Abstract Our aim in this paper is to establish some strong stability results for solutions of stochastic differential equations driven by a Riemann–Liouville multifractional Brownian motion. The latter is defined as a Gaussian non-stationary process with a Hurst parameter as a function of time. The results are obtained assuming that the pathwise uniqueness property holds and using Skorokhod’s selection theorem.
{"title":"Stability of stochastic differential equations driven by multifractional Brownian motion","authors":"Oussama El Barrimi, Y. Ouknine","doi":"10.1515/rose-2021-2055","DOIUrl":"https://doi.org/10.1515/rose-2021-2055","url":null,"abstract":"Abstract Our aim in this paper is to establish some strong stability results for solutions of stochastic differential equations driven by a Riemann–Liouville multifractional Brownian motion. The latter is defined as a Gaussian non-stationary process with a Hurst parameter as a function of time. The results are obtained assuming that the pathwise uniqueness property holds and using Skorokhod’s selection theorem.","PeriodicalId":43421,"journal":{"name":"Random Operators and Stochastic Equations","volume":"29 1","pages":"87 - 96"},"PeriodicalIF":0.4,"publicationDate":"2021-04-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/rose-2021-2055","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41492280","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 consider a class of random partial integro-differential equations with unbounded delay. Existence of mild solutions are investigated by using a random fixed point theorem with a stochastic domain combined with Schauder’s fixed point theorem and Grimmer’s resolvent operator theory. The results are obtained under Carathéodory conditions. Finally, an example is provided to illustrate our results.
{"title":"Existence results for a class of random delay integrodifferential equations","authors":"Amadou Diop, M. Diop, K. Ezzinbi","doi":"10.1515/rose-2021-2054","DOIUrl":"https://doi.org/10.1515/rose-2021-2054","url":null,"abstract":"Abstract In this paper, we consider a class of random partial integro-differential equations with unbounded delay. Existence of mild solutions are investigated by using a random fixed point theorem with a stochastic domain combined with Schauder’s fixed point theorem and Grimmer’s resolvent operator theory. The results are obtained under Carathéodory conditions. Finally, an example is provided to illustrate our results.","PeriodicalId":43421,"journal":{"name":"Random Operators and Stochastic Equations","volume":"29 1","pages":"79 - 86"},"PeriodicalIF":0.4,"publicationDate":"2021-04-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/rose-2021-2054","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47793120","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 anticipated backward stochastic differential equations driven by two mutually independent fractional Brownian motions. We essentially establish the existence and uniqueness of a solution in the case of Lipschitz coefficients and non-Lipschitz coefficients. The stochastic integral used throughout the paper is the divergence-type integral.
{"title":"Anticipated BSDEs driven by two mutually independent fractional Brownian motions with non-Lipschitz coefficients","authors":"Sadibou Aidara, Yaya Sagna","doi":"10.1515/rose-2020-2051","DOIUrl":"https://doi.org/10.1515/rose-2020-2051","url":null,"abstract":"Abstract This paper deals with a class of anticipated backward stochastic differential equations driven by two mutually independent fractional Brownian motions. We essentially establish the existence and uniqueness of a solution in the case of Lipschitz coefficients and non-Lipschitz coefficients. The stochastic integral used throughout the paper is the divergence-type integral.","PeriodicalId":43421,"journal":{"name":"Random Operators and Stochastic Equations","volume":"29 1","pages":"27 - 39"},"PeriodicalIF":0.4,"publicationDate":"2021-02-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/rose-2020-2051","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49154086","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 studies optimal time-consistent strategies for the mean-variance portfolio selection problem. Especially, we assume that the price processes of risky stocks are described by regime-switching SDEs. We consider a Markov-modulated state-dependent risk aversion and we formulate the problem in the game theoretic framework. Then, by solving a flow of forward-backward stochastic differential equations, an explicit representation as well as uniqueness results of an equilibrium solution are obtained.
