Pub Date : 2014-05-04DOI: 10.1080/17442508.2013.797423
Huai Xu, L. Tang
Let be an array of rowwise asymptotically almost negatively associated (NA) random variables which is stochastically dominated by a random variable X. Wang et al. [11] studied the complete convergence for arrays of rowwise asymptotically almost NA random variables under the condition that X has an index-order moment, which seems too strong. We will further study the complete convergence for arrays of rowwise asymptotically almost NA random variables under the condition that X has a power-order moment, which is weaker than index-order moment. Our results improve the corresponding ones of Wang et al. [11].
{"title":"On complete convergence for arrays of rowwise AANA random variables","authors":"Huai Xu, L. Tang","doi":"10.1080/17442508.2013.797423","DOIUrl":"https://doi.org/10.1080/17442508.2013.797423","url":null,"abstract":"Let be an array of rowwise asymptotically almost negatively associated (NA) random variables which is stochastically dominated by a random variable X. Wang et al. [11] studied the complete convergence for arrays of rowwise asymptotically almost NA random variables under the condition that X has an index-order moment, which seems too strong. We will further study the complete convergence for arrays of rowwise asymptotically almost NA random variables under the condition that X has a power-order moment, which is weaker than index-order moment. Our results improve the corresponding ones of Wang et al. [11].","PeriodicalId":49269,"journal":{"name":"Stochastics-An International Journal of Probability and Stochastic Processes","volume":"9 1","pages":"371 - 381"},"PeriodicalIF":0.9,"publicationDate":"2014-05-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73505755","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 : 2014-05-04DOI: 10.1080/17442508.2013.797424
Litan Yan, Bo Gao, Junfeng Liu
Let B be a bi-fractional Brownian motion with indices , and let be its local time process. We construct a Banach space of measurable functions such that the quadratic covariation and the integral exist provided . Moreover, the Bouleau–Yor identityholds for all .
{"title":"The Bouleau–Yor identity for a bi-fractional Brownian motion","authors":"Litan Yan, Bo Gao, Junfeng Liu","doi":"10.1080/17442508.2013.797424","DOIUrl":"https://doi.org/10.1080/17442508.2013.797424","url":null,"abstract":"Let B be a bi-fractional Brownian motion with indices , and let be its local time process. We construct a Banach space of measurable functions such that the quadratic covariation and the integral exist provided . Moreover, the Bouleau–Yor identityholds for all .","PeriodicalId":49269,"journal":{"name":"Stochastics-An International Journal of Probability and Stochastic Processes","volume":"15 1","pages":"382 - 414"},"PeriodicalIF":0.9,"publicationDate":"2014-05-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90617075","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 : 2014-05-04DOI: 10.1080/17442508.2013.837908
Y. Kitapbayev
The lookback option with fixed strike in the case of finite horizon was examined with help of the solution to the optimal stopping problem for a three-dimensional Markov process in [P. Gapeev, Discounted optimal stopping for maxima in diffusion models with finite horizon, Electron. J. Probab. 11 (2006), pp. 1031–1048]. The purpose of this paper was to illustrate another derivation of the solution in [P. Gapeev, Discounted optimal stopping for maxima in diffusion models with finite horizon, Electron. J. Probab. 11 (2006), pp. 1031–1048]. The key idea is to use the Girsanov change-of-measure theorem which allows to reduce the three-dimensional optimal stopping problem to a two-dimensional optimal stopping problem with a scaling strike. This approach simplifies the discussion and expressions for the arbitrage-free price and the rational exercise boundary. We derive a closed-form expression for the value function of the two-dimensional problem in terms of the optimal stopping boundary and show that the optimal stopping boundary itself can be characterized as the unique solution to a nonlinear integral equation. Using these results we obtain the arbitrage-free price and the rational exercise boundary of the option.
