Abstract In this paper, we consider a new telegraph process of Ornstein–Uhlenbeck type. The process is obtained by generalizing the telegraph process in a similar manner to how the Ornstein–Uhlenbeck process was obtained from the Wiener process, namely by adding a drift coefficient proportional to a displacement from the origin. This process was first introduced by Ratanov in [N. Ratanov, Ornstein–Uhlenbeck process of bounded variation, Methodol. Comput. Appl. Probab. 23 2021, 925–946]. We obtain the infinitesimal operator of this process and we present formulas for finding its stationary probability density. We consider both the symmetric and asymmetric cases.
{"title":"Stationary density function for a random evolution driven by a Markov-switching Ornstein–Uhlenbeck process with finite velocity","authors":"A. Pogorui, R. Rodríguez-Dagnino","doi":"10.1515/rose-2022-2075","DOIUrl":"https://doi.org/10.1515/rose-2022-2075","url":null,"abstract":"Abstract In this paper, we consider a new telegraph process of Ornstein–Uhlenbeck type. The process is obtained by generalizing the telegraph process in a similar manner to how the Ornstein–Uhlenbeck process was obtained from the Wiener process, namely by adding a drift coefficient proportional to a displacement from the origin. This process was first introduced by Ratanov in [N. Ratanov, Ornstein–Uhlenbeck process of bounded variation, Methodol. Comput. Appl. Probab. 23 2021, 925–946]. We obtain the infinitesimal operator of this process and we present formulas for finding its stationary probability density. We consider both the symmetric and asymmetric cases.","PeriodicalId":43421,"journal":{"name":"Random Operators and Stochastic Equations","volume":"30 1","pages":"113 - 120"},"PeriodicalIF":0.4,"publicationDate":"2022-04-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49663951","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 main probability G-density of the global law for random matrices whose entries are independent is founded.
摘要建立了条目独立的随机矩阵全局律的主概率G密度。
{"title":"The main probability G-density of the theory of non-Hermitian random matrices, VICTORIA transform, RESPECT and REFORM methods","authors":"V. Girko","doi":"10.1515/rose-2022-2071","DOIUrl":"https://doi.org/10.1515/rose-2022-2071","url":null,"abstract":"Abstract The main probability G-density of the global law for random matrices whose entries are independent is founded.","PeriodicalId":43421,"journal":{"name":"Random Operators and Stochastic Equations","volume":"30 1","pages":"39 - 69"},"PeriodicalIF":0.4,"publicationDate":"2022-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45045429","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 examples of the pencil law for two random matrices whose pairs of entries are independent are considered.
摘要考虑了两个元素对独立的随机矩阵的铅笔律的例子。
{"title":"The G-pencil law under G-Lindeberg condition. The canonical equation K_98 and G-logarithmic law","authors":"V. Girko, B. Shevchuk, L. Shevchuk","doi":"10.1515/rose-2022-2072","DOIUrl":"https://doi.org/10.1515/rose-2022-2072","url":null,"abstract":"Abstract The examples of the pencil law for two random matrices whose pairs of entries are independent are considered.","PeriodicalId":43421,"journal":{"name":"Random Operators and Stochastic Equations","volume":"30 1","pages":"71 - 84"},"PeriodicalIF":0.4,"publicationDate":"2022-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42275278","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 work we focus on a method for solving equations with a coefficient formally given in terms of the derivative of a continuous semimartingale. This generalizes the case of coefficients being the white noise. The idea for solving the equation is to find explicitly the inverse of the ill-posed differential operator, which boils down to finding the associated Green kernel. To find the kernel we give explicitly two homogeneous solutions in terms of the so-called Dolean–Dade exponential. The general idea to define rigorously differential operators lies on dealing with them through bilinear forms. We give several examples with explicit calculations.
