Pub Date : 2022-02-02DOI: 10.1080/07362994.2022.2029712
F. Delgado-Vences, J. J. Pavon-Español
Abstract In this paper, we study asymptotic properties of the maximum likelihood estimator (MLE) for the speed of a stochastic wave equation. We follow a well-known spectral approach to write the solution as a Fourier series, then we project the solution to a N-finite dimensional space and find the estimator as a function of the time and N. We then show consistency of the MLE using classical stochastic analysis. Afterward, we prove the asymptotic normality using the Malliavin–Stein method. We also study asymptotic properties of a discretized version of the MLE for the parameter. We provide this asymptotic analysis of the proposed estimator as the number of Fourier modes, N, used in the estimation and the observation time go to infinity. Finally, we illustrate the theoretical results with some numerical experiments.
{"title":"Statistical inference for a stochastic wave equation with Malliavin–Stein method","authors":"F. Delgado-Vences, J. J. Pavon-Español","doi":"10.1080/07362994.2022.2029712","DOIUrl":"https://doi.org/10.1080/07362994.2022.2029712","url":null,"abstract":"Abstract In this paper, we study asymptotic properties of the maximum likelihood estimator (MLE) for the speed of a stochastic wave equation. We follow a well-known spectral approach to write the solution as a Fourier series, then we project the solution to a N-finite dimensional space and find the estimator as a function of the time and N. We then show consistency of the MLE using classical stochastic analysis. Afterward, we prove the asymptotic normality using the Malliavin–Stein method. We also study asymptotic properties of a discretized version of the MLE for the parameter. We provide this asymptotic analysis of the proposed estimator as the number of Fourier modes, N, used in the estimation and the observation time go to infinity. Finally, we illustrate the theoretical results with some numerical experiments.","PeriodicalId":49474,"journal":{"name":"Stochastic Analysis and Applications","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2022-02-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42216719","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 : 2021-12-28DOI: 10.1080/07362994.2021.2018335
M. Abundo
Abstract For Ornstein-Uhlenbeck process starting from we highlight some results about the first-passage time of X(t) through zero and its first-passage area, that is the random area swept out by till its first-passage through zero. We study single and joint moments of the first-passage time and first-passage area, and their behaviors, as and moreover, we investigate the expected value of the time average of X(t) till the FPT, and the maximum displacement of
{"title":"The first-passage area of Ornstein-Uhlenbeck process revisited","authors":"M. Abundo","doi":"10.1080/07362994.2021.2018335","DOIUrl":"https://doi.org/10.1080/07362994.2021.2018335","url":null,"abstract":"Abstract For Ornstein-Uhlenbeck process starting from we highlight some results about the first-passage time of X(t) through zero and its first-passage area, that is the random area swept out by till its first-passage through zero. We study single and joint moments of the first-passage time and first-passage area, and their behaviors, as and moreover, we investigate the expected value of the time average of X(t) till the FPT, and the maximum displacement of","PeriodicalId":49474,"journal":{"name":"Stochastic Analysis and Applications","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2021-12-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44967875","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 : 2021-12-21DOI: 10.1080/07362994.2021.2013889
Subrata Golui, Chandan Pal
Abstract
We consider zero-sum stochastic games for controlled continuous time Markov processes on a general state space with risk-sensitive discounted cost criteria. The transition and cost rates are possibly unbounded. Under a stability assumption, we prove the existence of a saddle-point equilibrium in the class of Markov strategies and give a characterization in terms of the corresponding Hamilton-Jacobi-Isaacs (HJI) equation. Also, we illustrate our results and assumptions by an example.
{"title":"Continuous-time zero-sum games for markov decision processes with discounted risk-sensitive cost criterion on a general state space","authors":"Subrata Golui, Chandan Pal","doi":"10.1080/07362994.2021.2013889","DOIUrl":"https://doi.org/10.1080/07362994.2021.2013889","url":null,"abstract":"<p><b>Abstract</b></p><p>We consider zero-sum stochastic games for controlled continuous time Markov processes on a general state space with risk-sensitive discounted cost criteria. The transition and cost rates are possibly unbounded. Under a stability assumption, we prove the existence of a saddle-point equilibrium in the class of Markov strategies and give a characterization in terms of the corresponding Hamilton-Jacobi-Isaacs (HJI) equation. Also, we illustrate our results and assumptions by an example.</p>","PeriodicalId":49474,"journal":{"name":"Stochastic Analysis and Applications","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2021-12-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138516186","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 : 2021-12-16DOI: 10.1080/07362994.2021.2011317
Kamal Hiderah
Abstract In this article, we aim to obtain the existence and pathwise uniqueness of the solution to the one-dimensional stochastic differential equations involving the local time (SDELT) at point zero. The existence and pathwise uniqueness theorem for class of SDELT is established under the drift coefficient satisfies a one-sided Lipschitz condition plus the superlinear condition.
