Pub Date : 2021-08-01DOI: 10.1080/07362994.2021.1953386
N. Leonenko, E. Pirozzi
Abstract We consider some time-changed diffusion processes obtained by applying the Doob transformation rule to a time-changed Brownian motion. The time-change is obtained via the inverse of an α-stable subordinator. These processes are specified in terms of time-changed Gauss-Markov processes and fractional time-changed diffusions. A fractional pseudo-Fokker-Planck equation for such processes is given. We investigate their first passage time densities providing a generalized integral equation they satisfy and some transformation rules. First passage time densities for time-changed Brownian motion and Ornstein-Uhlenbeck processes are provided in several forms. Connections with closed form results and numerical evaluations through the level zero are given.
{"title":"First passage times for some classes of fractional time-changed diffusions","authors":"N. Leonenko, E. Pirozzi","doi":"10.1080/07362994.2021.1953386","DOIUrl":"https://doi.org/10.1080/07362994.2021.1953386","url":null,"abstract":"Abstract We consider some time-changed diffusion processes obtained by applying the Doob transformation rule to a time-changed Brownian motion. The time-change is obtained via the inverse of an α-stable subordinator. These processes are specified in terms of time-changed Gauss-Markov processes and fractional time-changed diffusions. A fractional pseudo-Fokker-Planck equation for such processes is given. We investigate their first passage time densities providing a generalized integral equation they satisfy and some transformation rules. First passage time densities for time-changed Brownian motion and Ornstein-Uhlenbeck processes are provided in several forms. Connections with closed form results and numerical evaluations through the level zero are given.","PeriodicalId":49474,"journal":{"name":"Stochastic Analysis and Applications","volume":"40 1","pages":"735 - 763"},"PeriodicalIF":1.3,"publicationDate":"2021-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/07362994.2021.1953386","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42539227","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 : 2021-07-25DOI: 10.1080/07362994.2021.1944876
Qun Liu, D. Jiang
Abstract In this paper, we analyze the salient features of a stochastic multigroup SEI epidemic model. We obtain sufficient criteria for the existence and uniqueness of an ergodic stationary distribution of positive solutions to the system by establishing a series of suitable Lyapunov functions. In a biological viewpoint, the existence of a stationary distribution indicates that the diseases will be prevalent and persistent in the long term. In addition, we make up adequate conditions for complete eradication and wiping out of the diseases. Some numerical simulations are presented to illustrate our main results.
{"title":"Dynamics of a stochastic multigroup SEI epidemic model","authors":"Qun Liu, D. Jiang","doi":"10.1080/07362994.2021.1944876","DOIUrl":"https://doi.org/10.1080/07362994.2021.1944876","url":null,"abstract":"Abstract In this paper, we analyze the salient features of a stochastic multigroup SEI epidemic model. We obtain sufficient criteria for the existence and uniqueness of an ergodic stationary distribution of positive solutions to the system by establishing a series of suitable Lyapunov functions. In a biological viewpoint, the existence of a stationary distribution indicates that the diseases will be prevalent and persistent in the long term. In addition, we make up adequate conditions for complete eradication and wiping out of the diseases. Some numerical simulations are presented to illustrate our main results.","PeriodicalId":49474,"journal":{"name":"Stochastic Analysis and Applications","volume":"40 1","pages":"623 - 656"},"PeriodicalIF":1.3,"publicationDate":"2021-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/07362994.2021.1944876","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49386229","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 : 2021-07-25DOI: 10.1080/07362994.2021.1950013
Yajuan Pan, Hui Jiang
ABSTRACT In this article, for some quadratic functionals of linear self-repelling diffusion process, we study the asymptotic properties, including the deviation inequalities and Cramér-type moderate deviations. The main methods consist of the deviation inequalities for multiple Wiener-Itô integrals, as well as the asymptotic analysis techiniques. As applications, (self-normalized) Cramér-type moderate deviations for the log-likelihood ratio process and drift parameter estimator are obtained.
