Pub Date : 2017-08-17DOI: 10.19195/0208-4147.39.2.1
W. Bryc, Raouf Fakhfakh, W. Mlotkowski
This paper studies variance functions of Cauchy–Stieltjes Kernel CSK families generated by compactly supported centered probability measures. We describe several operations that allow us to construct additional variance functions from known ones. We construct a class of examples which exhausts all cubic variance functions, and provide examples of polynomial variance functions of arbitrary degree. We also relate CSK families with polynomial variance functions to generalized orthogonality.Our main results are stated solely in terms of classical probability; some proofs rely on analytic machinery of free probability.
{"title":"Cauchy–Stieltjes families with polynomial variance functions and generalized orthogonality","authors":"W. Bryc, Raouf Fakhfakh, W. Mlotkowski","doi":"10.19195/0208-4147.39.2.1","DOIUrl":"https://doi.org/10.19195/0208-4147.39.2.1","url":null,"abstract":"This paper studies variance functions of Cauchy–Stieltjes Kernel CSK families generated by compactly supported centered probability measures. We describe several operations that allow us to construct additional variance functions from known ones. We construct a class of examples which exhausts all cubic variance functions, and provide examples of polynomial variance functions of arbitrary degree. We also relate CSK families with polynomial variance functions to generalized orthogonality.Our main results are stated solely in terms of classical probability; some proofs rely on analytic machinery of free probability.","PeriodicalId":48996,"journal":{"name":"Probability and Mathematical Statistics-Poland","volume":" ","pages":""},"PeriodicalIF":0.3,"publicationDate":"2017-08-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45616482","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}
The paper presents an elaboration of some results on Lin's conditions. A new proof of the fact that if densities of independent random variables $xi_1$ and $xi_2$ satisfy Lin's condition, the same is true for their product is presented. Also, it is shown that without the condition of independence, the statement is no longer valid.
{"title":"On Lin's condition for products of random variables with singular joint distribution","authors":"A. Il'inskii, Sofiya Ostrovska","doi":"10.15407/MAG15.01.079","DOIUrl":"https://doi.org/10.15407/MAG15.01.079","url":null,"abstract":"The paper presents an elaboration of some results on Lin's conditions. A new proof of the fact that if densities of independent random variables $xi_1$ and $xi_2$ satisfy Lin's condition, the same is true for their product is presented. Also, it is shown that without the condition of independence, the statement is no longer valid.","PeriodicalId":48996,"journal":{"name":"Probability and Mathematical Statistics-Poland","volume":" ","pages":""},"PeriodicalIF":0.3,"publicationDate":"2017-07-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49132209","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 : 2017-06-20DOI: 10.37190/0208-4147.40.1.9
N. Agram, Soukaina Douissi, A. Hilbert
We consider a mean-field optimal control problem for stochastic differential equations with delay driven by fractional Brownian motion with Hurst parameter greater than one half. Stochastic optimal control problems driven by fractional Brownian motion can not be studied using classical methods, because the fractional Brownian motion is neither a Markov process nor a semi-martingale. However, using the fractional White noise calculus combined with some special tools related to the differentiation for functions of measures, we establish and prove necessary and sufficient stochastic maximum principles. To illustrate our study, we consider two applications: we solve a problem of optimal consumption from a cash flow with delay and a linear-quadratique (LQ) problem with delay.
{"title":"Mean-field optimal control problem of SDDES driven by fractional Brownian Motion","authors":"N. Agram, Soukaina Douissi, A. Hilbert","doi":"10.37190/0208-4147.40.1.9","DOIUrl":"https://doi.org/10.37190/0208-4147.40.1.9","url":null,"abstract":"We consider a mean-field optimal control problem for stochastic differential equations with delay driven by fractional Brownian motion with Hurst parameter greater than one half. Stochastic optimal control problems driven by fractional Brownian motion can not be studied using classical methods, because the fractional Brownian motion is neither a Markov process nor a semi-martingale. However, using the fractional White noise calculus combined with some special tools related to the differentiation for functions of measures, we establish and prove necessary and sufficient stochastic maximum principles. To illustrate our study, we consider two applications: we solve a problem of optimal consumption from a cash flow with delay and a linear-quadratique (LQ) problem with delay.","PeriodicalId":48996,"journal":{"name":"Probability and Mathematical Statistics-Poland","volume":" ","pages":""},"PeriodicalIF":0.3,"publicationDate":"2017-06-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46650865","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 : 2017-02-02DOI: 10.19195/0208-4147.39.1.3
H. Tsukada
For symmetric Levy processes, if the local times exist, the Tanaka formula has already been constructed via the techniques in the potential theory by Salminen and Yor 2007. In this paper, we study the Tanaka formula for arbitrary strictly stable processes with index α ∈ 1, 2, including spectrally positive and negative cases in a framework of Ito’s stochastic calculus. Our approach to the existence of local times for such processes is different from that of Bertoin 1996.
