with the more conventional euler discretization method and demonstrate the faster convergence rate of the error in our method specifically our method achieves an o s 1 2 convergence rate where s is the total computational budget the convergence rate for the euler discretization method is o s 1 3 or slower depending on the model coefficients and option payoff function

an adaptive timestepping algorithm for stochastic differential equations we introduce a variable timestepping procedure using local error control for the pathwise strong numerical integration of a system of stochastic differential equations forced by a single wiener process the milstein method is used to advance the numerical solution and the stepsizes are determined via two local error estimates that roughly correspond to leading order deterministic and stochastic local error components one advantage of using two error controls is an increased flexibility that allows for the treatment of both drift and diffusion dominated regimes in a consistent manner numerical results are presented and the generalization of this approach to wider classes of problems and methods is discussed

standardized time series l p norm variance estimators for simulations this paper studies a class of estimators for the variance parameter of a stationary stochastic process the estimators are based on lp norms of standardized time series and they generalize previously studied estimators due to schruben we show that the new estimators have some desirable properties they are asymptotically unbiased and have low asymptotic variance we also illustrate empirically the performance of the lp norm estimators on various stochastic processes

multilevel monte carlo path simulation we show that multigrid ideas can be used to reduce the computational complexity of estimating an expected value arising from a stochastic differential equation using monte carlo path simulations in the simplest case of a lipschitz payoff and a euler discretisation the computational cost to achieve an accuracy of o e is reduced from o e 3 to o e 2 log e 2 the analysis is supported by numerical results showing significant computational savings

pairwise comparisons of generally correlated means abstract a commonly occurring statistical inference problem in practice is that of making simultaneous comparisons among three or more treatment means i 1 i k for certain experimental designs one way to do this the t method proposed by tukey 1953 constructs simultaneous confidence intervals for all pairwise differences of the treatment means i j 1 i j k i j the joint confidence level of these confidence intervals however depends in a very complicated fashion on the covariance structure of the treatment mean estimates i and therefore for designs where the covariance structure is not simple the joint confidence level is not readily apparent in general this is the case for any unbalanced design or for designs in which the treatment mean estimates i have unequal correlations tukey conjectured in 1953 that whatever the correlation structure of the treatment mean estimates i the t method would always provide a conservative set of confidence intervals that is that th

using standardized time series to estimate confidence intervals for the difference between two stationary stochastic processes this paper discusses confidence interval estimation for the difference between the means of two independent strictly stationary phi mixing stochastic processes the confidence intervals are based on batched subseries of the two time series and are obtained by using schruben s standardized time series method we compare these intervals with the classical result applied to the batch means and consider different assumptions on the variances and the number of observations the comparison results show that the four new estimators for each case have asymptotic properties that clearly dominate the classical estimator two applications of these intervals are 1 validating stationary discrete event simulation models and 2 comparing two alternative policies or system designs via simulation

the asymptotic efficiency of simulation estimators a decision theoretic framework is proposed for evaluating the efficiency of simulation estimators the framework includes the cost of obtaining the estimate as well as the cost of acting based on the estimate the cost of obtaining the estimate and the estimate itself are represented as realizations of jointly distributed stochastic processes in this context the efficiency of a simulation estimator based on a given computational budget is defined as the reciprocal of the risk the overall expected cost this framework is appealing philosophically but it is often difficult to apply in practice e g to compare the efficiency of two different estimators because only rarely can the efficiency associated with a given computational budget be calculated however a useful practical framework emerges in a large sample context when we consider the limiting behavior as the computational budget increases a limit theorem established for this model supports and extends a fairly well known efficiency principle

exact sampling of jump diffusions this paper develops a method for the exact simulation of a skeleton a hitting time and other functionals of a one dimensional jump diffusion with state dependent drift volatility jump intensity and jump size the method requires the drift function to be c1 the volatility function to be c2 and the jump intensity function to be locally bounded no further structure is imposed on these functions the method leads to unbiased simulation estimators of security prices transition densities hitting probabilities and other quantities numerical results illustrate its features