{"title":"Asymptotic properties for quadratic functionals of linear self-repelling diffusion process and applications","authors":"Yajuan Pan, Hui Jiang","doi":"10.1080/07362994.2021.1950013","DOIUrl":"https://doi.org/10.1080/07362994.2021.1950013","url":null,"abstract":"ABSTRACT In this article, for some quadratic functionals of linear self-repelling diffusion process, we study the asymptotic properties, including the deviation inequalities and Cramér-type moderate deviations. The main methods consist of the deviation inequalities for multiple Wiener-Itô integrals, as well as the asymptotic analysis techiniques. As applications, (self-normalized) Cramér-type moderate deviations for the log-likelihood ratio process and drift parameter estimator are obtained.","PeriodicalId":49474,"journal":{"name":"Stochastic Analysis and Applications","volume":"40 1","pages":"691 - 713"},"PeriodicalIF":1.3,"publicationDate":"2021-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/07362994.2021.1950013","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48322092","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 : 2021-07-21DOI: 10.1080/07362994.2022.2053541
M. Kov'acs, A. Lang, A. Petersson
Abstract Regularity estimates for an integral operator with a symmetric continuous kernel on a convex bounded domain are derived. The covariance of a mean-square continuous random field on the domain is an example of such an operator. The estimates are of the form of Hilbert–Schmidt norms of the integral operator and its square root, composed with fractional powers of an elliptic operator equipped with homogeneous boundary conditions of either Dirichlet or Neumann type. These types of estimates, which couple the regularity of the driving noise with the properties of the differential operator, have important implications for stochastic partial differential equations on bounded domains as well as their numerical approximations. The main tools used to derive the estimates are properties of reproducing kernel Hilbert spaces of functions on bounded domains along with Hilbert–Schmidt embeddings of Sobolev spaces. Both non-homogeneous and homogeneous kernels are considered. In the latter case, results in a general Schatten class norm are also provided. Important examples of homogeneous kernels covered by the results of the paper include the class of Matérn kernels.
{"title":"Hilbert–Schmidt regularity of symmetric integral operators on bounded domains with applications to SPDE approximations","authors":"M. Kov'acs, A. Lang, A. Petersson","doi":"10.1080/07362994.2022.2053541","DOIUrl":"https://doi.org/10.1080/07362994.2022.2053541","url":null,"abstract":"Abstract Regularity estimates for an integral operator with a symmetric continuous kernel on a convex bounded domain are derived. The covariance of a mean-square continuous random field on the domain is an example of such an operator. The estimates are of the form of Hilbert–Schmidt norms of the integral operator and its square root, composed with fractional powers of an elliptic operator equipped with homogeneous boundary conditions of either Dirichlet or Neumann type. These types of estimates, which couple the regularity of the driving noise with the properties of the differential operator, have important implications for stochastic partial differential equations on bounded domains as well as their numerical approximations. The main tools used to derive the estimates are properties of reproducing kernel Hilbert spaces of functions on bounded domains along with Hilbert–Schmidt embeddings of Sobolev spaces. Both non-homogeneous and homogeneous kernels are considered. In the latter case, results in a general Schatten class norm are also provided. Important examples of homogeneous kernels covered by the results of the paper include the class of Matérn kernels.","PeriodicalId":49474,"journal":{"name":"Stochastic Analysis and Applications","volume":"41 1","pages":"564 - 590"},"PeriodicalIF":1.3,"publicationDate":"2021-07-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44593391","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 : 2021-07-06DOI: 10.1080/07362994.2021.1990083
F. Biagini, Thomas Reitsam
Abstract We extend the super-replication theorem in a dynamic setting, both in the numéraire-based as well as in the numéraire-free setting. For this purpose, we generalize the notion of admissible strategies. In particular, we obtain a well-defined super-replication price process, which is right-continuous under some regularity assumptions.