{"title":"Tanaka formula for strictly stable processes","authors":"H. Tsukada","doi":"10.19195/0208-4147.39.1.3","DOIUrl":"https://doi.org/10.19195/0208-4147.39.1.3","url":null,"abstract":"For symmetric Levy processes, if the local times exist, the Tanaka formula has already been constructed via the techniques in the potential theory by Salminen and Yor 2007. In this paper, we study the Tanaka formula for arbitrary strictly stable processes with index α ∈ 1, 2, including spectrally positive and negative cases in a framework of Ito’s stochastic calculus. Our approach to the existence of local times for such processes is different from that of Bertoin 1996.","PeriodicalId":48996,"journal":{"name":"Probability and Mathematical Statistics-Poland","volume":"37 12","pages":""},"PeriodicalIF":0.3,"publicationDate":"2017-02-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41269771","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 : 2016-12-12DOI: 10.37190/0208-4147.40.1.3
Mateusz B. Majka
We extend some methods developed by Albeverio, Brze'{z}niak and Wu and we show how to apply them in order to prove existence of global strong solutions of stochastic differential equations with jumps, under a local one-sided Lipschitz condition on the drift (also known as a monotonicity condition) and a local Lipschitz condition on the diffusion and jump coefficients, while an additional global one-sided linear growth assumption is satisfied. Then we use these methods to prove existence of invariant measures for a broad class of such equations.
{"title":"A note on existence of global solutions and invariant measures for jump SDEs with locally one-sided Lipschitz drift","authors":"Mateusz B. Majka","doi":"10.37190/0208-4147.40.1.3","DOIUrl":"https://doi.org/10.37190/0208-4147.40.1.3","url":null,"abstract":"We extend some methods developed by Albeverio, Brze'{z}niak and Wu and we show how to apply them in order to prove existence of global strong solutions of stochastic differential equations with jumps, under a local one-sided Lipschitz condition on the drift (also known as a monotonicity condition) and a local Lipschitz condition on the diffusion and jump coefficients, while an additional global one-sided linear growth assumption is satisfied. Then we use these methods to prove existence of invariant measures for a broad class of such equations.","PeriodicalId":48996,"journal":{"name":"Probability and Mathematical Statistics-Poland","volume":"1 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2016-12-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"69996854","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 : 2016-11-21DOI: 10.19195/0208-4147.39.1.13
Mateusz Topolewski
We consider reflected backward stochastic differential equations, with two barriers, defined on probability spaces equipped with filtration satisfying only the usual assumptions of right-continuity and completeness. As for barriers, we assume that there are càdlàg processes of class D that are completely separated. We prove the existence and uniqueness of solutions for an integrable final condition and an integrable monotone generator. An application to the zero-sum Dynkin game is given.
{"title":"Reflected BSDEs with general filtration and two completely separated barriers","authors":"Mateusz Topolewski","doi":"10.19195/0208-4147.39.1.13","DOIUrl":"https://doi.org/10.19195/0208-4147.39.1.13","url":null,"abstract":"We consider reflected backward stochastic differential equations, with two barriers, defined on probability spaces equipped with filtration satisfying only the usual assumptions of right-continuity and completeness. As for barriers, we assume that there are càdlàg processes of class D that are completely separated. We prove the existence and uniqueness of solutions for an integrable final condition and an integrable monotone generator. An application to the zero-sum Dynkin game is given.","PeriodicalId":48996,"journal":{"name":"Probability and Mathematical Statistics-Poland","volume":"1 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2016-11-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68001762","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 : 2016-11-11DOI: 10.19195/0208-4147.39.2.8
T. T. Diu Tran
Let Zt q,H t∈[0,1]d denote a d-parameter Hermite random field of order q ≥ 1 and self-similarity parameter H = H₁, . . . ,Hd ∈ ½, 1d. This process is H-self-similar, has stationary increments and exhibits long-range dependence. Particular examples include fractional Brownian motion q = 1, d = 1, fractional Brownian sheet q = 1, d ≥ 2, the Rosenblatt process q = 2, d = 1 as well as the Rosenblatt sheet q = 2, d ≥ 2. For any q ≥ 2, d ≥ 1 and H ∈ ½, 1d we show in this paper that a proper renormalization of the quadratic variation of Zq,H converges in L2Ω to a standard d-parameter Rosenblatt random variable with self-similarity index H' = 1 + 2H − 2/q.