{"title":"Solving equations with semimartingale noise","authors":"Jonathan Gutierrez-Pavón, Carlos G. Pacheco","doi":"10.1515/rose-2021-2070","DOIUrl":"https://doi.org/10.1515/rose-2021-2070","url":null,"abstract":"Abstract In this work we focus on a method for solving equations with a coefficient formally given in terms of the derivative of a continuous semimartingale. This generalizes the case of coefficients being the white noise. The idea for solving the equation is to find explicitly the inverse of the ill-posed differential operator, which boils down to finding the associated Green kernel. To find the kernel we give explicitly two homogeneous solutions in terms of the so-called Dolean–Dade exponential. The general idea to define rigorously differential operators lies on dealing with them through bilinear forms. We give several examples with explicit calculations.","PeriodicalId":43421,"journal":{"name":"Random Operators and Stochastic Equations","volume":"30 1","pages":"33 - 38"},"PeriodicalIF":0.4,"publicationDate":"2022-01-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48218477","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}
A. Laadjel, J. Nieto, A. Ouahab, R. Rodríguez-López
Abstract In this paper, we present some random fixed point theorems in complete gauge spaces. We establish then a multivalued version of a Perov–Gheorghiu’s fixed point theorem in generalized gauge spaces. Finally, some examples are given to illustrate the results.
{"title":"Multivalued and random version of Perov fixed point theorem in generalized gauge spaces","authors":"A. Laadjel, J. Nieto, A. Ouahab, R. Rodríguez-López","doi":"10.1515/rose-2021-2068","DOIUrl":"https://doi.org/10.1515/rose-2021-2068","url":null,"abstract":"Abstract In this paper, we present some random fixed point theorems in complete gauge spaces. We establish then a multivalued version of a Perov–Gheorghiu’s fixed point theorem in generalized gauge spaces. Finally, some examples are given to illustrate the results.","PeriodicalId":43421,"journal":{"name":"Random Operators and Stochastic Equations","volume":"30 1","pages":"1 - 19"},"PeriodicalIF":0.4,"publicationDate":"2022-01-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44703902","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 This paper deals with a class of deplay backward stochastic differential equations driven by fractional Brownian motion (with Hurst parameter H greater than 1 2 {frac{1}{2}} ). In this type of equation, a generator at time t can depend not only on the present but also the past solutions. We essentially establish existence and uniqueness of a solution in the case of Lipschitz coefficients and non-Lipschitz coefficients. The stochastic integral used throughout this paper is the divergence-type integral.
{"title":"Deplay BSDEs driven by fractional Brownian motion","authors":"Sadibou Aidara, Ibrahima Sané","doi":"10.1515/rose-2021-2069","DOIUrl":"https://doi.org/10.1515/rose-2021-2069","url":null,"abstract":"Abstract This paper deals with a class of deplay backward stochastic differential equations driven by fractional Brownian motion (with Hurst parameter H greater than 1 2 {frac{1}{2}} ). In this type of equation, a generator at time t can depend not only on the present but also the past solutions. We essentially establish existence and uniqueness of a solution in the case of Lipschitz coefficients and non-Lipschitz coefficients. The stochastic integral used throughout this paper is the divergence-type integral.","PeriodicalId":43421,"journal":{"name":"Random Operators and Stochastic Equations","volume":"30 1","pages":"21 - 31"},"PeriodicalIF":0.4,"publicationDate":"2022-01-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43966204","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 derive a stability result for L 1 {L_{1}} and L ∞ {L_{infty}} perturbations of diffusions under weak regularity conditions on the coefficients. In particular, the drift terms we consider can be unbounded with at most linear growth, and the estimates reflect the transport of the initial condition by the unbounded drift through the corresponding flow. Our approach is based on the study of the distance in L 1 {L_{1}} - L 1 {L_{1}} metric between the transition densities of a given diffusion and the perturbed one using the McKean–Singer parametrix expansion. In the second part, we generalize the well-known result on the stability of diffusions with bounded coefficients to the case of at most linearly growing drift.
{"title":"L 1 and L ∞ stability of transition densities of perturbed diffusions","authors":"I. Bitter, V. Konakov","doi":"10.1515/rose-2021-2067","DOIUrl":"https://doi.org/10.1515/rose-2021-2067","url":null,"abstract":"Abstract In this paper, we derive a stability result for L 1 {L_{1}} and L ∞ {L_{infty}} perturbations of diffusions under weak regularity conditions on the coefficients. In particular, the drift terms we consider can be unbounded with at most linear growth, and the estimates reflect the transport of the initial condition by the unbounded drift through the corresponding flow. Our approach is based on the study of the distance in L 1 {L_{1}} - L 1 {L_{1}} metric between the transition densities of a given diffusion and the perturbed one using the McKean–Singer parametrix expansion. In the second part, we generalize the well-known result on the stability of diffusions with bounded coefficients to the case of at most linearly growing drift.","PeriodicalId":43421,"journal":{"name":"Random Operators and Stochastic Equations","volume":"29 1","pages":"287 - 308"},"PeriodicalIF":0.4,"publicationDate":"2021-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44174105","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 purpose of this work is to investigate the existence of solutions for a system of random differential equations involving the Riemann–Liouville fractional derivative. The existence result is established by means of a random abstract formulation to Sadovskii’s fixed point theorem principle [A. Baliki, J. J. Nieto, A. Ouahab and M. L. Sinacer, Random semilinear system of differential equations with impulses, Fixed Point Theory Appl. 2017 2017, Paper No. 27] combined with a technique based on vector-valued metrics and convergent to zero matrices. An example is also provided to illustrate our result.
