Pub Date : 2015-02-19DOI: 10.1080/17442508.2014.989526
Christophe Audouze, P. Nair
We consider finite element approximations of parabolic stochastic partial differential equations (SPDEs) in conjunction with the -weighted temporal discretization scheme. We study the stability of the numerical scheme and provide a priori error estimates, using a result of Galvis and Sarkis [Approximating infinity-dimensional stochastic Darcy's equations without uniform ellipticity, SIAM J. Numer. Anal. 47(5) (2009), pp. 3624–3651] on elliptic SPDEs.
我们考虑了抛物型随机偏微分方程(SPDEs)的有限元近似与加权时间离散方案。本文研究了数值格式的稳定性，并利用Galvis和Sarkis[近似无均匀椭圆的无限维随机达西方程，SIAM J. number]的结果提供了一个先验误差估计。论椭圆型SPDEs [j] .学报，47(5)(2009)，pp. 3624-3651。
{"title":"A priori error estimates for finite element approximations of parabolic stochastic partial differential equations with generalized random variables","authors":"Christophe Audouze, P. Nair","doi":"10.1080/17442508.2014.989526","DOIUrl":"https://doi.org/10.1080/17442508.2014.989526","url":null,"abstract":"We consider finite element approximations of parabolic stochastic partial differential equations (SPDEs) in conjunction with the -weighted temporal discretization scheme. We study the stability of the numerical scheme and provide a priori error estimates, using a result of Galvis and Sarkis [Approximating infinity-dimensional stochastic Darcy's equations without uniform ellipticity, SIAM J. Numer. Anal. 47(5) (2009), pp. 3624–3651] on elliptic SPDEs.","PeriodicalId":49269,"journal":{"name":"Stochastics-An International Journal of Probability and Stochastic Processes","volume":"241 1","pages":"537 - 561"},"PeriodicalIF":0.9,"publicationDate":"2015-02-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74963051","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2015-02-17DOI: 10.1080/17442508.2015.1012080
K. Bahlali, Antoine Hakassou, Y. Ouknine
The purpose of this paper is to study some properties of solutions to one-dimensional as well as multidimensional stochastic differential equations (SDEs in short) with super-linear growth and non-Lipschitz conditions on the coefficients. Taking inspiration from [K. Bahlali, E.H. Essaky, M. Hassani, and E. Pardoux Existence, uniqueness and stability of backward stochastic differential equation with locally monotone coefficient, C.R.A.S. Paris. 335(9) (2002), pp. 757–762; K. Bahlali, E. H. Essaky, and H. Hassani, Multidimensional BSDEs with super-linear growth coefficients: Application to degenerate systems of semilinear PDEs, C. R. Acad. Sci. Paris, Ser. I. 348 (2010), pp. 677-682; K. Bahlali, E. H. Essaky, and H. Hassani, p-Integrable solutions to multidimensional BSDEs and degenerate systems of PDEs with logarithmic nonlinearities, (2010). Available at arXiv:1007.2388v1 [math.PR]], we introduce a new local condition which ensures the pathwise uniqueness, as well as the non-contact property. We moreover show that the solution produces a stochastic flow of continuous maps and satisfies a large deviations principle of Freidlin–Wentzell type. Our conditions on the coefficients go beyond the existing ones in the literature. For instance, the coefficients are not assumed uniformly continuous and therefore cannot satisfy the classical Osgood condition. The drift coefficient could not be locally monotone and the diffusion is neither locally Lipschitz nor uniformly elliptic. Our conditions on the coefficients are, in some sense, near the best possible. Our results are sharp and mainly based on Gronwall lemma and the localization of the time parameter in concatenated intervals.