numerical analysis of stochastic schemes in geophysics we present and study the stability convergence and order of convergence of a numerical scheme used in geophysics namely the stochastic version of a deterministic implicit leapfrog scheme which has been developed for the approximation of the so called barotropic vorticity model two other schemes which might be useful in the context of geophysical applications are also introduced and discussed

steady vortex patches near a rotating flow with constant vorticity in a planar bounded domain abstract in this paper we study the possible equilibrium states of the 2d incompressible euler equations in a bounded domain by solving a suitable minimization problem we prove existence of a new type of steady euler flows in which the vorticity vanishes on a finite number of compact sets of small diameter and is a positive constant elsewhere the locations of those small compact sets are determined by the geometry of the domain

on the low frequency asymptotics in electromagnetic theory on considere les equations de maxwell harmoniques par rapport au temps avec des conditions aux limites on etudie le comportement asymptotique a basse frequence

robust sensitivity analysis for stochastic systems we study a worst case approach to measure the sensitivity to model misspecification in the performance analysis of stochastic systems the situation of interest is when only minimal parametric information is available on the form of the true model under this setting we post optimization programs that compute the worst case performance measures subject to constraints on the amount of model misspecification measured by kullback leibler divergence our main contribution is the development of infinitesimal approximations for these programs resulting in asymptotic expansions of their optimal values as the divergence shrinks to zero the coefficients of these expansions can be computed via simulation and are mathematically derived from the representation of the worst case models as changes of measure that satisfy a well defined class of functional fixed point equations

variable step size control in the numerical solution of stochastic differential equations we introduce a variable step size method for the numerical approximation of pathwise solutions to stochastic differential equations sdes the method which is dependent on a representation of bro

multiple comparisons with the best treatment abstract let 1 2 k be k 2 sources of observations treatments populations and suppose the goodness of treatment i is characterized by the size of an unknown real valued parameter i let k max1 i k i if i is preferred to j when i j the parameters i k i i 1 2 k reflect in an inverse sense the goodness of each treatment relative to the best treatment a general technique for obtaining simultaneous confidence intervals on the i is demonstrated with several examples this technique can be applied in any setting where comparison with control intervals can be computed regarding any j as the control these results have special importance in ranking and selection problems in that the process of generating upper bounds on the i generates traditional confidence statements of both the indifference zone and the subset selection schools simultaneously as established by hsu 1981

approximation of multiple stochastic integrals and its application to stochastic differential equations abstract as multiple stochastic integrals are not very easy to simulate we would like to treat them as solutions of systems of stochastic differential equations and solve them successively and recursively approximated by the stochastic taylor expansion as a chen series in terms of a philip hall basis or lyndon basis we can save sufficient values of multiple stochastic integrals with independent sample paths in a look up table for future use the table can be used to implement high order schemes to solve stochastic differential equations numerically a numerical example will be shown to illustrate the efficiency

on nonnegative unbiased estimators we study the existence of algorithms generating almost surely non negative unbiased estimators of some quantities of practical interest our work is motivated by recent exact approximation methods to compute integrals with respect to any target distribution up to any arbitrary precision these techniques can be implemented as long as non negative unbiased estimators of the target density evaluations are available in particular we show that given a non constant function f from r to r and real valued unbiased estimators xn of r there is no algorithm yielding almost surely non negative unbiased estimators of f even if xn is itself a s non negative then there is no algorithm yielding a s non negative unbiased estimators of f if f is not increasing however if the support of xn is a known interval of r then such algorithms exist as long as the function f satisfies some polynomial bound away from zero the results are discussed in the light of several typical situations corresponding to different functions f

general multilevel monte carlo methods for pricing discretely monitored asian options abstract we describe general multilevel monte carlo methods that estimate the price of an asian option monitored at m fixed dates for a variety of processes that can be simulated exactly we prove that for the same computational cost our method yields an unbiased estimator with variance lower than the variance of the standard monte carlo estimator by a factor of order m we show how to combine our approach with the milstein scheme for processes driven by scalar stochastic differential equations and with the euler scheme for processes driven by multidimensional stochastic differential equations numerical experiments confirm that our method outperforms the conventional monte carlo algorithm by a factor proportional to m