{"title":"Asymptotic behavior for quadratic variations of non-Gaussian multiparameter Hermite random fields","authors":"T. T. Diu Tran","doi":"10.19195/0208-4147.39.2.8","DOIUrl":"https://doi.org/10.19195/0208-4147.39.2.8","url":null,"abstract":"Let Zt q,H t∈[0,1]d denote a d-parameter Hermite random field of order q ≥ 1 and self-similarity parameter H = H₁, . . . ,Hd ∈ ½, 1d. This process is H-self-similar, has stationary increments and exhibits long-range dependence. Particular examples include fractional Brownian motion q = 1, d = 1, fractional Brownian sheet q = 1, d ≥ 2, the Rosenblatt process q = 2, d = 1 as well as the Rosenblatt sheet q = 2, d ≥ 2. For any q ≥ 2, d ≥ 1 and H ∈ ½, 1d we show in this paper that a proper renormalization of the quadratic variation of Zq,H converges in L2Ω to a standard d-parameter Rosenblatt random variable with self-similarity index H' = 1 + 2H − 2/q.","PeriodicalId":48996,"journal":{"name":"Probability and Mathematical Statistics-Poland","volume":"1 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2016-11-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68002033","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 : 2016-10-10DOI: 10.19195/0208-4147.38.1.10
Natalia Soja-Kukieła
Tu wpisz tekst
在此处输入文本
{"title":"Extremes of multidimensional stationary Gaussian random fields","authors":"Natalia Soja-Kukieła","doi":"10.19195/0208-4147.38.1.10","DOIUrl":"https://doi.org/10.19195/0208-4147.38.1.10","url":null,"abstract":"Tu wpisz tekst","PeriodicalId":48996,"journal":{"name":"Probability and Mathematical Statistics-Poland","volume":"1 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2016-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68000140","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 : 2016-10-02DOI: 10.37190/0208-4147.41.1.3
Z. Palmowski, Longmin Wang
In this paper we identify the asymptotic tail of the distribution of the exit time $tau_C$ from a cone $C$ of an isotropic $alpha$-self-similar Markov process $X_t$ with a skew-product structure, that is $X_t$ is a product of its radial process and independent time changed angular component $Theta_t$. Under some additional regularity assumptions, the angular process $Theta_t$ killed on exiting from the cone $C$ has the transition density that could be expressed in terms of a complete set of orthogonal eigenfunctions with corresponding eigenvalues of an appropriate generator. Using this fact and some asymptotic properties of the exponential functional of a killed L'evy process related with Lamperti representation of the radial process, we prove that $$mathbb{P}_x(tau_C>t)sim h(x)t^{-kappa_1}$$ as $trightarrowinfty$ for $h$ and $kappa_1$ identified explicitly. The result extends the work of DeBlassie (1988) and Ba~nuelos and Smits (1997) concerning the Brownian motion.
{"title":"On th exact asymptotics of exit time from a cone of an isotropic alpha-self-similar Markov process with a skew-product structure","authors":"Z. Palmowski, Longmin Wang","doi":"10.37190/0208-4147.41.1.3","DOIUrl":"https://doi.org/10.37190/0208-4147.41.1.3","url":null,"abstract":"In this paper we identify the asymptotic tail of the distribution of the exit time $tau_C$ from a cone $C$ of an isotropic $alpha$-self-similar Markov process $X_t$ with a skew-product structure, that is $X_t$ is a product of its radial process and independent time changed angular component $Theta_t$. Under some additional regularity assumptions, the angular process $Theta_t$ killed on exiting from the cone $C$ has the transition density that could be expressed in terms of a complete set of orthogonal eigenfunctions with corresponding eigenvalues of an appropriate generator. Using this fact and some asymptotic properties of the exponential functional of a killed L'evy process related with Lamperti representation of the radial process, we prove that $$mathbb{P}_x(tau_C>t)sim h(x)t^{-kappa_1}$$ as $trightarrowinfty$ for $h$ and $kappa_1$ identified explicitly. The result extends the work of DeBlassie (1988) and Ba~nuelos and Smits (1997) concerning the Brownian motion.","PeriodicalId":48996,"journal":{"name":"Probability and Mathematical Statistics-Poland","volume":"1 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2016-10-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"69996923","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 : 2016-09-09DOI: 10.19195/0208-4147.38.2.11
Vjekoslav Kovavc, K. vSkreb
We give an explicit formula for one possible Bellman function associated with the Lp boundedness of dyadic paraproducts regarded as bilinear operators or trilinear forms. Then we apply the same Bellman function in various other settings, to give self-contained alternative proofs of the estimates for several classical operators. These include the martingale paraproducts of Bañuelos and Bennett and the paraproducts with respect to the heat flows.
{"title":"Bellman functions and L^p estimates for paraproducts","authors":"Vjekoslav Kovavc, K. vSkreb","doi":"10.19195/0208-4147.38.2.11","DOIUrl":"https://doi.org/10.19195/0208-4147.38.2.11","url":null,"abstract":"We give an explicit formula for one possible Bellman function associated with the Lp boundedness of dyadic paraproducts regarded as bilinear operators or trilinear forms. Then we apply the same Bellman function in various other settings, to give self-contained alternative proofs of the estimates for several classical operators. These include the martingale paraproducts of Bañuelos and Bennett and the paraproducts with respect to the heat flows.","PeriodicalId":48996,"journal":{"name":"Probability and Mathematical Statistics-Poland","volume":"7 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2016-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68000950","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}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：