本文的目的是研究具有超线性增长和系数非lipschitz条件的一维和多维随机微分方程(简称SDEs)解的一些性质。从[K.]李建军，李建军，李建军，等。带局部单调系数的倒向随机微分方程的存在性、唯一性和稳定性，数学学报，335(9)(2002)，pp. 757-762;李建军，李建军，李建军。基于超线性增长系数的多维偏微分方程及其在半线性偏微分方程退化系统中的应用。巴黎,爵士。I. 348 (2010)， pp. 677-682;K. Bahlali, E. H. Essaky和H. Hassani，具有对数非线性的多维BSDEs和退化系统的p-可积解，(2010)。可在arXiv:1007.2388v1[数学。PR]]，我们引入了一个新的局部条件，保证了路径唯一性和非接触性。此外，我们还证明了该解产生连续映射的随机流，并满足Freidlin-Wentzell型的大偏差原理。我们的系数条件超出了文献中已有的条件。例如，不假设系数一致连续，因此不能满足经典的奥斯良条件。漂移系数不可能是局部单调的，扩散既不是局部利普希茨的，也不是均匀椭圆的。我们的系数条件，在某种意义上，接近最好的可能。我们的结果是清晰的，主要基于Gronwall引理和时间参数在串联区间的局部化。
{"title":"A class of stochastic differential equations with super-linear growth and non-Lipschitz coefficients","authors":"K. Bahlali, Antoine Hakassou, Y. Ouknine","doi":"10.1080/17442508.2015.1012080","DOIUrl":"https://doi.org/10.1080/17442508.2015.1012080","url":null,"abstract":"The purpose of this paper is to study some properties of solutions to one-dimensional as well as multidimensional stochastic differential equations (SDEs in short) with super-linear growth and non-Lipschitz conditions on the coefficients. Taking inspiration from [K. Bahlali, E.H. Essaky, M. Hassani, and E. Pardoux Existence, uniqueness and stability of backward stochastic differential equation with locally monotone coefficient, C.R.A.S. Paris. 335(9) (2002), pp. 757–762; K. Bahlali, E. H. Essaky, and H. Hassani, Multidimensional BSDEs with super-linear growth coefficients: Application to degenerate systems of semilinear PDEs, C. R. Acad. Sci. Paris, Ser. I. 348 (2010), pp. 677-682; K. Bahlali, E. H. Essaky, and H. Hassani, p-Integrable solutions to multidimensional BSDEs and degenerate systems of PDEs with logarithmic nonlinearities, (2010). Available at arXiv:1007.2388v1 [math.PR]], we introduce a new local condition which ensures the pathwise uniqueness, as well as the non-contact property. We moreover show that the solution produces a stochastic flow of continuous maps and satisfies a large deviations principle of Freidlin–Wentzell type. Our conditions on the coefficients go beyond the existing ones in the literature. For instance, the coefficients are not assumed uniformly continuous and therefore cannot satisfy the classical Osgood condition. The drift coefficient could not be locally monotone and the diffusion is neither locally Lipschitz nor uniformly elliptic. Our conditions on the coefficients are, in some sense, near the best possible. Our results are sharp and mainly based on Gronwall lemma and the localization of the time parameter in concatenated intervals.","PeriodicalId":49269,"journal":{"name":"Stochastics-An International Journal of Probability and Stochastic Processes","volume":"66 1","pages":"806 - 847"},"PeriodicalIF":0.9,"publicationDate":"2015-02-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83838255","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2015-01-13DOI: 10.1080/17442508.2014.939977
A. Osȩkowski
We establish sharp weak type and logarithmic estimates for the diameter of the stopped Brownian motion. Then, using standard embedding theorems, we extend the results to the case of general real-valued continuous-path martingales. The proof rests on finding of the solutions to the corresponding three-dimensional optimal stopping problems.
{"title":"Estimates for the diameter of a martingale","authors":"A. Osȩkowski","doi":"10.1080/17442508.2014.939977","DOIUrl":"https://doi.org/10.1080/17442508.2014.939977","url":null,"abstract":"We establish sharp weak type and logarithmic estimates for the diameter of the stopped Brownian motion. Then, using standard embedding theorems, we extend the results to the case of general real-valued continuous-path martingales. The proof rests on finding of the solutions to the corresponding three-dimensional optimal stopping problems.","PeriodicalId":49269,"journal":{"name":"Stochastics-An International Journal of Probability and Stochastic Processes","volume":"26 1","pages":"235 - 256"},"PeriodicalIF":0.9,"publicationDate":"2015-01-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73438242","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2015-01-02DOI: 10.1080/17442508.2014.931959
A. Shen, H. Zhu, R. Wu, Y. Zhang
In this paper, we study the complete convergence for weighted sums of linearly negative quadrant dependent (LNQD) random variables based on the exponential bounds. In addition, we present some complete convergence for arrays of rowwise LNQD random variables.