{"title":"A dynamic version of the super-replication theorem under proportional transaction costs","authors":"F. Biagini, Thomas Reitsam","doi":"10.1080/07362994.2021.1990083","DOIUrl":"https://doi.org/10.1080/07362994.2021.1990083","url":null,"abstract":"Abstract We extend the super-replication theorem in a dynamic setting, both in the numéraire-based as well as in the numéraire-free setting. For this purpose, we generalize the notion of admissible strategies. In particular, we obtain a well-defined super-replication price process, which is right-continuous under some regularity assumptions.","PeriodicalId":49474,"journal":{"name":"Stochastic Analysis and Applications","volume":"41 1","pages":"80 - 101"},"PeriodicalIF":1.3,"publicationDate":"2021-07-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49599869","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 : 2021-06-30DOI: 10.1080/07362994.2021.2021091
L. Beghin, M. Caputo
Abstract We consider here convolution operators, in the Caputo sense, with nonsingular kernels. We prove that the solutions to some integro-differential equations with such operators (acting on the space variable) coincide with the transition densities of a particular class of Lévy subordinators (i.e. compound Poisson processes with non-negative jumps). We then extend these results to the case where the kernels of the operators have random parameters, with given distribution. This assumption allows greater flexibility in the choice of the kernel’s parameters and, consequently, of the jumps’ density function.
{"title":"Stochastic applications of Caputo-type convolution operators with nonsingular kernels","authors":"L. Beghin, M. Caputo","doi":"10.1080/07362994.2021.2021091","DOIUrl":"https://doi.org/10.1080/07362994.2021.2021091","url":null,"abstract":"Abstract We consider here convolution operators, in the Caputo sense, with nonsingular kernels. We prove that the solutions to some integro-differential equations with such operators (acting on the space variable) coincide with the transition densities of a particular class of Lévy subordinators (i.e. compound Poisson processes with non-negative jumps). We then extend these results to the case where the kernels of the operators have random parameters, with given distribution. This assumption allows greater flexibility in the choice of the kernel’s parameters and, consequently, of the jumps’ density function.","PeriodicalId":49474,"journal":{"name":"Stochastic Analysis and Applications","volume":"41 1","pages":"377 - 393"},"PeriodicalIF":1.3,"publicationDate":"2021-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43431785","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 : 2021-06-28DOI: 10.1080/07362994.2021.1942046
M. Mariani, Osei K. Tweneboah
Abstract The main task of this paper is to model the dependency and effects of the Lehman Brothers financial collapse event using a superposed and coupled Ornstein-Uhlenbeck type system of stochastic differential equations driven by a Lévy process. The development of these types of efficient models to correctly quantify and predict the sample paths of these kinds of time series is essential since it helps prevent losses or maximize profits in the field of financial modeling. The results obtained from this study suggest that the solutions of the stochastic models provide a good fit to the high frequency financial stock market data since it captures realistic dependence structures. In addition, the estimated model parameters are useful for making inferences and predicting these types of events.
{"title":"Modeling high frequency stock market data by using stochastic models","authors":"M. Mariani, Osei K. Tweneboah","doi":"10.1080/07362994.2021.1942046","DOIUrl":"https://doi.org/10.1080/07362994.2021.1942046","url":null,"abstract":"Abstract The main task of this paper is to model the dependency and effects of the Lehman Brothers financial collapse event using a superposed and coupled Ornstein-Uhlenbeck type system of stochastic differential equations driven by a Lévy process. The development of these types of efficient models to correctly quantify and predict the sample paths of these kinds of time series is essential since it helps prevent losses or maximize profits in the field of financial modeling. The results obtained from this study suggest that the solutions of the stochastic models provide a good fit to the high frequency financial stock market data since it captures realistic dependence structures. In addition, the estimated model parameters are useful for making inferences and predicting these types of events.","PeriodicalId":49474,"journal":{"name":"Stochastic Analysis and Applications","volume":"40 1","pages":"573 - 588"},"PeriodicalIF":1.3,"publicationDate":"2021-06-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/07362994.2021.1942046","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42735285","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 : 2021-06-28DOI: 10.1080/07362994.2021.1942917
K. K. Kataria, P. Vellaisamy, Vijay Kumar
Abstract In this paper, we obtain a probabilistic relationship between the exponential Bell polynomials and the weighted sums of independent Poisson random variables. A recently established probabilistic connection between the Adomian polynomials and independent Poisson random variables can be derived from the obtained relationship. This result has importance because any known identity for the exponential Bell polynomials will generate a new identity for the Poisson random variables. We use the obtained relationship to derive several new identities for the joint distribution of weighted sums of independent Poisson random variables. Few examples are provided that substantiate the obtained identities.