{"title":"Continuous-time mean-variance portfolio selection with regime-switching financial market: Time-consistent solution","authors":"I. Alia, F. Chighoub","doi":"10.1515/rose-2020-2050","DOIUrl":"https://doi.org/10.1515/rose-2020-2050","url":null,"abstract":"Abstract This paper studies optimal time-consistent strategies for the mean-variance portfolio selection problem. Especially, we assume that the price processes of risky stocks are described by regime-switching SDEs. We consider a Markov-modulated state-dependent risk aversion and we formulate the problem in the game theoretic framework. Then, by solving a flow of forward-backward stochastic differential equations, an explicit representation as well as uniqueness results of an equilibrium solution are obtained.","PeriodicalId":43421,"journal":{"name":"Random Operators and Stochastic Equations","volume":"29 1","pages":"11 - 25"},"PeriodicalIF":0.4,"publicationDate":"2021-01-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/rose-2020-2050","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47120980","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 Probabilistic metric spaces are metric structures having uncertainty built within their geometry, which has made them into an appropriate context for modelling many real life problems. Theoretical studies on these structures have also appeared extensively. This paper is intended for some development of fixed point theory in probabilistic metric spaces, which is an active area of contemporary research. We define a new contraction mapping in such spaces and show that the contraction has a unique fixed point if such spaces are G-complete with an arbitrary choice of a continuous t-norm. With a minimum t-norm, the result is further extended in any complete probabilistic metric space. The contraction is defined with the help of a control function which is different from several other control functions used in probabilistic fixed point theory by other authors. The methodology of the proof is new. An illustrative example is given. The present work is a part of probabilistic analysis.
{"title":"Probabilistic contraction under a control function","authors":"B. Choudhury, Vandana Tiwari, T. Som, P. Saha","doi":"10.1515/rose-2020-2049","DOIUrl":"https://doi.org/10.1515/rose-2020-2049","url":null,"abstract":"Abstract Probabilistic metric spaces are metric structures having uncertainty built within their geometry, which has made them into an appropriate context for modelling many real life problems. Theoretical studies on these structures have also appeared extensively. This paper is intended for some development of fixed point theory in probabilistic metric spaces, which is an active area of contemporary research. We define a new contraction mapping in such spaces and show that the contraction has a unique fixed point if such spaces are G-complete with an arbitrary choice of a continuous t-norm. With a minimum t-norm, the result is further extended in any complete probabilistic metric space. The contraction is defined with the help of a control function which is different from several other control functions used in probabilistic fixed point theory by other authors. The methodology of the proof is new. An illustrative example is given. The present work is a part of probabilistic analysis.","PeriodicalId":43421,"journal":{"name":"Random Operators and Stochastic Equations","volume":"29 1","pages":"1 - 10"},"PeriodicalIF":0.4,"publicationDate":"2021-01-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/rose-2020-2049","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41524482","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 consider various approximation properties for systems driven by a McKean–Vlasov stochastic differential equations (MVSDEs) with continuous coefficients, for which pathwise uniqueness holds. We prove that the solution of such equations is stable with respect to small perturbation of initial conditions, parameters and driving processes. Moreover, the unique strong solutions may be constructed by an effective approximation procedure. Finally, we show that the set of bounded uniformly continuous coefficients for which the corresponding MVSDE have a unique strong solution is a set of second category in the sense of Baire.
{"title":"Stability and prevalence of McKean–Vlasov stochastic differential equations with non-Lipschitz coefficients","authors":"Mohamed Amine Mezerdi, N. Khelfallah","doi":"10.1515/rose-2021-2053","DOIUrl":"https://doi.org/10.1515/rose-2021-2053","url":null,"abstract":"Abstract We consider various approximation properties for systems driven by a McKean–Vlasov stochastic differential equations (MVSDEs) with continuous coefficients, for which pathwise uniqueness holds. We prove that the solution of such equations is stable with respect to small perturbation of initial conditions, parameters and driving processes. Moreover, the unique strong solutions may be constructed by an effective approximation procedure. Finally, we show that the set of bounded uniformly continuous coefficients for which the corresponding MVSDE have a unique strong solution is a set of second category in the sense of Baire.","PeriodicalId":43421,"journal":{"name":"Random Operators and Stochastic Equations","volume":"29 1","pages":"67 - 78"},"PeriodicalIF":0.4,"publicationDate":"2021-01-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/rose-2021-2053","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46579573","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 the present paper, we consider reflected backward stochastic differential equations when the reflecting obstacle is not necessarily right-continuous in a general filtration that supports a one-dimensional Brownian motion and an independent Poisson random measure. We prove the existence and uniqueness of a predictable solution for such equations under the stochastic Lipschitz coefficient by using the predictable Mertens decomposition.