{"title":"On the lookback option with fixed strike","authors":"Y. Kitapbayev","doi":"10.1080/17442508.2013.837908","DOIUrl":"https://doi.org/10.1080/17442508.2013.837908","url":null,"abstract":"The lookback option with fixed strike in the case of finite horizon was examined with help of the solution to the optimal stopping problem for a three-dimensional Markov process in [P. Gapeev, Discounted optimal stopping for maxima in diffusion models with finite horizon, Electron. J. Probab. 11 (2006), pp. 1031–1048]. The purpose of this paper was to illustrate another derivation of the solution in [P. Gapeev, Discounted optimal stopping for maxima in diffusion models with finite horizon, Electron. J. Probab. 11 (2006), pp. 1031–1048]. The key idea is to use the Girsanov change-of-measure theorem which allows to reduce the three-dimensional optimal stopping problem to a two-dimensional optimal stopping problem with a scaling strike. This approach simplifies the discussion and expressions for the arbitrage-free price and the rational exercise boundary. We derive a closed-form expression for the value function of the two-dimensional problem in terms of the optimal stopping boundary and show that the optimal stopping boundary itself can be characterized as the unique solution to a nonlinear integral equation. Using these results we obtain the arbitrage-free price and the rational exercise boundary of the option.","PeriodicalId":49269,"journal":{"name":"Stochastics-An International Journal of Probability and Stochastic Processes","volume":"177 1","pages":"510 - 526"},"PeriodicalIF":0.9,"publicationDate":"2014-05-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73350492","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 : 2014-05-04DOI: 10.1080/17442508.2013.841695
Qi He, G. Yin
Motivated by problems arising in time-dependent queues and dynamic systems with random environment, this work develops moderate deviations principles for dynamic systems driven by a fast-varying non-homogeneous Markov chain in continuous time. A distinct feature is that the Markov chain is time dependent or inhomogeneous, so are the dynamic systems. Under irreducibility of the non-homogeneous Markov chain, moderate deviations of a non-homogeneous functional are established first. With the help of a martingale problem formulation and a functional central limit theorem for the two timescale system, both upper and lower bounds of moderate deviations are obtained for the rapidly fluctuating Markovian systems. Then applications to queueing systems and dynamic systems modulated by a fast-varying Markov chain are examined.
{"title":"Moderate deviations for time-varying dynamic systems driven by non-homogeneous Markov chains with Two-time Scales","authors":"Qi He, G. Yin","doi":"10.1080/17442508.2013.841695","DOIUrl":"https://doi.org/10.1080/17442508.2013.841695","url":null,"abstract":"Motivated by problems arising in time-dependent queues and dynamic systems with random environment, this work develops moderate deviations principles for dynamic systems driven by a fast-varying non-homogeneous Markov chain in continuous time. A distinct feature is that the Markov chain is time dependent or inhomogeneous, so are the dynamic systems. Under irreducibility of the non-homogeneous Markov chain, moderate deviations of a non-homogeneous functional are established first. With the help of a martingale problem formulation and a functional central limit theorem for the two timescale system, both upper and lower bounds of moderate deviations are obtained for the rapidly fluctuating Markovian systems. Then applications to queueing systems and dynamic systems modulated by a fast-varying Markov chain are examined.","PeriodicalId":49269,"journal":{"name":"Stochastics-An International Journal of Probability and Stochastic Processes","volume":"20 1","pages":"527 - 550"},"PeriodicalIF":0.9,"publicationDate":"2014-05-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86691172","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 : 2014-05-04DOI: 10.1080/17442508.2013.801971
M. Guo
Negatively associated (NA) random variables are a more general class of random variables which include a set of independent random variables and have been applied to many practical fields. In this paper, the complete moment convergence of weighted sums for arrays of row-wise NA random variables is investigated. Some sufficient conditions for complete moment convergence of weighted sums for arrays of row-wise NA random variables are established. Moreover, under the weaker conditions, we extend the results of Baek et al. [J. Korean Stat. Soc. 37 (2008), pp. 73–80] and Sung [Abstr. Appl. Anal. 2011 (2011)]. As an application, the complete moment convergence of moving average processes based on an NA random sequence is obtained, which improves the result of Li and Zhang [Stat. Probab. Lett. 70 (2004), pp. 191–197 ].
{"title":"On complete moment convergence of weighted sums for arrays of row-wise negatively associated random variables","authors":"M. Guo","doi":"10.1080/17442508.2013.801971","DOIUrl":"https://doi.org/10.1080/17442508.2013.801971","url":null,"abstract":"Negatively associated (NA) random variables are a more general class of random variables which include a set of independent random variables and have been applied to many practical fields. In this paper, the complete moment convergence of weighted sums for arrays of row-wise NA random variables is investigated. Some sufficient conditions for complete moment convergence of weighted sums for arrays of row-wise NA random variables are established. Moreover, under the weaker conditions, we extend the results of Baek et al. [J. Korean Stat. Soc. 37 (2008), pp. 73–80] and Sung [Abstr. Appl. Anal. 2011 (2011)]. As an application, the complete moment convergence of moving average processes based on an NA random sequence is obtained, which improves the result of Li and Zhang [Stat. Probab. Lett. 70 (2004), pp. 191–197 ].","PeriodicalId":49269,"journal":{"name":"Stochastics-An International Journal of Probability and Stochastic Processes","volume":"21 1","pages":"415 - 428"},"PeriodicalIF":0.9,"publicationDate":"2014-05-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73834505","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 : 2014-04-07DOI: 10.1080/17442508.2014.956458
L. Abbas-Turki, I. Karatzas, Qinghua Li
This paper solves a Bayes sequential impulse control problem for a diffusion, whose drift has an unobservable parameter with a change point. The partially observed problem is reformulated into one with full observations, via a change of probability measure which removes the drift. The optimal impulse controls can be expressed in terms of the solutions and the current values of a Markov process adapted to the observation filtration. We shall illustrate the application of our results using the Longstaff–Schwartz algorithm for multiple optimal stopping times in a geometric Brownian motion stock price model with drift uncertainty.