{"title":"Complete convergence for weighted sums of LNQD random variables","authors":"A. Shen, H. Zhu, R. Wu, Y. Zhang","doi":"10.1080/17442508.2014.931959","DOIUrl":"https://doi.org/10.1080/17442508.2014.931959","url":null,"abstract":"In this paper, we study the complete convergence for weighted sums of linearly negative quadrant dependent (LNQD) random variables based on the exponential bounds. In addition, we present some complete convergence for arrays of rowwise LNQD random variables.","PeriodicalId":49269,"journal":{"name":"Stochastics-An International Journal of Probability and Stochastic Processes","volume":"56 1","pages":"160 - 169"},"PeriodicalIF":0.9,"publicationDate":"2015-01-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76263481","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2015-01-02DOI: 10.1080/17442508.2014.914514
Jingtao Shi, Zhen Wu
This paper is concerned with backward stochastic differential equations with Markov switching driven by Brownian motion and Poisson random measure. The motivation is a constrained stochastic Riccati equation derived from a stochastic linear quadratic optimal control problem with both Poisson and Markovian jumps. The existence and uniqueness of an adapted solution under global Lipschitz condition on the generator is obtained. The continuous dependence of the solution on parameters is proved. Two comparison theorems are also derived by a generalized Girsanov transformation theorem.
{"title":"Backward stochastic differential equations with Markov switching driven by Brownian motion and Poisson random measure","authors":"Jingtao Shi, Zhen Wu","doi":"10.1080/17442508.2014.914514","DOIUrl":"https://doi.org/10.1080/17442508.2014.914514","url":null,"abstract":"This paper is concerned with backward stochastic differential equations with Markov switching driven by Brownian motion and Poisson random measure. The motivation is a constrained stochastic Riccati equation derived from a stochastic linear quadratic optimal control problem with both Poisson and Markovian jumps. The existence and uniqueness of an adapted solution under global Lipschitz condition on the generator is obtained. The continuous dependence of the solution on parameters is proved. Two comparison theorems are also derived by a generalized Girsanov transformation theorem.","PeriodicalId":49269,"journal":{"name":"Stochastics-An International Journal of Probability and Stochastic Processes","volume":"47 1","pages":"1 - 29"},"PeriodicalIF":0.9,"publicationDate":"2015-01-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89157940","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2015-01-02DOI: 10.1080/17442508.2014.932051
P. Di Tella, H. Engelbert
We investigate the predictable representation property (PRP) in the frame of Lévy processes. To give a general definition of the PRP, we make use of the theory of stable subspaces. Let L be a Lévy process with Lévy measure . The main result is that any total system in leads to a family of martingales with the PRP.
{"title":"On the predictable representation property of martingales associated with Lévy processes","authors":"P. Di Tella, H. Engelbert","doi":"10.1080/17442508.2014.932051","DOIUrl":"https://doi.org/10.1080/17442508.2014.932051","url":null,"abstract":"We investigate the predictable representation property (PRP) in the frame of Lévy processes. To give a general definition of the PRP, we make use of the theory of stable subspaces. Let L be a Lévy process with Lévy measure . The main result is that any total system in leads to a family of martingales with the PRP.","PeriodicalId":49269,"journal":{"name":"Stochastics-An International Journal of Probability and Stochastic Processes","volume":"33 1","pages":"170 - 184"},"PeriodicalIF":0.9,"publicationDate":"2015-01-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73495711","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2015-01-02DOI: 10.1080/17442508.2014.914516
J. Bao, C. Yuan
In this paper, by the weak convergence method, based on a variational representation for positive functionals of a Poisson random measure and Brownian motion, we establish uniform large deviation principles (LDPs) for a class of neutral stochastic differential equations driven by jump processes. As a byproduct, we also obtain uniform LDPs for neutral stochastic differential delay equations which, in particular, allow the coefficients to be highly nonlinear with respect to the delay argument.
{"title":"Large deviations for neutral functional SDEs with jumps","authors":"J. Bao, C. Yuan","doi":"10.1080/17442508.2014.914516","DOIUrl":"https://doi.org/10.1080/17442508.2014.914516","url":null,"abstract":"In this paper, by the weak convergence method, based on a variational representation for positive functionals of a Poisson random measure and Brownian motion, we establish uniform large deviation principles (LDPs) for a class of neutral stochastic differential equations driven by jump processes. As a byproduct, we also obtain uniform LDPs for neutral stochastic differential delay equations which, in particular, allow the coefficients to be highly nonlinear with respect to the delay argument.","PeriodicalId":49269,"journal":{"name":"Stochastics-An International Journal of Probability and Stochastic Processes","volume":"25 1","pages":"48 - 70"},"PeriodicalIF":0.9,"publicationDate":"2015-01-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74105404","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2015-01-02DOI: 10.1080/17442508.2014.915972
T. T. da Silva, M. Fragoso
The subject matter of this paper is the so-called jump-type Fleming–Viot process. The main result shows that the density of the process has a representation as the solution of a stochastic partial differential equation. When reduced to the Fleming–Viot process, our result recovers the result of N. Konno and T. Shiga [Stochastic partial differential equations for some measure-valued diffusions, Probab. Theory Relat. Fields 79 (1988), pp. 201–225].
