Abstract It is known that the block-based version of the bootstrap method can be used for distributional parameter estimation of dependent data. One of the advantages of this method is that it improves mean square errors. The paper makes two contributions. First, we consider the moving blocking bootstrap method for estimation of parameters of the autoregressive model. For each block, the parameters are estimated based on the modified maximum likelihood method. Second, we provide a method for model selection, Vuong’s test and tracking interval, i.e. for selecting the optimal model for the innovation’s distribution. Our analysis provides analytic results on the asymptotic distribution of the bootstrap estimators and also computational results via simulations. Some properties of the moving blocking bootstrap method are investigated through Monte Carlo simulation. This simulation study shows that, sometimes, Vuong’s test based on the modified maximum likelihood method is not able to distinguish between the two models; Vuong’s test based on the moving blocking bootstrap selects one of the competing models as optimal model. We have studied real data, the S&P500 data, and select optimal model for this data based on the theoretical results.
{"title":"Bootstrap choice of non-nested autoregressive model with non-normal innovations","authors":"Sedigheh Zamani Mehreyan","doi":"10.1515/mcma-2023-2010","DOIUrl":"https://doi.org/10.1515/mcma-2023-2010","url":null,"abstract":"Abstract It is known that the block-based version of the bootstrap method can be used for distributional parameter estimation of dependent data. One of the advantages of this method is that it improves mean square errors. The paper makes two contributions. First, we consider the moving blocking bootstrap method for estimation of parameters of the autoregressive model. For each block, the parameters are estimated based on the modified maximum likelihood method. Second, we provide a method for model selection, Vuong’s test and tracking interval, i.e. for selecting the optimal model for the innovation’s distribution. Our analysis provides analytic results on the asymptotic distribution of the bootstrap estimators and also computational results via simulations. Some properties of the moving blocking bootstrap method are investigated through Monte Carlo simulation. This simulation study shows that, sometimes, Vuong’s test based on the modified maximum likelihood method is not able to distinguish between the two models; Vuong’s test based on the moving blocking bootstrap selects one of the competing models as optimal model. We have studied real data, the S&P500 data, and select optimal model for this data based on the theoretical results.","PeriodicalId":46576,"journal":{"name":"Monte Carlo Methods and Applications","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2023-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46015548","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-06-01DOI: 10.1515/mcma-2023-frontmatter2
{"title":"Frontmatter","authors":"","doi":"10.1515/mcma-2023-frontmatter2","DOIUrl":"https://doi.org/10.1515/mcma-2023-frontmatter2","url":null,"abstract":"","PeriodicalId":46576,"journal":{"name":"Monte Carlo Methods and Applications","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135673784","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract The paper gives new results for the Milstein scheme of stochastic differential equations. We show that (i) the Milstein scheme holds as a weak approximation in total variation sense and is given by second-order polynomials of Brownian motion without using iterated integrals under non-commutative vector fields; (ii) the accuracy of the Milstein scheme is better than that of the Euler–Maruyama scheme in an asymptotic sense. In particular, we prove d TV ( X T ε , X ¯ T ε , Mil , ( n ) ) ≤ C ε 2 / n d_{mathrm{TV}}(X_{T}^{varepsilon},bar{X}_{T}^{varepsilon,mathrm{Mil},(n)})leq Cvarepsilon^{2}/n and d TV ( X T ε , X ¯ T ε , EM , ( n ) ) ≤ C ε / n d_{mathrm{TV}}(X_{T}^{varepsilon},bar{X}_{T}^{varepsilon,mathrm{EM},(n)})leq Cvarepsilon/n , where d TV d_{mathrm{TV}} is the total variation distance, X ε X^{varepsilon} is a solution of a stochastic differential equation with a small parameter 𝜀, and X ¯ ε , Mil , ( n ) bar{X}^{varepsilon,mathrm{Mil},(n)} and X ¯ ε , EM , ( n ) bar{X}^{varepsilon,mathrm{EM},(n)} are the Milstein scheme without iterated integrals and the Euler–Maruyama scheme, respectively. In computational aspect, the scheme is useful to estimate probability distribution functions by a simple simulation without Lévy area computation. Numerical examples demonstrate the validity of the method.
