Abstract In this paper, we will study an important property on the regularity of the trajectories of the stochastic flow generated by a famous model in finance. More precisely, we prove the differentiability with respect to initial data of the solution of the stochastic differential equation associated with this model based on Gronwall’s lemma, Itô’s isometry and Burkholder–Davis–Gundy’s and Hölder’s inequalities. This is the main motivation of our research.
{"title":"On the stochastic flow generated by the one default model in one-dimensional case","authors":"Yamina Khatir, Fatima Benziadi, A. Kandouci","doi":"10.1515/rose-2022-2093","DOIUrl":"https://doi.org/10.1515/rose-2022-2093","url":null,"abstract":"Abstract In this paper, we will study an important property on the regularity of the trajectories of the stochastic flow generated by a famous model in finance. More precisely, we prove the differentiability with respect to initial data of the solution of the stochastic differential equation associated with this model based on Gronwall’s lemma, Itô’s isometry and Burkholder–Davis–Gundy’s and Hölder’s inequalities. This is the main motivation of our research.","PeriodicalId":43421,"journal":{"name":"Random Operators and Stochastic Equations","volume":"31 1","pages":"9 - 23"},"PeriodicalIF":0.4,"publicationDate":"2023-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46925508","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 will try to weaken the hypothesis imposed by Hu and Peng. We will be concerned with finding the solution of locally monotone BSDEs associated to fBm. As an auxiliary step, we study the existence and uniqueness of a solution to the monotone backward SDEs associated to fBm. Then we connect these two kinds of fractional backward SDEs with the corresponding semilinear partial differential equations (PDEs for short).
{"title":"Fractional backward SDEs with locally monotone coefficient and application to PDEs","authors":"M. A. Saouli","doi":"10.1515/rose-2022-2095","DOIUrl":"https://doi.org/10.1515/rose-2022-2095","url":null,"abstract":"Abstract In this work, we will try to weaken the hypothesis imposed by Hu and Peng. We will be concerned with finding the solution of locally monotone BSDEs associated to fBm. As an auxiliary step, we study the existence and uniqueness of a solution to the monotone backward SDEs associated to fBm. Then we connect these two kinds of fractional backward SDEs with the corresponding semilinear partial differential equations (PDEs for short).","PeriodicalId":43421,"journal":{"name":"Random Operators and Stochastic Equations","volume":"31 1","pages":"25 - 45"},"PeriodicalIF":0.4,"publicationDate":"2023-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48665850","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 Here we consider a more flexible class of asymmetric mixture normal distribution and investigate some of its important statistical properties. We define its location-scale extension and discuss the method of maximum likelihood for estimating its parameters. Two real life data sets are considered for illustrating the usefulness of the model. Further, a simulation study is carried out for examining the efficiency of maximum likelihood estimators of the parameters of the distribution.
{"title":"On a flexible class of asymmetric mixture normal distribution and its applications","authors":"C. Satheesh Kumar, G. V. Anila","doi":"10.1515/rose-2022-2092","DOIUrl":"https://doi.org/10.1515/rose-2022-2092","url":null,"abstract":"Abstract Here we consider a more flexible class of asymmetric mixture normal distribution and investigate some of its important statistical properties. We define its location-scale extension and discuss the method of maximum likelihood for estimating its parameters. Two real life data sets are considered for illustrating the usefulness of the model. Further, a simulation study is carried out for examining the efficiency of maximum likelihood estimators of the parameters of the distribution.","PeriodicalId":43421,"journal":{"name":"Random Operators and Stochastic Equations","volume":"31 1","pages":"1 - 8"},"PeriodicalIF":0.4,"publicationDate":"2023-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49477867","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 study the stochastic zero-sum differential game in finite horizon in a general case. We first prove that the BSDE associated with a specific generator (the Hamiltonian function for the game) has a unique solution. Then we characterize the value function as that solution to prove the existence of a saddle point for the game. Finally, in the Markovian framework, we show that the value function is the unique viscosity solution for the related partial differential equation.