{"title":"On the differential equation satisfied by the random measure density of a jump-type Fleming–Viot process","authors":"T. T. da Silva, M. Fragoso","doi":"10.1080/17442508.2014.915972","DOIUrl":"https://doi.org/10.1080/17442508.2014.915972","url":null,"abstract":"The subject matter of this paper is the so-called jump-type Fleming–Viot process. The main result shows that the density of the process has a representation as the solution of a stochastic partial differential equation. When reduced to the Fleming–Viot process, our result recovers the result of N. Konno and T. Shiga [Stochastic partial differential equations for some measure-valued diffusions, Probab. Theory Relat. Fields 79 (1988), pp. 201–225].","PeriodicalId":49269,"journal":{"name":"Stochastics-An International Journal of Probability and Stochastic Processes","volume":"68 1","pages":"71 - 84"},"PeriodicalIF":0.9,"publicationDate":"2015-01-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90576016","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2015-01-02DOI: 10.1080/17442508.2014.924938
N. Dung
In this article, we consider a class of stochastic Volterra integro-differential equations with infinite delay and impulsive effects, driven by fractional Brownian motion with the Hurst index in a Hilbert space. The cases of Lipschitz and bounded impulses are studied separately. The existence and uniqueness of mild solutions are proved by using different fixed-point theorems. An example is given to illustrate the theory.
{"title":"Stochastic Volterra integro-differential equations driven by fractional Brownian motion in a Hilbert space","authors":"N. Dung","doi":"10.1080/17442508.2014.924938","DOIUrl":"https://doi.org/10.1080/17442508.2014.924938","url":null,"abstract":"In this article, we consider a class of stochastic Volterra integro-differential equations with infinite delay and impulsive effects, driven by fractional Brownian motion with the Hurst index in a Hilbert space. The cases of Lipschitz and bounded impulses are studied separately. The existence and uniqueness of mild solutions are proved by using different fixed-point theorems. An example is given to illustrate the theory.","PeriodicalId":49269,"journal":{"name":"Stochastics-An International Journal of Probability and Stochastic Processes","volume":"8 1","pages":"142 - 159"},"PeriodicalIF":0.9,"publicationDate":"2015-01-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78525239","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2015-01-02DOI: 10.1080/17442508.2014.914515
R. Farnoosh, Mahboubeh Aalaei, M. Ebrahimi
The present study establishes an accurate and efficient algorithm based on Monte Carlo (MC) simulation for solving high dimensional linear systems of algebraic equations (LSAEs) and two-dimensional Fredholm integral equations of the second kind (FIESK). This new combined numerical-probabilistic algorithm is based on Jacobi over-relaxation method and MC simulation in conjunction with the iterative refinement technique to find the unique solution of the large sparse LSAEs. It has an excellent accuracy, low cost and simple structure. Theoretical results are established to justify the convergence of the algorithm. To confirm the accuracy and efficiency of the present work, the proposed algorithm is used for solving and LSAEs. Furthermore, the algorithm is coupled with Galerkin's method to illustrate the power and effectiveness of the proposed algorithm for solving two-dimensional FIESK.
{"title":"Combined probabilistic algorithm for solving high dimensional problems","authors":"R. Farnoosh, Mahboubeh Aalaei, M. Ebrahimi","doi":"10.1080/17442508.2014.914515","DOIUrl":"https://doi.org/10.1080/17442508.2014.914515","url":null,"abstract":"The present study establishes an accurate and efficient algorithm based on Monte Carlo (MC) simulation for solving high dimensional linear systems of algebraic equations (LSAEs) and two-dimensional Fredholm integral equations of the second kind (FIESK). This new combined numerical-probabilistic algorithm is based on Jacobi over-relaxation method and MC simulation in conjunction with the iterative refinement technique to find the unique solution of the large sparse LSAEs. It has an excellent accuracy, low cost and simple structure. Theoretical results are established to justify the convergence of the algorithm. To confirm the accuracy and efficiency of the present work, the proposed algorithm is used for solving and LSAEs. Furthermore, the algorithm is coupled with Galerkin's method to illustrate the power and effectiveness of the proposed algorithm for solving two-dimensional FIESK.","PeriodicalId":49269,"journal":{"name":"Stochastics-An International Journal of Probability and Stochastic Processes","volume":"28 1","pages":"30 - 47"},"PeriodicalIF":0.9,"publicationDate":"2015-01-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74850404","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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