{"title":"Total variation bound for Milstein scheme without iterated integrals","authors":"Toshihiro Yamada","doi":"10.2139/ssrn.4333285","DOIUrl":"https://doi.org/10.2139/ssrn.4333285","url":null,"abstract":"Abstract The paper gives new results for the Milstein scheme of stochastic differential equations. We show that (i) the Milstein scheme holds as a weak approximation in total variation sense and is given by second-order polynomials of Brownian motion without using iterated integrals under non-commutative vector fields; (ii) the accuracy of the Milstein scheme is better than that of the Euler–Maruyama scheme in an asymptotic sense. In particular, we prove d TV ( X T ε , X ¯ T ε , Mil , ( n ) ) ≤ C ε 2 / n d_{mathrm{TV}}(X_{T}^{varepsilon},bar{X}_{T}^{varepsilon,mathrm{Mil},(n)})leq Cvarepsilon^{2}/n and d TV ( X T ε , X ¯ T ε , EM , ( n ) ) ≤ C ε / n d_{mathrm{TV}}(X_{T}^{varepsilon},bar{X}_{T}^{varepsilon,mathrm{EM},(n)})leq Cvarepsilon/n , where d TV d_{mathrm{TV}} is the total variation distance, X ε X^{varepsilon} is a solution of a stochastic differential equation with a small parameter 𝜀, and X ¯ ε , Mil , ( n ) bar{X}^{varepsilon,mathrm{Mil},(n)} and X ¯ ε , EM , ( n ) bar{X}^{varepsilon,mathrm{EM},(n)} are the Milstein scheme without iterated integrals and the Euler–Maruyama scheme, respectively. In computational aspect, the scheme is useful to estimate probability distribution functions by a simple simulation without Lévy area computation. Numerical examples demonstrate the validity of the method.","PeriodicalId":46576,"journal":{"name":"Monte Carlo Methods and Applications","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2023-05-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47464087","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract The paper gives new results for the Milstein scheme of stochastic differential equations. We show that (i) the Milstein scheme holds as a weak approximation in total variation sense and is given by second-order polynomials of Brownian motion without using iterated integrals under non-commutative vector fields; (ii) the accuracy of the Milstein scheme is better than that of the Euler–Maruyama scheme in an asymptotic sense. In particular, we prove dTV(XTε,X¯Tε,Mil,(n))≤Cε2/n d_{mathrm{TV}}(X_{T}^{varepsilon},bar{X}_{T}^{varepsilon,mathrm{Mil},(n)})leq Cvarepsilon^{2}/n and dTV(XTε,X¯Tε,EM,(n))≤Cε/n d_{mathrm{TV}}(X_{T}^{varepsilon},bar{X}_{T}^{varepsilon,mathrm{EM},(n)})leq Cvarepsilon/n , where dTV d_{mathrm{TV}} is the total variation distance, Xε X^{varepsilon} is a solution of a stochastic differential equation with a small parameter 𝜀, and X¯ε,Mil,(n) bar{X}^{varepsilon,mathrm{Mil},(n)} and
{"title":"Total variation bound for Milstein scheme without iterated integrals","authors":"Toshihiro Yamada","doi":"10.1515/mcma-2023-2007","DOIUrl":"https://doi.org/10.1515/mcma-2023-2007","url":null,"abstract":"Abstract The paper gives new results for the Milstein scheme of stochastic differential equations. We show that (i) the Milstein scheme holds as a weak approximation in total variation sense and is given by second-order polynomials of Brownian motion without using iterated integrals under non-commutative vector fields; (ii) the accuracy of the Milstein scheme is better than that of the Euler–Maruyama scheme in an asymptotic sense. In particular, we prove <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mrow> <m:msub> <m:mi>d</m:mi> <m:mi>TV</m:mi> </m:msub> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:msubsup> <m:mi>X</m:mi> <m:mi>T</m:mi> <m:mi>ε</m:mi> </m:msubsup> <m:mo>,</m:mo> <m:msubsup> <m:mover accent=\"true\"> <m:mi>X</m:mi> <m:mo>¯</m:mo> </m:mover> <m:mi>T</m:mi> <m:mrow> <m:mi>ε</m:mi> <m:mo>,</m:mo> <m:mi>Mil</m:mi> <m:mo>,</m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>n</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> </m:msubsup> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> <m:mo>≤</m:mo> <m:mrow> <m:mrow> <m:mi>C</m:mi> <m:mo></m:mo> <m:msup> <m:mi>ε</m:mi> <m:mn>2</m:mn> </m:msup> </m:mrow> <m:mo>/</m:mo> <m:mi>n</m:mi> </m:mrow> </m:mrow> </m:math> d_{mathrm{TV}}(X_{T}^{varepsilon},bar{X}_{T}^{varepsilon,mathrm{Mil},(n)})leq Cvarepsilon^{2}/n and <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mrow> <m:msub> <m:mi>d</m:mi> <m:mi>TV</m:mi> </m:msub> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:msubsup> <m:mi>X</m:mi> <m:mi>T</m:mi> <m:mi>ε</m:mi> </m:msubsup> <m:mo>,</m:mo> <m:msubsup> <m:mover accent=\"true\"> <m:mi>X</m:mi> <m:mo>¯</m:mo> </m:mover> <m:mi>T</m:mi> <m:mrow> <m:mi>ε</m:mi> <m:mo>,</m:mo> <m:mi>EM</m:mi> <m:mo>,</m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>n</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> </m:msubsup> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> <m:mo>≤</m:mo> <m:mrow> <m:mrow> <m:mi>C</m:mi> <m:mo></m:mo> <m:mi>ε</m:mi> </m:mrow> <m:mo>/</m:mo> <m:mi>n</m:mi> </m:mrow> </m:mrow> </m:math> d_{mathrm{TV}}(X_{T}^{varepsilon},bar{X}_{T}^{varepsilon,mathrm{EM},(n)})leq Cvarepsilon/n , where <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mi>d</m:mi> <m:mi>TV</m:mi> </m:msub> </m:math> d_{mathrm{TV}} is the total variation distance, <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mi>X</m:mi> <m:mi>ε</m:mi> </m:msup> </m:math> X^{varepsilon} is a solution of a stochastic differential equation with a small parameter 𝜀, and <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mover accent=\"true\"> <m:mi>X</m:mi> <m:mo>¯</m:mo> </m:mover> <m:mrow> <m:mi>ε</m:mi> <m:mo>,</m:mo> <m:mi>Mil</m:mi> <m:mo>,</m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>n</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> </m:msup> </m:math> bar{X}^{varepsilon,mathrm{Mil},(n)} and <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mover acc","PeriodicalId":46576,"journal":{"name":"Monte Carlo Methods and Applications","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-05-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135996653","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract Finite-dimensional (FD) models X d ( t ) X_{d}(t) , i.e., deterministic functions of time and finite sets of 𝑑 random variables, are constructed for stationary and nonstationary Gaussian processes X ( t ) X(t) with continuous samples defined on a bounded time interval [ 0 , τ ] [0,tau] . The basis functions of these FD models are finite sets of eigenfunctions of the correlation functions of X ( t ) X(t) and of trigonometric functions. Numerical illustrations are presented for a stationary Gaussian process X ( t ) X(t) with exponential correlation function and a nonstationary version of this process obtained by time distortion. It was found that the FD models are consistent with the theoretical results in the sense that their samples approach the target samples as the stochastic dimension is increased.