{"title":"Stochastic zero-sum differential games and backward stochastic differential equations","authors":"Khalid Oufdil","doi":"10.1515/rose-2022-2097","DOIUrl":"https://doi.org/10.1515/rose-2022-2097","url":null,"abstract":"Abstract In this paper, we study the stochastic zero-sum differential game in finite horizon in a general case. We first prove that the BSDE associated with a specific generator (the Hamiltonian function for the game) has a unique solution. Then we characterize the value function as that solution to prove the existence of a saddle point for the game. Finally, in the Markovian framework, we show that the value function is the unique viscosity solution for the related partial differential equation.","PeriodicalId":43421,"journal":{"name":"Random Operators and Stochastic Equations","volume":"31 1","pages":"65 - 86"},"PeriodicalIF":0.4,"publicationDate":"2023-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43790705","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 study random displacements models with a long-range particle-media interaction potential 𝔲 ( r , θ ) = 𝔣 ( θ ) r - A {mathfrak{u}(r,theta)=mathfrak{f}(theta)r^{-A}} in polar coordinates, with a smooth function 𝔣 {mathfrak{f}} which can be sign-indefinite. Spectral and dynamical localization, with an asymptotically exponential decay of eigenfunction correlators, is proved under the optimal condition A > d {A>d} .
摘要研究了具有长程粒子-介质相互作用势𝔲(r， θ)= (θ)r - a {mathfrak{u}(r，theta)=mathfrak{f}(theta)r^{- a}}的极坐标系下随机位移模型，该模型具有符号无定的光滑函数{mathfrak{f}}。在最优条件A>d {A>d}下，证明了特征函数相关器具有渐近指数衰减的谱和动态局部化。
{"title":"An optimal result on localization in random displacements models","authors":"V. Chulaevsky","doi":"10.1515/rose-2022-2091","DOIUrl":"https://doi.org/10.1515/rose-2022-2091","url":null,"abstract":"Abstract We study random displacements models with a long-range particle-media interaction potential 𝔲 ( r , θ ) = 𝔣 ( θ ) r - A {mathfrak{u}(r,theta)=mathfrak{f}(theta)r^{-A}} in polar coordinates, with a smooth function 𝔣 {mathfrak{f}} which can be sign-indefinite. Spectral and dynamical localization, with an asymptotically exponential decay of eigenfunction correlators, is proved under the optimal condition A > d {A>d} .","PeriodicalId":43421,"journal":{"name":"Random Operators and Stochastic Equations","volume":"30 1","pages":"301 - 314"},"PeriodicalIF":0.4,"publicationDate":"2022-11-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42992956","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}
{"title":"The estimators G 9 and G 10 for the solutions of~the Kolmogorov–Wiener filter","authors":"V. Girko, B. Shevchuk, L. Shevchuk","doi":"10.1515/rose-2022-2090","DOIUrl":"https://doi.org/10.1515/rose-2022-2090","url":null,"abstract":"Abstract The limit theorems for the estimators G 9 {G_{9}} and G 10 {G_{10}} for the solutions of the Kolmogorov–Wiener filter are proved.","PeriodicalId":43421,"journal":{"name":"Random Operators and Stochastic Equations","volume":"30 1","pages":"295 - 300"},"PeriodicalIF":0.4,"publicationDate":"2022-11-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48972747","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 study the asymptotic behavior of a solution of a mixed differential equation driven by independent fractional Brownian motion with Hurst index H ∈ ( 0 ; 1 ) {Hin(0;1)} and compensated Poisson process. This study consists in determining the uniform Freidlin–Wentzell estimates in a temporal distribution space.
