Pub Date : 2015-04-10DOI: 10.1080/17442508.2014.989528
A. Joulin, Solym Mawaki Manou-Abi
We establish a convex ordering between stochastic integrals driven by strictly α-stable processes with index α ∈ (1,2). Our approach is based on the forward–backward stochastic calculus for martingales together with a suitable decomposition of stable stochastic integrals.
{"title":"A note on convex ordering for stable stochastic integrals","authors":"A. Joulin, Solym Mawaki Manou-Abi","doi":"10.1080/17442508.2014.989528","DOIUrl":"https://doi.org/10.1080/17442508.2014.989528","url":null,"abstract":"We establish a convex ordering between stochastic integrals driven by strictly α-stable processes with index α ∈ (1,2). Our approach is based on the forward–backward stochastic calculus for martingales together with a suitable decomposition of stable stochastic integrals.","PeriodicalId":49269,"journal":{"name":"Stochastics-An International Journal of Probability and Stochastic Processes","volume":"85 1","pages":"592 - 603"},"PeriodicalIF":0.9,"publicationDate":"2015-04-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84043971","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 : 2015-04-09DOI: 10.1080/17442508.2014.989527
K. Kubilius
Orey suggested the definition of an index for a Gaussian process with stationary increments which determines various properties of the sample paths of this process. We provide an extension of the definition of the Orey index towards a second-order stochastic process which may not have stationary increments and estimate the Orey index towards a Gaussian process from discrete observations of its sample paths.
{"title":"On estimation of the extended Orey index for Gaussian processes","authors":"K. Kubilius","doi":"10.1080/17442508.2014.989527","DOIUrl":"https://doi.org/10.1080/17442508.2014.989527","url":null,"abstract":"Orey suggested the definition of an index for a Gaussian process with stationary increments which determines various properties of the sample paths of this process. We provide an extension of the definition of the Orey index towards a second-order stochastic process which may not have stationary increments and estimate the Orey index towards a Gaussian process from discrete observations of its sample paths.","PeriodicalId":49269,"journal":{"name":"Stochastics-An International Journal of Probability and Stochastic Processes","volume":"42 1","pages":"562 - 591"},"PeriodicalIF":0.9,"publicationDate":"2015-04-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85770710","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 : 2015-03-04DOI: 10.1080/17442508.2014.942307
C. Macci, B. Pacchiarotti
We consider independent geometric distributed random variables which satisfy suitable hypotheses. We study large and moderate deviations for their empirical means, and we illustrate applications of the large deviation results for the weak record values of i.i.d. discrete random variables.
{"title":"Asymptotic results for empirical means of independent geometric distributed random variables","authors":"C. Macci, B. Pacchiarotti","doi":"10.1080/17442508.2014.942307","DOIUrl":"https://doi.org/10.1080/17442508.2014.942307","url":null,"abstract":"We consider independent geometric distributed random variables which satisfy suitable hypotheses. We study large and moderate deviations for their empirical means, and we illustrate applications of the large deviation results for the weak record values of i.i.d. discrete random variables.","PeriodicalId":49269,"journal":{"name":"Stochastics-An International Journal of Probability and Stochastic Processes","volume":"55 1","pages":"308 - 325"},"PeriodicalIF":0.9,"publicationDate":"2015-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89825969","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 : 2015-03-04DOI: 10.1080/17442508.2014.944913
D. Yuan, Shunjing Li, B. Tao
From the ordinary notion of -mixing for a sequence of random variables, a new concept called conditionally -mixing is proposed. That conditionally -mixing neither implies nor is implied by -mixing is illustrated by examples. Certain criteria for checking conditionally -mixing property as well as basic properties are derived, and several conditional covariance inequalities are obtained. By means of these properties and inequalities, a conditional central limit theorem stated in terms of conditional characteristic functions is established.