{"title":"Monte Carlo estimates of extremes of stationary/nonstationary Gaussian processes","authors":"M. Grigoriu","doi":"10.1515/mcma-2023-2006","DOIUrl":"https://doi.org/10.1515/mcma-2023-2006","url":null,"abstract":"Abstract Finite-dimensional (FD) models X d ( t ) X_{d}(t) , i.e., deterministic functions of time and finite sets of 𝑑 random variables, are constructed for stationary and nonstationary Gaussian processes X ( t ) X(t) with continuous samples defined on a bounded time interval [ 0 , τ ] [0,tau] . The basis functions of these FD models are finite sets of eigenfunctions of the correlation functions of X ( t ) X(t) and of trigonometric functions. Numerical illustrations are presented for a stationary Gaussian process X ( t ) X(t) with exponential correlation function and a nonstationary version of this process obtained by time distortion. It was found that the FD models are consistent with the theoretical results in the sense that their samples approach the target samples as the stochastic dimension is increased.","PeriodicalId":46576,"journal":{"name":"Monte Carlo Methods and Applications","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2023-05-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44816176","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract In this paper, we construct stochastic simulation algorithms for solving an elastostatics problem governed by the Lamé equation. Two different stochastic simulation methods are suggested: (1) a method based on a random walk on spheres, which is iteratively applied to anisotropic diffusion equations that are related through the mixed second-order derivatives (this method is meshless and can be applied to boundary value problems for complicated domains); (2) a randomized algorithm for solving large systems of linear algebraic equations that is the core of this method. It needs a mesh formation, but even for very fine grids, the algorithm shows a high efficiency. Both methods are scalable and can be easily parallelized.
{"title":"Two stochastic algorithms for solving elastostatics problems governed by the Lamé equation","authors":"A. Kireeva, Ivan Aksyuk, K. Sabelfeld","doi":"10.1515/mcma-2023-2008","DOIUrl":"https://doi.org/10.1515/mcma-2023-2008","url":null,"abstract":"Abstract In this paper, we construct stochastic simulation algorithms for solving an elastostatics problem governed by the Lamé equation. Two different stochastic simulation methods are suggested: (1) a method based on a random walk on spheres, which is iteratively applied to anisotropic diffusion equations that are related through the mixed second-order derivatives (this method is meshless and can be applied to boundary value problems for complicated domains); (2) a randomized algorithm for solving large systems of linear algebraic equations that is the core of this method. It needs a mesh formation, but even for very fine grids, the algorithm shows a high efficiency. Both methods are scalable and can be easily parallelized.","PeriodicalId":46576,"journal":{"name":"Monte Carlo Methods and Applications","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2023-05-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47878109","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract In this paper, we introduce a new algorithm for estimating the lower bounds for the star discrepancy of any arbitrary point sets in [ 0 , 1 ] s [0,1]^{s} . Computing the exact star discrepancy is known to be an NP-hard problem, so we have been looking for effective approximation algorithms. The star discrepancy can be thought of as the maximum of a function called the local discrepancy, and we will develop approximation algorithms to maximize this function. Our algorithm is analogous to the random walk algorithm described in one of our previous papers [M. Alsolami and M. Mascagni, A random walk algorithm to estimate a lower bound of the star discrepancy, Monte Carlo Methods Appl. 28 (2022), 4, 341–348.]. We add a statistical technique to the random walk algorithm by implementing the Metropolis algorithm in random walks on each chosen dimension to accept or reject this movement. We call this Metropolis random walk algorithm. In comparison to all previously known techniques, our new algorithm is superior, especially in high dimensions. Also, it can quickly determine the precise value of the star discrepancy in most of our data sets of various sizes and dimensions, or at least the lower bounds of the star discrepancy.