{"title":"Large deviation principle for a mixed fractional and jump diffusion process","authors":"R. Diatta, C. Manga, A. Diédhiou","doi":"10.1515/rose-2022-2083","DOIUrl":"https://doi.org/10.1515/rose-2022-2083","url":null,"abstract":"Abstract We study the asymptotic behavior of a solution of a mixed differential equation driven by independent fractional Brownian motion with Hurst index H ∈ ( 0 ; 1 ) {Hin(0;1)} and compensated Poisson process. This study consists in determining the uniform Freidlin–Wentzell estimates in a temporal distribution space.","PeriodicalId":43421,"journal":{"name":"Random Operators and Stochastic Equations","volume":"30 1","pages":"241 - 249"},"PeriodicalIF":0.4,"publicationDate":"2022-11-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48055738","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 formulate a specific kind of reflected backward doubly stochastic differential equation with two barriers not necessarily right continuous. We prove the existence and uniqueness of the solution under Mokobodzki’s condition on the barriers and a Lipschitz driver through a Picard’s iteration method in an appropriate Banach space. Moreover, we show that the solution of such equations is characterized in terms of the value function of an extension of the corresponding stochastic Dynkin game.
{"title":"Two-barriers reflected backward doubly SDEs beyond right continuity","authors":"M. Marzougue","doi":"10.1515/rose-2022-2089","DOIUrl":"https://doi.org/10.1515/rose-2022-2089","url":null,"abstract":"Abstract In this paper, we formulate a specific kind of reflected backward doubly stochastic differential equation with two barriers not necessarily right continuous. We prove the existence and uniqueness of the solution under Mokobodzki’s condition on the barriers and a Lipschitz driver through a Picard’s iteration method in an appropriate Banach space. Moreover, we show that the solution of such equations is characterized in terms of the value function of an extension of the corresponding stochastic Dynkin game.","PeriodicalId":43421,"journal":{"name":"Random Operators and Stochastic Equations","volume":"30 1","pages":"271 - 293"},"PeriodicalIF":0.4,"publicationDate":"2022-10-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47404700","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 will establish the existence and the Hölder regularity of the local time of the Brownian bridge. Our results are obtained by using a result on Malliavin calculus in [K. Es-Sebaiy, D. Nualart, Y. Ouknine and C. A. Tudor, Occupation densities for certain processes related to fractional Brownian motion, Stochastics 82 2010, 1–3, 133–147] for a Gaussian process with an absolutely continuous random drift, jointly with the classical approach based on the concept of local nondeterminism for Gaussian processes introduced by Berman [S. M. Berman, Local nondeterminism and local times of Gaussian processes, Indiana Univ. Math. J. 23 1973/74, 69–94].
{"title":"On the existence and the Hölder regularity of the local time of the Brownian bridge","authors":"O. Allaoui, A. Sghir, S. Hadiri","doi":"10.1515/rose-2022-2087","DOIUrl":"https://doi.org/10.1515/rose-2022-2087","url":null,"abstract":"Abstract In this paper, we will establish the existence and the Hölder regularity of the local time of the Brownian bridge. Our results are obtained by using a result on Malliavin calculus in [K. Es-Sebaiy, D. Nualart, Y. Ouknine and C. A. Tudor, Occupation densities for certain processes related to fractional Brownian motion, Stochastics 82 2010, 1–3, 133–147] for a Gaussian process with an absolutely continuous random drift, jointly with the classical approach based on the concept of local nondeterminism for Gaussian processes introduced by Berman [S. M. Berman, Local nondeterminism and local times of Gaussian processes, Indiana Univ. Math. J. 23 1973/74, 69–94].","PeriodicalId":43421,"journal":{"name":"Random Operators and Stochastic Equations","volume":"30 1","pages":"259 - 270"},"PeriodicalIF":0.4,"publicationDate":"2022-10-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41754210","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 One generalization of the G-density of the global law for random matrices whose entries are independent is founded
摘要建立了项独立随机矩阵全局律G密度的一个推广
{"title":"One generalization of the main probability G-density","authors":"V. Girko","doi":"10.1515/rose-2022-2086","DOIUrl":"https://doi.org/10.1515/rose-2022-2086","url":null,"abstract":"Abstract One generalization of the 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":"251 - 257"},"PeriodicalIF":0.4,"publicationDate":"2022-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46516297","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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