{"title":"Impulse control of a diffusion with a change point","authors":"L. Abbas-Turki, I. Karatzas, Qinghua Li","doi":"10.1080/17442508.2014.956458","DOIUrl":"https://doi.org/10.1080/17442508.2014.956458","url":null,"abstract":"This paper solves a Bayes sequential impulse control problem for a diffusion, whose drift has an unobservable parameter with a change point. The partially observed problem is reformulated into one with full observations, via a change of probability measure which removes the drift. The optimal impulse controls can be expressed in terms of the solutions and the current values of a Markov process adapted to the observation filtration. We shall illustrate the application of our results using the Longstaff–Schwartz algorithm for multiple optimal stopping times in a geometric Brownian motion stock price model with drift uncertainty.","PeriodicalId":49269,"journal":{"name":"Stochastics-An International Journal of Probability and Stochastic Processes","volume":"48 1","pages":"382 - 408"},"PeriodicalIF":0.9,"publicationDate":"2014-04-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85362636","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 : 2014-03-04DOI: 10.1080/17442508.2013.775287
J. Tugaut
The aim of this work is to establish the results for a particular class of inhomogeneous processes, the McKean–Vlasov diffusions. Such diffusions correspond to the hydrodynamical limit of an interacting particle system. In convex landscapes, existence and uniqueness of the invariant probability is a well-known result. However, previous results state the nonuniqueness of the invariant probabilities under nonconvexity assumptions. Here, we prove that there exists a phase transition. Below a critical value, there are exactly three invariant probabilities and above another critical value, there is exactly one. Under simple assumptions, these critical values coincide and it is characterized by a simple implicit equation. We also investigate other cases in which phase transitions occur. Finally, we provide numerical estimations of the critical values.
{"title":"Phase transitions of McKean–Vlasov processes in double-wells landscape","authors":"J. Tugaut","doi":"10.1080/17442508.2013.775287","DOIUrl":"https://doi.org/10.1080/17442508.2013.775287","url":null,"abstract":"The aim of this work is to establish the results for a particular class of inhomogeneous processes, the McKean–Vlasov diffusions. Such diffusions correspond to the hydrodynamical limit of an interacting particle system. In convex landscapes, existence and uniqueness of the invariant probability is a well-known result. However, previous results state the nonuniqueness of the invariant probabilities under nonconvexity assumptions. Here, we prove that there exists a phase transition. Below a critical value, there are exactly three invariant probabilities and above another critical value, there is exactly one. Under simple assumptions, these critical values coincide and it is characterized by a simple implicit equation. We also investigate other cases in which phase transitions occur. Finally, we provide numerical estimations of the critical values.","PeriodicalId":49269,"journal":{"name":"Stochastics-An International Journal of Probability and Stochastic Processes","volume":"2 1","pages":"257 - 284"},"PeriodicalIF":0.9,"publicationDate":"2014-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85130992","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 : 2014-03-04DOI: 10.1080/17442508.2013.775284
S. Bonaccorsi, G. Ziglio
We study a nonlinear partial differential equation of the calculus of variation in a bounded domain, perturbed by noise; we allow stochastic boundary conditions that depend on the time derivative of the solution on the boundary. This work provides the existence and uniqueness of the solution and it shows the existence of an ergodic invariant measure for the corresponding transition semigroup; furthermore, under suitable additional assumptions, uniqueness and strong asymptotic stability of the invariant measure are proved.
{"title":"A variational approach to stochastic nonlinear diffusion problems with dynamical boundary conditions","authors":"S. Bonaccorsi, G. Ziglio","doi":"10.1080/17442508.2013.775284","DOIUrl":"https://doi.org/10.1080/17442508.2013.775284","url":null,"abstract":"We study a nonlinear partial differential equation of the calculus of variation in a bounded domain, perturbed by noise; we allow stochastic boundary conditions that depend on the time derivative of the solution on the boundary. This work provides the existence and uniqueness of the solution and it shows the existence of an ergodic invariant measure for the corresponding transition semigroup; furthermore, under suitable additional assumptions, uniqueness and strong asymptotic stability of the invariant measure are proved.","PeriodicalId":49269,"journal":{"name":"Stochastics-An International Journal of Probability and Stochastic Processes","volume":"45 1","pages":"218 - 233"},"PeriodicalIF":0.9,"publicationDate":"2014-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86497081","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 : 2014-03-04DOI: 10.1080/17442508.2013.775286
Jing Wu
The subject of the paper is to find existence conditions of weak solutions to multivalued stochastic differential equations with discontinuous coefficients. First we prove that a non-exploding solution exists when the drift coefficient b satisfies linear growth and the diffusion coefficient σ is uniformly elliptic. On this basis, we continue to obtain a solution (up to the explosion time) in the weak sense under certain local integrability, improving the result of Rozkosz and Słomiński.