{"title":"A Metropolis random walk algorithm to estimate a lower bound of the star discrepancy","authors":"Maryam Alsolami, M. Mascagni","doi":"10.1515/mcma-2023-2005","DOIUrl":"https://doi.org/10.1515/mcma-2023-2005","url":null,"abstract":"Abstract In this paper, we introduce a new algorithm for estimating the lower bounds for the star discrepancy of any arbitrary point sets in [ 0 , 1 ] s [0,1]^{s} . Computing the exact star discrepancy is known to be an NP-hard problem, so we have been looking for effective approximation algorithms. The star discrepancy can be thought of as the maximum of a function called the local discrepancy, and we will develop approximation algorithms to maximize this function. Our algorithm is analogous to the random walk algorithm described in one of our previous papers [M. Alsolami and M. Mascagni, A random walk algorithm to estimate a lower bound of the star discrepancy, Monte Carlo Methods Appl. 28 (2022), 4, 341–348.]. We add a statistical technique to the random walk algorithm by implementing the Metropolis algorithm in random walks on each chosen dimension to accept or reject this movement. We call this Metropolis random walk algorithm. In comparison to all previously known techniques, our new algorithm is superior, especially in high dimensions. Also, it can quickly determine the precise value of the star discrepancy in most of our data sets of various sizes and dimensions, or at least the lower bounds of the star discrepancy.","PeriodicalId":46576,"journal":{"name":"Monte Carlo Methods and Applications","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2023-05-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44369916","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Hadjer Nita, Fairouz Afroun, M. Cherfaoui, D. Aïssani
Abstract In this work, we considered the parametric estimation of the characteristics of the M / M / 1 / N {M/M/1/N} waiting model with Bernoulli feedback. Through a Monte-Carlo simulation study, we have illustrated the effect of the estimation of the starting parameters of the considered waiting system on the statistical properties of its performance measures estimates, when these latter are obtained using the plug-in method. In addition, several types of convergence (bias, variance, MSE, in law) of these performance measure estimators have also been showed by simulation.
{"title":"Statistical analysis of the estimates of some stationary performances of the unreliable M/M/1/N queue with Bernoulli feedback","authors":"Hadjer Nita, Fairouz Afroun, M. Cherfaoui, D. Aïssani","doi":"10.1515/mcma-2023-2004","DOIUrl":"https://doi.org/10.1515/mcma-2023-2004","url":null,"abstract":"Abstract In this work, we considered the parametric estimation of the characteristics of the M / M / 1 / N {M/M/1/N} waiting model with Bernoulli feedback. Through a Monte-Carlo simulation study, we have illustrated the effect of the estimation of the starting parameters of the considered waiting system on the statistical properties of its performance measures estimates, when these latter are obtained using the plug-in method. In addition, several types of convergence (bias, variance, MSE, in law) of these performance measure estimators have also been showed by simulation.","PeriodicalId":46576,"journal":{"name":"Monte Carlo Methods and Applications","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2023-05-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41551785","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract The partitioning algorithm is an iterative procedure that computes explicitly the steady-state probability of a finite Markov chain 𝑋. In this paper, we propose to adapt this algorithm to the case where the state space E := C ∪ D E:=Ccup D is composed of a continuous part 𝐶 and a finite part 𝐷 such that C ∩ D = ∅ Ccap D=emptyset . In this case, the steady-state probability 𝜋 of 𝑋 is a convex combination of two steady-state probabilities π C pi_{C} and π D pi_{D} of two Markov chains on 𝐶 and 𝐷 respectively. The obtained algorithm allows to compute explicitly π D pi_{D} . If π C pi_{C} cannot be computed explicitly, our algorithm approximates it by numerical resolution of successive integral equations. Some numerical examples are studied to show the usefulness and proper functioning of our proposal.