control variate models of common random numbers for multiple comparisons with the best using common random numbers crn in simulation experiment design is known to reduce the variance of estimators of differences in system performance however when more than two systems are compared exact simultaneous statistical inference in conjunction with crn is typically impossible we introduce control variate models of crn that permit exact statistical inference specifically multiple comparisons with the best these models explain the effect of crn via a linear regression of the simulation output on control variates that are functions of the simulation inputs we establish theoretically and illustrate empirically that the control variate models lead to sharper statistical inference in the sense that the probability of detecting differences in systems performance is increased

bayesian optimization via simulation with pairwise sampling and correlated prior beliefs this paper addresses discrete optimization via simulation we show that allowing for both a correlated prior distribution on the means e g with discrete kriging models and sampling correlation e g with common random numbers or crn can significantly improve the ability to quickly identify the best alternative these two correlations are brought together for the first time in a highly sequential knowledge gradient sampling algorithm which chooses points to sample using a bayesian value of information voi criterion we provide almost sure convergence guarantees as the number of samples grows without bound when parameters are known and provide approximations that allow practical implementation including a novel use of the voi s gradient rather than the response surface s gradient we demonstrate that crn leads to improved optimization performance for voi based algorithms in sequential sampling environments with a combinatorial number of alternatives and costly samples

constrained simultaneous confidence intervals for multiple comparisons with the best for comparing k treatment effects 01 02 ok often the parameters of primary interest are 0i maxeioi0 i 1 k in this article we develop constrained 100p two sided simultaneous confidence intervals for 0i maxj i0j which we refer to as constrained mcb intervals it turns out that the lower bounds of the intervals imply indifference zone selection inference and the upper bounds of these intervals imply subset selection inference each given at the same confidence level 100p as mcb we also extend our

simulation output analysis using standardized time series the method of standardized time series sts was proposed by schruben as an approach for constructing asymptotic confidence intervals for the steady state mean from a single simulation run the sts method cancels out the variance constant while other methods attempt to consistently estimate the variance constant our goal in this paper is to generalize the sts method and to study some of its basic properties starting from a functional central limit theorem fclt for the sample mean of the simulated process a class of mappings of c 0 1 to is identified each of which leads to a sts confidence interval one of these mappings leads to the batch means method a lower bound is obtained for the expected length of the asymptotic as the run size becomes large sts confidence intervals this lower bound is not attained but can be approached arbitrarily closely by sts confidence intervals methods that consistently estimate the variance constant do realize this lower bound the variance of the length of a sts confidence interval is of larger order in the run length than is that for the regenerative method

exact simulation of the sabr model the stochastic alpha beta rho sabr model becomes popular in the financial industry because it is capable of providing good fits to various types of implied volatility curves observed in the marketplace however no analytical solution to the sabr model exists that can be simulated directly this paper explores the possibility of exact simulation for the sabr model our contribution is threefold i we propose an exact simulation method for the forward price and its volatility in two special but practically interesting cases i e when the elasticity 1 or when 1 and the price and volatility processes are instantaneously uncorrelated primary difficulties involved are how to simulate two random variables whose distributions can be expressed in terms of the hartman watson and the noncentral chi squared distribution functions respectively two novel simulation schemes are proposed to achieve numerical accuracy efficiency and stability one stems from numerical laplace inversion and asian option literature and the other is based on recent developments in evaluating the noncentral chi squared distribution functions in a robust way numerical examples demonstrate that our method is fast and accurate under various market environments ii when 1 but the price and volatility processes are correlated our simulation method becomes a semi exact one numerical results suggest that it is still quite accurate when the time horizon is not long e g no greater than one year for long time horizons a piecewise semi exact simulation scheme is developed that reduces the biases substantially iii for european option pricing under the sabr model we propose a conditional simulation method which reduces the variance of the plain simulation significantly e g by more than 99 the e companion is available at https doi org 10 1287 opre 2017 1617