{"title":"Some preliminary results on conditionally ψ-mixing sequences of random variables","authors":"D. Yuan, Shunjing Li, B. Tao","doi":"10.1080/17442508.2014.944913","DOIUrl":"https://doi.org/10.1080/17442508.2014.944913","url":null,"abstract":"From the ordinary notion of -mixing for a sequence of random variables, a new concept called conditionally -mixing is proposed. That conditionally -mixing neither implies nor is implied by -mixing is illustrated by examples. Certain criteria for checking conditionally -mixing property as well as basic properties are derived, and several conditional covariance inequalities are obtained. By means of these properties and inequalities, a conditional central limit theorem stated in terms of conditional characteristic functions is established.","PeriodicalId":49269,"journal":{"name":"Stochastics-An International Journal of Probability and Stochastic Processes","volume":"43 1","pages":"326 - 346"},"PeriodicalIF":0.9,"publicationDate":"2015-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90214012","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 : 2015-03-04DOI: 10.1080/17442508.2014.939976
Yongchao Zhang
From the viewpoint of stochastic programming, we rigorously analyse entry and exit decisions of a project which were proposed by Dixit [A. Dixit, Entry and exit decisions under uncertainty, J. Polit. Econ. 97 (1989), pp. 620–638]. In this article, instead of assuming that the costs are constant in classical research, we assume that they are linear with respect to the price of the commodity produced by the project. Under this assumption, we obtain a condition which guarantees that investing in the project is worthless; besides, the project may be terminated when the commodity price is greater than a certain value. In contrast, there are no such results provided that the costs are constant. Moreover, we provide an explicit solution of entry and exit decisions if the project is worthy to be invested in.
{"title":"Entry and exit decisions with linear costs under uncertainty","authors":"Yongchao Zhang","doi":"10.1080/17442508.2014.939976","DOIUrl":"https://doi.org/10.1080/17442508.2014.939976","url":null,"abstract":"From the viewpoint of stochastic programming, we rigorously analyse entry and exit decisions of a project which were proposed by Dixit [A. Dixit, Entry and exit decisions under uncertainty, J. Polit. Econ. 97 (1989), pp. 620–638]. In this article, instead of assuming that the costs are constant in classical research, we assume that they are linear with respect to the price of the commodity produced by the project. Under this assumption, we obtain a condition which guarantees that investing in the project is worthless; besides, the project may be terminated when the commodity price is greater than a certain value. In contrast, there are no such results provided that the costs are constant. Moreover, we provide an explicit solution of entry and exit decisions if the project is worthy to be invested in.","PeriodicalId":49269,"journal":{"name":"Stochastics-An International Journal of Probability and Stochastic Processes","volume":"12 1","pages":"209 - 234"},"PeriodicalIF":0.9,"publicationDate":"2015-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78952609","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 : 2015-03-04DOI: 10.1080/17442508.2014.938075
Y. Miao, Guangyu Yang, G. Stoica
The aim of this note is to establish the Baum–Katz type rate of convergence in the Marcinkiewicz–Zygmund strong law of large numbers for martingales, which improves the recent works of Stoica [Series of moderate deviation probabilities for martingales, J. Math. Anal. Appl. 336 (2005), pp. 759–763; Baum–Katz–Nagaev type results for martingales, J. Math. Anal. Appl. 336 (2007), pp. 1489–1492; A note on the rate of convergence in the strong law of large numbers for martingales, J. Math. Anal. Appl. 381 (2011), pp. 910–913]. Furthermore, we also study some relevant limit behaviours for the uniform mixing process. Under some uniform mixing conditions, the sufficient and necessary condition of the convergence of the martingale series is established.
{"title":"On the rate of convergence in the strong law of large numbers for martingales","authors":"Y. Miao, Guangyu Yang, G. Stoica","doi":"10.1080/17442508.2014.938075","DOIUrl":"https://doi.org/10.1080/17442508.2014.938075","url":null,"abstract":"The aim of this note is to establish the Baum–Katz type rate of convergence in the Marcinkiewicz–Zygmund strong law of large numbers for martingales, which improves the recent works of Stoica [Series of moderate deviation probabilities for martingales, J. Math. Anal. Appl. 336 (2005), pp. 759–763; Baum–Katz–Nagaev type results for martingales, J. Math. Anal. Appl. 336 (2007), pp. 1489–1492; A note on the rate of convergence in the strong law of large numbers for martingales, J. Math. Anal. Appl. 381 (2011), pp. 910–913]. Furthermore, we also study some relevant limit behaviours for the uniform mixing process. Under some uniform mixing conditions, the sufficient and necessary condition of the convergence of the martingale series is established.","PeriodicalId":49269,"journal":{"name":"Stochastics-An International Journal of Probability and Stochastic Processes","volume":"60 1","pages":"185 - 198"},"PeriodicalIF":0.9,"publicationDate":"2015-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77194501","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 : 2015-03-04DOI: 10.1080/17442508.2014.939979
F. Dufour, T. Prieto-Rumeau
We consider a discrete-time Markov decision process with Borel state and action spaces, and possibly unbounded cost function. We assume that the Markov transition kernel is absolutely continuous with respect to some probability measure . By replacing this probability measure with its empirical distribution for a sample of size n, we obtain a finite state space control problem, which is used to provide an approximation of the optimal value and an optimal policy of the original control model. We impose Lipschitz continuity properties on the control model and its associated density functions. We measure the accuracy of the approximation of the optimal value and an optimal policy by means of a non-asymptotic concentration inequality based on the 1-Wasserstein distance between and . Obtaining numerically the solution of the approximating control model is discussed and an application to an inventory management problem is presented.