{"title":"Existence and pathwise uniqueness of solutions for stochastic differential equations involving the local time at point zero","authors":"Kamal Hiderah","doi":"10.1080/07362994.2021.2011317","DOIUrl":"https://doi.org/10.1080/07362994.2021.2011317","url":null,"abstract":"Abstract In this article, we aim to obtain the existence and pathwise uniqueness of the solution to the one-dimensional stochastic differential equations involving the local time (SDELT) at point zero. The existence and pathwise uniqueness theorem for class of SDELT is established under the drift coefficient satisfies a one-sided Lipschitz condition plus the superlinear condition.","PeriodicalId":49474,"journal":{"name":"Stochastic Analysis and Applications","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2021-12-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46440122","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 : 2021-12-09DOI: 10.1080/07362994.2021.2002164
Mohamad S. Alwan, Xinzhi Liu, Taghreed G. Sugati, Humeyra Kiyak
Abstract This article addresses a class of large-scale stochastic impulsive systems with time delay and time-varying input disturbance having bounded magnitude. The main interest is to develop sufficient conditions for the input-to-state stability (ISS) and stabilization in the presence of impulsive effects. The method of Razumikhin–Lyapunov function is used to develop the ISS and stabilization properties. Later, these results are applied to a class of control systems where the controller actuators are susceptible to failures. It should be noted that our results are delay independent, and the designed reliable controller is robust with respect to the actuator failures and to the system uncertainties. It is also observed that if the isolated continuous system is ISS and subjected to bounded impulsive effects, then the resulting impulsive system preserves the ISS property. Moreover, if the isolated continuous subsystems are all ISS and the interconnection amongst them is bounded from above, then the impulsive interconnected system is ISS provided that the degree of stability of each subsystem is larger than the magnitude of interconnection. If the underlying continuous system is unstable, then the input-to-state stabilization of the impulsive system is guaranteed if the stabilizing impulses are applied to the system frequently. As an implication to these results, if the input disturbance is zero, then the input-to-state stability (or stabilization) reduces to the stability (or stabilization) of the equilibrium state of the underlying disturbance-free system. A numerical example and simulations are provided to illustrate the proposed results.
{"title":"Input-to-state stability for large-scale stochastic impulsive systems with state delay","authors":"Mohamad S. Alwan, Xinzhi Liu, Taghreed G. Sugati, Humeyra Kiyak","doi":"10.1080/07362994.2021.2002164","DOIUrl":"https://doi.org/10.1080/07362994.2021.2002164","url":null,"abstract":"Abstract This article addresses a class of large-scale stochastic impulsive systems with time delay and time-varying input disturbance having bounded magnitude. The main interest is to develop sufficient conditions for the input-to-state stability (ISS) and stabilization in the presence of impulsive effects. The method of Razumikhin–Lyapunov function is used to develop the ISS and stabilization properties. Later, these results are applied to a class of control systems where the controller actuators are susceptible to failures. It should be noted that our results are delay independent, and the designed reliable controller is robust with respect to the actuator failures and to the system uncertainties. It is also observed that if the isolated continuous system is ISS and subjected to bounded impulsive effects, then the resulting impulsive system preserves the ISS property. Moreover, if the isolated continuous subsystems are all ISS and the interconnection amongst them is bounded from above, then the impulsive interconnected system is ISS provided that the degree of stability of each subsystem is larger than the magnitude of interconnection. If the underlying continuous system is unstable, then the input-to-state stabilization of the impulsive system is guaranteed if the stabilizing impulses are applied to the system frequently. As an implication to these results, if the input disturbance is zero, then the input-to-state stability (or stabilization) reduces to the stability (or stabilization) of the equilibrium state of the underlying disturbance-free system. A numerical example and simulations are provided to illustrate the proposed results.","PeriodicalId":49474,"journal":{"name":"Stochastic Analysis and Applications","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2021-12-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43401083","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 : 2021-12-09DOI: 10.1080/07362994.2021.2007777
Adina Oprisan
Abstract A large deviation principle (LDP) for a class of additive functionals of semi-Markov processes and their associated Markov renewal processes is studied via an almost sure functional central limit theorem. The rate function corresponding to the deviations from the paths of the corresponding empirical processes with logarithmic averaging is determined as a relative entropy with respect to the Wiener measure on A martingale decomposition for additive functionals of Markov renewal processes is employed.
{"title":"Large deviation principle for additive functionals of semi-Markov processes","authors":"Adina Oprisan","doi":"10.1080/07362994.2021.2007777","DOIUrl":"https://doi.org/10.1080/07362994.2021.2007777","url":null,"abstract":"Abstract A large deviation principle (LDP) for a class of additive functionals of semi-Markov processes and their associated Markov renewal processes is studied via an almost sure functional central limit theorem. The rate function corresponding to the deviations from the paths of the corresponding empirical processes with logarithmic averaging is determined as a relative entropy with respect to the Wiener measure on A martingale decomposition for additive functionals of Markov renewal processes is employed.","PeriodicalId":49474,"journal":{"name":"Stochastic Analysis and Applications","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2021-12-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43705828","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 : 2021-11-16DOI: 10.1080/07362994.2022.2108450
Fausto Colantoni, M. D’Ovidio
Abstract In this paper we deal with some open problems concerned with Gamma subordinators. In particular, we first provide a representation for the moments of the inverse gamma subordinator. Then, we focus on λ-potentials and we study the governing equations associated with Gamma subordinators and inverse processes. Such representations are given in terms of higher transcendental functions, also known as Volterra functions.