an investigation of finite sample behavior of confidence interval estimators we investigate the small sample behavior and convergence properties of confidence interval estimators cies for the mean of a stationary discrete process we consider cies arising from nonoverlapping batch means overlapping batch means and standardized time series all of which are commonly used in discrete event simulation the performance measures of interest are the coverage probability and the expected value and variance of the half length we use empirical and analytical methods to make detailed comparisons regarding the behavior of the cies for a variety of stochastic processes all the cies under study are asymptotically valid however they are usually invalid for small sample sizes we find that for small samples the bias of the variance parameter estimator figures significantly in cie coverage performance the less bias the better a secondary role is played by the marginal distribution of the stationary process we also point out that some cies require fewer observations before manifesting

on δ equivalence with the best in k sample models in recent work we introduced a general a weak and a strong partitioning principles for the construction of multiple decision procedures as multiple testing or selection procedures partitioning principles can be viewed as natural extensions of the closure principle and sometimes yield more powerful decision procedures in this article we consider the problem of establishing equivalence with the best with respect to k treatment means where equivalence is defined in terms of a threshold value 0 we reformulate the original selection problem as a multiple testing problem and develop various step down and step up procedures by applying the closure principle and partitioning principles the new step down procedure is shown to provide a uniform improvement over procedures currently in use moreover we propose some projection methods that yield confidence intervals being compatible with stepwise tests and selection procedures

efficient estimation and filtering for multivariate jump diffusions abstract this paper develops estimators of the transition density filters and parameters of multivariate jump diffusion models the drift volatility jump intensity and jump magnitude are allowed to be state dependent and non affine it is not necessary to diagonalize the volatility matrix our density and filter estimators converge at the canonical rate typically associated with exact monte carlo estimation our parameter estimators have the same asymptotic distribution as maximum likelihood estimators which are often intractable for the class of models we consider the results of this paper enable the empirical analysis of previously intractable models of asset prices and economic time series

efficient simulation of clustering jumps with cir intensity we introduce a broad family of generalised self exciting point processes with cir type intensities and we develop associated algorithms for their exact simulation the underlying models are extensions of the classical hawkes process which already has numerous applications in modelling the arrival of events with clustering or contagion effect in finance economics and many other fields interestingly we find that the cir type intensity together with its point process can be sequentially decomposed into simple random variables which immediately leads to a very efficient simulation scheme our algorithms are also pretty accurate and flexible they can be easily extended to further incorporate externally excited jumps or to a multidimensional framework some typical numerical examples and comparisons with other well known schemes are reported in detail in addition a simple application for modelling a portfolio loss process is presented the online appendix is available at https doi org 10 1287 op

two stage stopping procedures based on standardized time series we propose some new two stage stopping procedures to construct absolute width and relative width confidence intervals for a simulation estimator of the steady state mean of a stochastic process the procedures are based on the method of standardized time series proposed by schruben and on stein s two stage sampling scheme we prove that our two stage procedures give rise to asymptotically valid confidence intervals as the prescribed length of the confidence interval approaches zero and the size of the first stage grows to infinity the sole assumption required is that the stochastic process satisfy a functional central limit theorem

confidence interval estimation using standardized time series observations of a stationary stochastic process can be transformed into a standardized time series this paper presents a lemma giving the asymptotic properties of this standardized series under quite general conditions in particular the conditions are satisfied by stationary discrete event simulations confidence intervals can be constructed using this lemma for illustration we develop two easily computed interval estimators for the process mean when independent replications of the series are available such as in computer simulation experiments these interval estimators may be combined with the classical confidence interval estimator these interval estimators also tend to compensate for simulation initialization bias if the sign of the bias is known in an empirical study using three elementary simulated processes the interval estimators presented here compare favorably with the classical interval estimator in a recent paper goldsman and schruben goldsman d l schruben 1982 asymptotic properties of some confidence interval estimators tech rep 544 school of o r i e cornell university ithaca ny show that the asymptotic properties of the confidence intervals presented in this paper strictly dominate those of classical confidence intervals