{"title":"On existence of solutions of multivalued stochastic differential equations with discontinuous coefficients","authors":"Jing Wu","doi":"10.1080/17442508.2013.775286","DOIUrl":"https://doi.org/10.1080/17442508.2013.775286","url":null,"abstract":"The subject of the paper is to find existence conditions of weak solutions to multivalued stochastic differential equations with discontinuous coefficients. First we prove that a non-exploding solution exists when the drift coefficient b satisfies linear growth and the diffusion coefficient σ is uniformly elliptic. On this basis, we continue to obtain a solution (up to the explosion time) in the weak sense under certain local integrability, improving the result of Rozkosz and Słomiński.","PeriodicalId":49269,"journal":{"name":"Stochastics-An International Journal of Probability and Stochastic Processes","volume":"5 1","pages":"234 - 256"},"PeriodicalIF":0.9,"publicationDate":"2014-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78129663","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 : 2014-03-04DOI: 10.1080/17442508.2013.850203
M. Kobylanski, M. Quenez, Marc Roger de Campagnolle
and j and z are left limited along stopping times at T with fjðT 2Þ 1⁄4 zðT 2Þ} 1⁄4 Y a.s. (and the same remark holds in the case of processes, that is Theorem B.4). Similarly, in Lemma 4.6, the first equality holds a.s. on ft , T} and the second equality on fu , T}. Also, at the end of the proof of Theorem 4.3, the two inequalities concerning E1⁄2Jðu2Þ1C and E1⁄2J0ðu2Þ1C are precised as follows: E1⁄2Jðu2Þ1C # 1⁄2E1⁄2J0ðu2Þ1C þ E1⁄2jðuÞ1C>fu,T} þ E1⁄2jðT Þ1C>fu1⁄4T} ; E1⁄2J 0ðu2Þ1C # 1⁄2E1⁄2Jðu2Þ1C 2 E1⁄2zðuÞ1C>fu,T} 2 E1⁄2zðT Þ1C>fu1⁄4T} ;
而j和z则随着在T处的停止时间而被限定为fj / T 2Þ 1⁄4 z / T 2Þ} 1⁄4 Y a.s.(对于过程也是如此，即定理B.4)。同样，在引理4.6中，第一个等式在ft, T}上成立，第二个等式在fu, T}上成立。同样，在定理4.3的证明的最后，关于E1⁄2Jðu2Þ1C和E1⁄2J0ðu2Þ1C的两个不等式被精确地表述为:E1⁄2Jðu2Þ1C # 1⁄2E1⁄2J0ðu2Þ1C þ E1⁄2jðuÞ1C>fu,T} þ E1⁄2j / T Þ1C>fu1⁄4T};E1⁄2J 0ðu2Þ1C # 1⁄2E1⁄2Jðu2Þ1C 2E1⁄2zðuÞ1C>fu,T} 2E1⁄2z / T Þ1C>fu1⁄4T};
{"title":"Dynkin games in a general framework","authors":"M. Kobylanski, M. Quenez, Marc Roger de Campagnolle","doi":"10.1080/17442508.2013.850203","DOIUrl":"https://doi.org/10.1080/17442508.2013.850203","url":null,"abstract":"and j and z are left limited along stopping times at T with fjðT 2Þ 1⁄4 zðT 2Þ} 1⁄4 Y a.s. (and the same remark holds in the case of processes, that is Theorem B.4). Similarly, in Lemma 4.6, the first equality holds a.s. on ft , T} and the second equality on fu , T}. Also, at the end of the proof of Theorem 4.3, the two inequalities concerning E1⁄2Jðu2Þ1C and E1⁄2J0ðu2Þ1C are precised as follows: E1⁄2Jðu2Þ1C # 1⁄2E1⁄2J0ðu2Þ1C þ E1⁄2jðuÞ1C>fu,T} þ E1⁄2jðT Þ1C>fu1⁄4T} ; E1⁄2J 0ðu2Þ1C # 1⁄2E1⁄2Jðu2Þ1C 2 E1⁄2zðuÞ1C>fu,T} 2 E1⁄2zðT Þ1C>fu1⁄4T} ;","PeriodicalId":49269,"journal":{"name":"Stochastics-An International Journal of Probability and Stochastic Processes","volume":"28 1","pages":"370 - 370"},"PeriodicalIF":0.9,"publicationDate":"2014-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87657814","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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