分划算法是显式计算有限马尔可夫链稳态概率的迭代过程𝑋。在本文中，我们提出将该算法应用于状态空间E:=C∪D E:=C cup D由连续部分和有限部分𝐷组成，使得C∩D=∅C cap D= emptyset。在这种情况下，稳态概率𝑋分别是两条马尔可夫链的两个稳态概率π C pi ＿C{和π }D pi ＿D{的凸组合。得到的算法允许显式计算π D }pi ＿D{。如果π C }pi ＿C{不能显式计算，我们的算法通过连续积分方程的数值解析近似它。通过数值算例分析，说明了该方法的有效性和良好的功能。}
{"title":"Computation of the steady-state probability of Markov chain evolving on a mixed state space","authors":"Az-eddine Zakrad, A. Nasroallah","doi":"10.1515/mcma-2023-2003","DOIUrl":"https://doi.org/10.1515/mcma-2023-2003","url":null,"abstract":"Abstract The partitioning algorithm is an iterative procedure that computes explicitly the steady-state probability of a finite Markov chain 𝑋. In this paper, we propose to adapt this algorithm to the case where the state space E := C ∪ D E:=Ccup D is composed of a continuous part 𝐶 and a finite part 𝐷 such that C ∩ D = ∅ Ccap D=emptyset . In this case, the steady-state probability 𝜋 of 𝑋 is a convex combination of two steady-state probabilities π C pi_{C} and π D pi_{D} of two Markov chains on 𝐶 and 𝐷 respectively. The obtained algorithm allows to compute explicitly π D pi_{D} . If π C pi_{C} cannot be computed explicitly, our algorithm approximates it by numerical resolution of successive integral equations. Some numerical examples are studied to show the usefulness and proper functioning of our proposal.","PeriodicalId":46576,"journal":{"name":"Monte Carlo Methods and Applications","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2023-03-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43124785","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Guilhem Balvet, J. Minier, C. Henry, Y. Roustan, M. Ferrand
Abstract The purpose of this paper is to propose a time-step-robust cell-to-cell integration of particle trajectories in 3-D unstructured meshes in particle/mesh Lagrangian stochastic methods. The main idea is to dynamically update the mean fields used in the time integration by splitting, for each particle, the time step into sub-steps such that each of these sub-steps corresponds to particle cell residence times. This reduces the spatial discretization error. Given the stochastic nature of the models, a key aspect is to derive estimations of the residence times that do not anticipate the future of the Wiener process. To that effect, the new algorithm relies on a virtual particle, attached to each stochastic one, whose mean conditional behavior provides free-of-statistical-bias predictions of residence times. After consistency checks, this new algorithm is validated on two representative test cases: particle dispersion in a statistically uniform flow and particle dynamics in a non-uniform flow.
{"title":"A time-step-robust algorithm to compute particle trajectories in 3-D unstructured meshes for Lagrangian stochastic methods","authors":"Guilhem Balvet, J. Minier, C. Henry, Y. Roustan, M. Ferrand","doi":"10.1515/mcma-2023-2002","DOIUrl":"https://doi.org/10.1515/mcma-2023-2002","url":null,"abstract":"Abstract The purpose of this paper is to propose a time-step-robust cell-to-cell integration of particle trajectories in 3-D unstructured meshes in particle/mesh Lagrangian stochastic methods. The main idea is to dynamically update the mean fields used in the time integration by splitting, for each particle, the time step into sub-steps such that each of these sub-steps corresponds to particle cell residence times. This reduces the spatial discretization error. Given the stochastic nature of the models, a key aspect is to derive estimations of the residence times that do not anticipate the future of the Wiener process. To that effect, the new algorithm relies on a virtual particle, attached to each stochastic one, whose mean conditional behavior provides free-of-statistical-bias predictions of residence times. After consistency checks, this new algorithm is validated on two representative test cases: particle dispersion in a statistically uniform flow and particle dynamics in a non-uniform flow.","PeriodicalId":46576,"journal":{"name":"Monte Carlo Methods and Applications","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2023-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48076657","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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