product expansion for stochastic jump diffusions and its application to numerical approximation abstract we derive a product expansion of the exponential lie series in terms of a philip hall basis for the chen series corresponding to the stochastic jump diffusion as in sussmann in c i byrnes and a lindquist eds theory and applications of nonlinear control systems north holland amsterdam 1986 pp 323 335 for the deterministic case based on the expansion we establish the stratonovich taylor hall sth schemes such that each scheme involves only the minimum number of multiple stochastic integrals which can be regarded as systems of stochastic differential equations and approximated by a lower order scheme with an appropriate step size to ensure the necessary accuracy mean square convergence of the sth schemes is shown and numerical examples are provided to illustrate the results

the law of the euler scheme for stochastic differential equations i convergence rate of the distribution function we study the approximation problem ofef x t byef x t n where x t is the solution of a stochastic differential equation x t n is defined by the euler discretization scheme with stept n andf is a given function for smoothf s talay and tubaro have shown that the erroref x t f x t n can be expanded in powers of 1 n which permits to construct romberg extrapolation precedures to accelerate the convergence rate here we prove that the expansion exists also whenf is only supposed measurable and bounded under an additional nondegeneracy condition of hormander type for the infinitesimal generator of x t to obtain this result we use the stochastic variations calculus in the second part of this work we will consider the density of the law ofx t n and compare it to the density of the law ofx t

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 validity of fully sequential selection procedures for steady state simulation we present fully sequential procedures for steady state simulation that are designed to select the best of a finite number of simulated systems when best is defined by the largest or smallest long run average performance we also provide a framework for establishing the asymptotic validity of such procedures and prove the validity of our procedures an example based on the m m 1 queue is given

high order discretization schemes for the cir process application to affine term structure and heston models this paper presents weak second and third order schemes for the cox ingersoll ross cir process without any restriction on its parameters at the same time it gives a general recursive construction method to get weak second order schemes that extends the one introduced by ninomiya and victoir cite nv combining these both results this allows to propose a second order scheme for more general affine diffusions simulation examples are given to illustrate the convergence of these schemes on cir and heston models algorithms are stated in a pseudocode language

convergence of numerical schemes for the solution of parabolic stochastic partial differential equations we consider the numerical solution of the stochastic partial differential equation u t 2 u x 2 u w x t where w is space time white noise using finite differences for this equation gyongy has obtained an estimate of the rate of convergence for a simple scheme based on integrals of w over a rectangular grid we investigate the extent to which this order of convergence can be improved and find that better approximations are possible for the case of additive noise u 1 if we wish to estimate space averages of the solution rather than pointwise estimates or if we are permitted to generate other functionals of the noise but for multiplicative noise u u we show that no such improvements are possible

a variational principle for three dimensional water waves over beltrami flows abstract we consider steady three dimensional gravity capillary water waves with vorticity propagating on water of finite depth we prove a variational principle for doubly periodic waves with relative velocities given by beltrami vector fields under general assumptions on the wave profile

control variate remedies other than common random numbers control variates is the most promising variance reduction technique in terms of its potential for widespread use control variates is applicable in single or multiple response simulation it does not require altering the simulation run in any way and any stochastic simulation contains potential control variates a rich theory of control variates has been developed in recent years most of this theory assumes a specific probabilistic structure for the simulation output process usually joint normality of the response and the control variates when these assumptions are not satisfied desirable properties of the estimator such as unbiasedness may be lost a number of remedies for violations of the assumptions have been proposed including jackknifing and splitting however there has been no systematic analytical and empirical evaluation of these remedies this paper presents such an evaluation including evaluation of the small sample statistical properties of the proposed remedies

using common random numbers and control variates in multiple comparison procedures this paper considers the determination of the relative merits of two or more system designs via stochastic simulation experiments by constructing simultaneous interval estimates of certain differences in expected performance tukey s all pairwise comparisons procedure hsu s multiple comparisons with the best procedure and dunnett s multiple comparisons with a control procedure are standard methods for making such comparisons we propose refinements for all three procedures through the use of two variance reduction techniques common random numbers and control variates we show that the proposed procedures are better than the standard multiple comparison procedures in the sense that they have a larger probability of containing the true difference and at the same time not containing zero when a difference exists