{"title":"Approximation of average cost Markov decision processes using empirical distributions and concentration inequalities","authors":"F. Dufour, T. Prieto-Rumeau","doi":"10.1080/17442508.2014.939979","DOIUrl":"https://doi.org/10.1080/17442508.2014.939979","url":null,"abstract":"We consider a discrete-time Markov decision process with Borel state and action spaces, and possibly unbounded cost function. We assume that the Markov transition kernel is absolutely continuous with respect to some probability measure . By replacing this probability measure with its empirical distribution for a sample of size n, we obtain a finite state space control problem, which is used to provide an approximation of the optimal value and an optimal policy of the original control model. We impose Lipschitz continuity properties on the control model and its associated density functions. We measure the accuracy of the approximation of the optimal value and an optimal policy by means of a non-asymptotic concentration inequality based on the 1-Wasserstein distance between and . Obtaining numerically the solution of the approximating control model is discussed and an application to an inventory management problem is presented.","PeriodicalId":49269,"journal":{"name":"Stochastics-An International Journal of Probability and Stochastic Processes","volume":"58 1","pages":"273 - 307"},"PeriodicalIF":0.9,"publicationDate":"2015-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83987273","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 : 2015-03-04DOI: 10.1080/17442508.2014.939978
M. Guo, Dongjin Zhu, Yong Ren
In this paper, the complete qth moment convergence of weighted sums for arrays of rowwise negatively associated (NA) random variables is investigated. By using moment inequality and truncation methods, some general results on complete qth moment convergence of weighted sums for arrays of rowwise NA random variables are obtained. As their applications, we not only generalize and extend the corresponding results of Baek et al. [On the complete convergence of weighted sums for arrays of negatively associated variables, J. Korean Stat. Soc. 37 (2008), pp. 73–80], Liang [Complete convergence for weighted sums of negatively associated random variables, Stat. Probab. Lett. 48 (2000), pp. 317–325 and Liang et al. [Complete moment convergence for sums of negatively associated random variables, Acta Math. Sin. English Ser. 26 (2010), pp. 419–432], but also greatly simplify their proofs.
本文研究了行负相关随机变量阵列加权和的完全第q阶收敛性。利用矩不等式和截断方法，得到了行NA随机变量数组加权和的完全第q阶矩收敛性的一些一般结果。作为它们的应用，我们不仅推广和扩展了Baek等人的相应结果[关于负相关变量数组的加权和的完全收敛，J. Korean Stat. Soc. 37 (2008)， pp. 73-80]， Liang[负相关随机变量的加权和的完全收敛，Stat. Probab.]。左48 (2000)，pp. 317-325和Liang等。[负相关随机变量和的完全矩收敛，数学学报。]罪。英文卷26(2010)，页419-432]，但也大大简化了他们的证明。
{"title":"Complete qth moment convergence of weighted sums for arrays of rowwise negatively associated random variables","authors":"M. Guo, Dongjin Zhu, Yong Ren","doi":"10.1080/17442508.2014.939978","DOIUrl":"https://doi.org/10.1080/17442508.2014.939978","url":null,"abstract":"In this paper, the complete qth moment convergence of weighted sums for arrays of rowwise negatively associated (NA) random variables is investigated. By using moment inequality and truncation methods, some general results on complete qth moment convergence of weighted sums for arrays of rowwise NA random variables are obtained. As their applications, we not only generalize and extend the corresponding results of Baek et al. [On the complete convergence of weighted sums for arrays of negatively associated variables, J. Korean Stat. Soc. 37 (2008), pp. 73–80], Liang [Complete convergence for weighted sums of negatively associated random variables, Stat. Probab. Lett. 48 (2000), pp. 317–325 and Liang et al. [Complete moment convergence for sums of negatively associated random variables, Acta Math. Sin. English Ser. 26 (2010), pp. 419–432], but also greatly simplify their proofs.","PeriodicalId":49269,"journal":{"name":"Stochastics-An International Journal of Probability and Stochastic Processes","volume":"9 1","pages":"257 - 272"},"PeriodicalIF":0.9,"publicationDate":"2015-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73007282","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 : 2015-03-04DOI: 10.1080/17442508.2014.939975
Wenzhi Yang, Shuhe Hu
It is known that the dependence structure of pairwise negative quadrant dependent (NQD) random variables is weaker than those of negatively associated random variables and negatively orthant dependent random variables. In this article, we investigate the moving average process which is based on the pairwise NQD random variables. The complete moment convergence and the integrability of the supremum are presented for this moving average process. The results imply complete convergence and the Marcinkiewicz–Zygmund-type strong law of large numbers for pairwise NQD sequences.