{"title":"A probabilistic interpretation of the Bell polynomials","authors":"K. K. Kataria, P. Vellaisamy, Vijay Kumar","doi":"10.1080/07362994.2021.1942917","DOIUrl":"https://doi.org/10.1080/07362994.2021.1942917","url":null,"abstract":"Abstract In this paper, we obtain a probabilistic relationship between the exponential Bell polynomials and the weighted sums of independent Poisson random variables. A recently established probabilistic connection between the Adomian polynomials and independent Poisson random variables can be derived from the obtained relationship. This result has importance because any known identity for the exponential Bell polynomials will generate a new identity for the Poisson random variables. We use the obtained relationship to derive several new identities for the joint distribution of weighted sums of independent Poisson random variables. Few examples are provided that substantiate the obtained identities.","PeriodicalId":49474,"journal":{"name":"Stochastic Analysis and Applications","volume":"40 1","pages":"610 - 622"},"PeriodicalIF":1.3,"publicationDate":"2021-06-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/07362994.2021.1942917","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45899585","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 : 2021-06-23DOI: 10.1080/07362994.2021.2022495
Khalifa Es-Sebaiy
Abstract In the present paper we study the asymptotic behavior of the auto-covariance function for Ornstein–Uhlenbeck (OU) processes driven by Gaussian noises with stationary and non-stationary increments and for Hermite OU processes. Our results are generalizations of the corresponding results of Cheridito et al. and Kaarakka and Salminen.
{"title":"Gaussian and hermite Ornstein–Uhlenbeck processes","authors":"Khalifa Es-Sebaiy","doi":"10.1080/07362994.2021.2022495","DOIUrl":"https://doi.org/10.1080/07362994.2021.2022495","url":null,"abstract":"Abstract In the present paper we study the asymptotic behavior of the auto-covariance function for Ornstein–Uhlenbeck (OU) processes driven by Gaussian noises with stationary and non-stationary increments and for Hermite OU processes. Our results are generalizations of the corresponding results of Cheridito et al. and Kaarakka and Salminen.","PeriodicalId":49474,"journal":{"name":"Stochastic Analysis and Applications","volume":"41 1","pages":"394 - 423"},"PeriodicalIF":1.3,"publicationDate":"2021-06-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47093016","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 : 2021-06-22DOI: 10.1080/07362994.2022.2055574
Krzysztof Bisewski, E. Hashorva, G. Shevchenko
Abstract Motivated by the classical harmonic mean formula, estabished by Aldous in 1989, we investigate the relation between the sojourn time and supremum of a random process and extend the harmonic mean formula for general stochastically continuous X. We discuss two applications concerning the continuity of distribution of supremum of X and representations of classical Pickands constants.
{"title":"The harmonic mean formula for random processes","authors":"Krzysztof Bisewski, E. Hashorva, G. Shevchenko","doi":"10.1080/07362994.2022.2055574","DOIUrl":"https://doi.org/10.1080/07362994.2022.2055574","url":null,"abstract":"Abstract Motivated by the classical harmonic mean formula, estabished by Aldous in 1989, we investigate the relation between the sojourn time and supremum of a random process and extend the harmonic mean formula for general stochastically continuous X. We discuss two applications concerning the continuity of distribution of supremum of X and representations of classical Pickands constants.","PeriodicalId":49474,"journal":{"name":"Stochastic Analysis and Applications","volume":"41 1","pages":"591 - 603"},"PeriodicalIF":1.3,"publicationDate":"2021-06-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47760944","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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