{"title":"On the inverse gamma subordinator","authors":"Fausto Colantoni, M. D’Ovidio","doi":"10.1080/07362994.2022.2108450","DOIUrl":"https://doi.org/10.1080/07362994.2022.2108450","url":null,"abstract":"Abstract In this paper we deal with some open problems concerned with Gamma subordinators. In particular, we first provide a representation for the moments of the inverse gamma subordinator. Then, we focus on λ-potentials and we study the governing equations associated with Gamma subordinators and inverse processes. Such representations are given in terms of higher transcendental functions, also known as Volterra functions.","PeriodicalId":49474,"journal":{"name":"Stochastic Analysis and Applications","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2021-11-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41333341","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 : 2021-11-16DOI: 10.1080/07362994.2022.2079530
Zenghu Li, Fei Pu
Abstract Let be the density of one-dimensional super-Brownian motion starting from Lebesgue measure. Using the Laplace functional of super-Brownian motion, we prove that as the normalized spatial integral converges jointly in (t, x) to Brownian sheet in distribution.
{"title":"Gaussian fluctuation for spatial average of super-Brownian motion","authors":"Zenghu Li, Fei Pu","doi":"10.1080/07362994.2022.2079530","DOIUrl":"https://doi.org/10.1080/07362994.2022.2079530","url":null,"abstract":"Abstract Let be the density of one-dimensional super-Brownian motion starting from Lebesgue measure. Using the Laplace functional of super-Brownian motion, we prove that as the normalized spatial integral converges jointly in (t, x) to Brownian sheet in distribution.","PeriodicalId":49474,"journal":{"name":"Stochastic Analysis and Applications","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2021-11-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48463333","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 : 2021-11-01DOI: 10.1080/07362994.2021.1972008
R. Dhoyer, C. Tudor
Abstract We consider the stochastic heat equation driven by a Rosenblatt sheet and we study the limit behavior in distribution of the spatial average of the solution. By analyzing the cumulants of the solution, we prove that the spatial average converges weakly, in the space of continuous functions, to a Rosenblatt process.
{"title":"Spatial average for the solution to the heat equation with Rosenblatt noise","authors":"R. Dhoyer, C. Tudor","doi":"10.1080/07362994.2021.1972008","DOIUrl":"https://doi.org/10.1080/07362994.2021.1972008","url":null,"abstract":"Abstract We consider the stochastic heat equation driven by a Rosenblatt sheet and we study the limit behavior in distribution of the spatial average of the solution. By analyzing the cumulants of the solution, we prove that the spatial average converges weakly, in the space of continuous functions, to a Rosenblatt process.","PeriodicalId":49474,"journal":{"name":"Stochastic Analysis and Applications","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2021-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45914674","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 : 2021-10-27DOI: 10.1080/07362994.2021.1988856
P. T. Huong, P. Kloeden, Doan Thai Son
Abstract In the first part of this paper, we establish the well-posedness for solutions of Caputo stochastic fractional differential equations (for short Caputo SFDE) of order in Lp spaces with whose coefficients satisfy a standard Lipschitz condition. More precisely, we first show a result on the existence and uniqueness of solutions, next we show the continuous dependence of solutions on the initial values and on the fractional exponent α. The second part of this paper is devoted to studying the regularity in time for solutions of Caputo SFDE. As a consequence, we obtain that a solution of Caputo SFDE has a δ-Hölder continuous version for any The main ingredient in the proof is to use a temporally weighted norm and the Burkholder-Davis-Gundy inequality.
{"title":"Well-posedness and regularity for solutions of caputo stochastic fractional differential equations in Lp spaces","authors":"P. T. Huong, P. Kloeden, Doan Thai Son","doi":"10.1080/07362994.2021.1988856","DOIUrl":"https://doi.org/10.1080/07362994.2021.1988856","url":null,"abstract":"Abstract In the first part of this paper, we establish the well-posedness for solutions of Caputo stochastic fractional differential equations (for short Caputo SFDE) of order in Lp spaces with whose coefficients satisfy a standard Lipschitz condition. More precisely, we first show a result on the existence and uniqueness of solutions, next we show the continuous dependence of solutions on the initial values and on the fractional exponent α. The second part of this paper is devoted to studying the regularity in time for solutions of Caputo SFDE. As a consequence, we obtain that a solution of Caputo SFDE has a δ-Hölder continuous version for any The main ingredient in the proof is to use a temporally weighted norm and the Burkholder-Davis-Gundy inequality.","PeriodicalId":49474,"journal":{"name":"Stochastic Analysis and Applications","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2021-10-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48575840","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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