three dimensional steady subsonic euler flows in bounded nozzles abstract the existence and uniqueness of three dimensional steady subsonic euler flows in rectangular nozzles were obtained when prescribing normal component of momentum at both the entrance and exit if in addition the normal component of the voriticity and the variation of bernoulli s function at the entrance are both zero then there exists a unique subsonic potential flow when the magnitude of the normal component of the momentum is less than a critical number as the magnitude of the normal component of the momentum approaches the critical number the associated flows converge to a subsonic sonic flow furthermore when the normal component of vorticity and the variation of bernoulli function are both small the existence and uniqueness of subsonic euler flows with non zero vorticity are established the proof of these results is based on a new formulation for the euler system a priori estimate for nonlinear elliptic equations with nonlinear boundary conditions detailed study for a linear div curl system and delicate estimate for the transport equations

unbiased estimators and multilevel monte carlo multilevel monte carlo mlmc and recently proposed unbiased estimators are closely related this connection is elaborated by presenting a new general class of unbiased estimators which admits previous debiasing schemes as special cases new lower variance estimators are proposed which are stratified versions of earlier unbiased schemes under general conditions essentially when mlmc admits the canonical square root monte carlo error rate the proposed new schemes are shown to be asymptotically as efficient as mlmc both in terms of variance and cost the experiments demonstrate that the variance reduction provided by the new schemes can be substantial

higher order time integration of coulomb collisions in a plasma using langevin equations the extension of langevin equation monte carlo algorithms for coulomb collisions from the conventional euler maruyama time integration to the next higher order of accuracy the milstein scheme has been developed implemented and tested this extension proceeds via a formulation of the angular scattering directly as stochastic differential equations in the fixed frame spherical coordinate velocity variables results from the numerical implementation show the expected improvement o dt vs o dt 1 2 in the strong convergence rate both for the speed v and angular components of the scattering an important result is that this improved convergence is achieved for the angular component of the scattering if and only if the area integral terms in the milstein scheme are included the resulting milstein scheme is of value as a step towards algorithms with both improved accuracy and efficiency these include both algorithms with improved convergence in the averages weak convergence and multi time level schemes the latter have been shown to give a greatly reduced cost for a given overall error level when compared with conventional monte carlo schemes and their performance is improved considerably when the milstein algorithm is used for the underlying time advance versus the euler maruyama algorithm a new method for sampling the area integrals is given which is a simplification of an earlier direct method and which retains high accuracy this method while being useful in its own right because of its relative simplicity is also expected to considerably reduce the computational requirements for the direct conditional sampling of the area integrals that is needed for adaptive strong integration

simulated likelihood estimators for discretely observed jump diffusions abstract this paper develops an unbiased monte carlo approximation to the transition density of a jump diffusion process with state dependent drift volatility jump intensity and jump magnitude the approximation is used to construct a likelihood estimator of the parameters of a jump diffusion observed at fixed time intervals that need not be short the estimator is asymptotically unbiased for any sample size it has the same large sample asymptotic properties as the true but uncomputable likelihood estimator numerical results illustrate its properties

using common random numbers for indifference zone selection and multiple comparisons in simulation we present a general recipe for constructing experiment design and analysis procedures that simultaneously provide indifference zone selection and multiple comparison inference for choosing the best among k simulated systems we then exhibit two such procedures that exploit the variance reduction technique of common random numbers to reduce the sample size required to attain a fixed precision one procedure is based on the bonferroni inequality and is guaranteed to be statistically conservative the other procedure is exact under a specific dependence structure but may be slightly liberal otherwise both are easy to apply requiring only simple calculations and tabled constants we illustrate the procedures with a numerical example