{"title":"Complete moment convergence of pairwise NQD random variables","authors":"Wenzhi Yang, Shuhe Hu","doi":"10.1080/17442508.2014.939975","DOIUrl":"https://doi.org/10.1080/17442508.2014.939975","url":null,"abstract":"It is known that the dependence structure of pairwise negative quadrant dependent (NQD) random variables is weaker than those of negatively associated random variables and negatively orthant dependent random variables. In this article, we investigate the moving average process which is based on the pairwise NQD random variables. The complete moment convergence and the integrability of the supremum are presented for this moving average process. The results imply complete convergence and the Marcinkiewicz–Zygmund-type strong law of large numbers for pairwise NQD sequences.","PeriodicalId":49269,"journal":{"name":"Stochastics-An International Journal of Probability and Stochastic Processes","volume":"50 1","pages":"199 - 208"},"PeriodicalIF":0.9,"publicationDate":"2015-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88375822","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 : 2015-02-20DOI: 10.1080/17442508.2014.991325
Christoph Belak, Olaf Menkens, Jörn Sass
We study optimal asset allocation in a crash-threatened financial market with proportional transaction costs. The market is assumed to be either in a normal state, in which the risky asset follows a geometric Brownian motion, or in a crash state, in which the price of the risky asset can suddenly drop by a certain relative amount. We only assume the maximum number and the maximum relative size of the crashes to be given and do not make any assumptions about their distributions. For every investment strategy, we identify the worst-case scenario in the sense that the expected utility of terminal wealth is minimized. The objective is then to determine the investment strategy which yields the highest expected utility in its worst-case scenario. We solve the problem for utility functions with constant relative risk aversion using a stochastic control approach. We characterize the value function as the unique viscosity solution of a second-order nonlinear partial differential equation. The optimal strategies are characterized by time-dependent free boundaries which we compute numerically. The numerical examples suggest that it is not optimal to invest any wealth in the risky asset close to the investment horizon, while a long position in the risky asset is optimal if the remaining investment period is sufficiently large.
{"title":"Worst-case portfolio optimization with proportional transaction costs","authors":"Christoph Belak, Olaf Menkens, Jörn Sass","doi":"10.1080/17442508.2014.991325","DOIUrl":"https://doi.org/10.1080/17442508.2014.991325","url":null,"abstract":"We study optimal asset allocation in a crash-threatened financial market with proportional transaction costs. The market is assumed to be either in a normal state, in which the risky asset follows a geometric Brownian motion, or in a crash state, in which the price of the risky asset can suddenly drop by a certain relative amount. We only assume the maximum number and the maximum relative size of the crashes to be given and do not make any assumptions about their distributions. For every investment strategy, we identify the worst-case scenario in the sense that the expected utility of terminal wealth is minimized. The objective is then to determine the investment strategy which yields the highest expected utility in its worst-case scenario. We solve the problem for utility functions with constant relative risk aversion using a stochastic control approach. We characterize the value function as the unique viscosity solution of a second-order nonlinear partial differential equation. The optimal strategies are characterized by time-dependent free boundaries which we compute numerically. The numerical examples suggest that it is not optimal to invest any wealth in the risky asset close to the investment horizon, while a long position in the risky asset is optimal if the remaining investment period is sufficiently large.","PeriodicalId":49269,"journal":{"name":"Stochastics-An International Journal of Probability and Stochastic Processes","volume":"47 1","pages":"623 - 663"},"PeriodicalIF":0.9,"publicationDate":"2015-02-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89086770","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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