{"title":"Predictable solution for reflected BSDEs when the obstacle is not right-continuous","authors":"M. Marzougue, M. El Otmani","doi":"10.1515/rose-2020-2045","DOIUrl":"https://doi.org/10.1515/rose-2020-2045","url":null,"abstract":"Abstract In the present paper, we consider reflected backward stochastic differential equations when the reflecting obstacle is not necessarily right-continuous in a general filtration that supports a one-dimensional Brownian motion and an independent Poisson random measure. We prove the existence and uniqueness of a predictable solution for such equations under the stochastic Lipschitz coefficient by using the predictable Mertens decomposition.","PeriodicalId":43421,"journal":{"name":"Random Operators and Stochastic Equations","volume":"28 1","pages":"269 - 279"},"PeriodicalIF":0.4,"publicationDate":"2020-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/rose-2020-2045","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49496266","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 investigate the stability of functionals and trajectories of two different, independent, time-inhomogeneous, discrete-time Markov chains on a general state space. We obtain various stability estimates such as an estimate for a difference in expectations of functionals, L 2 {L_{2}} stability, and a probability of large deviations. The key condition that is used is the minorization condition on the whole space. We consider different limitations on the functional and on the proximity of two chains. We use the coupling method as a primary technique in our proofs.
{"title":"Stability of functionals of perturbed Markov chains under the condition of uniform minorization","authors":"V. Golomoziy","doi":"10.1515/rose-2020-2043","DOIUrl":"https://doi.org/10.1515/rose-2020-2043","url":null,"abstract":"Abstract In this paper, we investigate the stability of functionals and trajectories of two different, independent, time-inhomogeneous, discrete-time Markov chains on a general state space. We obtain various stability estimates such as an estimate for a difference in expectations of functionals, L 2 {L_{2}} stability, and a probability of large deviations. The key condition that is used is the minorization condition on the whole space. We consider different limitations on the functional and on the proximity of two chains. We use the coupling method as a primary technique in our proofs.","PeriodicalId":43421,"journal":{"name":"Random Operators and Stochastic Equations","volume":"28 1","pages":"237 - 251"},"PeriodicalIF":0.4,"publicationDate":"2020-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/rose-2020-2043","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43690609","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 Using the coupling method and Girsanov theorem, we prove a Harnack-type inequality for a stochastic differential equation with non-Lipschitz drift and driven by a fractional Brownian motion with Hurst parameter H < 1 2 {H
摘要利用耦合方法和Girsanov定理，证明了Hurst参数H<12{H
{"title":"Harnack-type inequality for linear fractional stochastic equations","authors":"B. Boufoussi, S. Mouchtabih","doi":"10.1515/rose-2020-2046","DOIUrl":"https://doi.org/10.1515/rose-2020-2046","url":null,"abstract":"Abstract Using the coupling method and Girsanov theorem, we prove a Harnack-type inequality for a stochastic differential equation with non-Lipschitz drift and driven by a fractional Brownian motion with Hurst parameter H < 1 2 {H<frac{1}{2}} . We also investigate this inequality for a stochastic differential equation driven by an additive fractional Brownian sheet.","PeriodicalId":43421,"journal":{"name":"Random Operators and Stochastic Equations","volume":"28 1","pages":"281 - 290"},"PeriodicalIF":0.4,"publicationDate":"2020-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/rose-2020-2046","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46352523","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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