摘要本文的目的是研究一类含有Riemann-Liouville分数阶导数的随机微分方程组解的存在性。通过对Sadovskii不动点定理原理的一个随机抽象表述，建立了存在性结果。Baliki, J. J. Nieto, a . Ouahab和M. L. Sinacer，随机半线性脉冲微分方程系统，不动点理论应用，2017,No. 27]。最后给出了一个例子来说明我们的结果。
{"title":"Coupled fractional differential systems with random effects in Banach spaces","authors":"O. Zentar, M. Ziane, S. Khelifa","doi":"10.1515/rose-2021-2064","DOIUrl":"https://doi.org/10.1515/rose-2021-2064","url":null,"abstract":"Abstract The purpose of this work is to investigate the existence of solutions for a system of random differential equations involving the Riemann–Liouville fractional derivative. The existence result is established by means of a random abstract formulation to Sadovskii’s fixed point theorem principle [A. Baliki, J. J. Nieto, A. Ouahab and M. L. Sinacer, Random semilinear system of differential equations with impulses, Fixed Point Theory Appl. 2017 2017, Paper No. 27] combined with a technique based on vector-valued metrics and convergent to zero matrices. An example is also provided to illustrate our result.","PeriodicalId":43421,"journal":{"name":"Random Operators and Stochastic Equations","volume":"0 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2021-10-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41356211","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 We provide a large deviation principle on the stochastic differential equations with reflecting Wentzel boundary condition if δ ε {frac{delta}{varepsilon}} tends to 0 when the two parameters δ (homogenization parameter) and ε (the large deviations parameter) tend to zero. Here, we suppose that the homogenization parameter converges sufficiently quickly more than the large deviations parameter. Furthermore, we will make explicit the associated rate function.
{"title":"Small double limit with reflecting Wentzel boundary condition","authors":"Ibrahima Sané, A. Diédhiou","doi":"10.1515/rose-2021-2066","DOIUrl":"https://doi.org/10.1515/rose-2021-2066","url":null,"abstract":"Abstract We provide a large deviation principle on the stochastic differential equations with reflecting Wentzel boundary condition if δ ε {frac{delta}{varepsilon}} tends to 0 when the two parameters δ (homogenization parameter) and ε (the large deviations parameter) tend to zero. Here, we suppose that the homogenization parameter converges sufficiently quickly more than the large deviations parameter. Furthermore, we will make explicit the associated rate function.","PeriodicalId":43421,"journal":{"name":"Random Operators and Stochastic Equations","volume":"29 1","pages":"279 - 286"},"PeriodicalIF":0.4,"publicationDate":"2021-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48675819","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 We investigate the asymptotic properties of maximum likelihood estimators of the drift parameters for the fractional Vasicek model driven by a sub-fractional Brownian motion.
研究了由次分数阶布朗运动驱动的分数阶Vasicek模型漂移参数的极大似然估计的渐近性质。
{"title":"Maximum likelihood estimation for sub-fractional Vasicek model","authors":"B. Prakasa Rao","doi":"10.1515/rose-2021-2065","DOIUrl":"https://doi.org/10.1515/rose-2021-2065","url":null,"abstract":"Abstract We investigate the asymptotic properties of maximum likelihood estimators of the drift parameters for the fractional Vasicek model driven by a sub-fractional Brownian motion.","PeriodicalId":43421,"journal":{"name":"Random Operators and Stochastic Equations","volume":"29 1","pages":"265 - 277"},"PeriodicalIF":0.4,"publicationDate":"2021-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47762263","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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