efficient strong integrators for linear stochastic systems we present numerical schemes for the strong solution of linear stochastic differential equations driven by an arbitrary number of wiener processes these schemes are based on the neumann stochastic taylor and magnus expansions first we consider the case when the governing linear diffusion vector fields commute with each other but not with the linear drift vector field we prove that numerical methods based on the magnus expansion are more accurate in the mean square sense than corresponding stochastic taylor integration schemes second we derive the maximal rate of convergence for arbitrary multidimensional stochastic integrals approximated by their conditional expectations consequently for general nonlinear stochastic differential equations with noncommuting vector fields we deduce explicit formulae for the relation between error and computational costs for methods of arbitrary order third we consider the consequences in two numerical studies one of which is an application arising in stochastic linear quadratic optimal control

runge kutta methods for the strong approximation of solutions of stochastic differential equations some new stochastic runge kutta srk methods for the strong approximation of solutions of stochastic differential equations sdes with improved efficiency are introduced their convergence is proved by applying multicolored rooted tree analysis order conditions for the coefficients of explicit and implicit srk methods are calculated as the main novelty order 1 0 strong srk methods with significantly reduced computational complexity for ito as well as for stratonovich sdes with a multidimensional driving wiener process are presented where the number of stages is independent of the dimension of the wiener process further an order 1 0 strong srk method customized for ito sdes with commutative noise is introduced finally some order 1 5 strong srk methods for sdes with scalar diagonal and additive noise are proposed all introduced srk methods feature significantly reduced computational complexity compared to well known schemes

filtered likelihood for point processes abstract point processes are widely used in finance and economics to model the timing of defaults market transactions unemployment spells births and a range of other events we develop and analyze likelihood estimators for the parameters of a marked point process and incompletely observed explanatory factors that influence the arrival intensity and mark distribution we establish an approximation to the likelihood and analyze the convergence and large sample properties of the associated estimators numerical results illustrate the behavior of our estimators

optimal unbiased estimation for expected cumulative discounted cost abstract we consider estimating an expected infinite horizon cumulative discounted cost reward contingent on an underlying stochastic process by monte carlo simulation an unbiased estimator based on truncating the cumulative cost at a random horizon is proposed explicit forms for the optimal distributions of the random horizon are given and explicit expressions for the optimal random truncation level are obtained leading to a full analysis of the bias variance tradeoff when comparing this new class of randomized estimators with traditional fixed truncation estimators moreover we characterize when the optimal randomized estimator is preferred over a fixed truncation estimator by considering the tradeoff between bias and variance this comparison provides guidance on when to choose randomized estimators over fixed truncation estimators in practice numerical experiments substantiate the theoretical results

an invariance principle for certain dependent sequences let x 1 x 2 cdots be a strictly stationary second order sequence which is associated i e is such that any two coordinatewise nondecreasing functions of the x i s of finite variance are nonnegatively correlated if sum j operatorname cov x 1 x j infty then the partial sum processes w n t defined in the usual way so that w n m n x 1 cdots x m me x 1 sqrt n for m 1 2 cdots converge in distribution on c lbrack 0 t rbrack to a wiener process this result is based on two general theorems concerning associated random variables which are of independent interest

sdelab a package for solving stochastic differential equations in matlab we introduce sdelab a package for solving stochastic differential equations sdes within matlab sdelab features explicit and implicit integrators for a general class of ito and stratonovich sdes including milstein s method sophisticated algorithms for iterated stochastic integrals and flexible plotting facilities

localization and exact simulation of brownian motion driven stochastic differential equations generating sample paths of stochastic differential equations sde using the monte carlo method finds wide applications in financial engineering discretization is a popular approximate approach to generating those paths it is easy to implement but prone to simulation bias this paper presents a new simulation scheme to exactly generate samples for sdes the key observation is that the law of a general sde can be decomposed into a product of the law of standard brownian motion and the law of a doubly stochastic poisson process an acceptance rejection algorithm is devised based on the combination of this decomposition and a localization technique the numerical results corroborates that the mean square error of the proposed method is in the order of ot 1 2 which is superior to the conventional discretization schemes furthermore the proposed method also can generate exact samples for sde with boundaries which the discretization schemes usually find difficulty in dealing with