Pub Date : 2021-09-05DOI: 10.1080/07362994.2021.1967761
Priya Singh, S. Saha Ray
Abstract This article proposes two efficient methods to solve nonlinear stochastic Itô–Volterra integral equations. The shifted Jacobi spectral Galerkin method and shifted Jacobi operational matrix method have been applied to solve these equations. The presented methods convert this equation to a system of nonlinear algebraic equations and then Newton’s method have been implemented to solve the obtained algebraic equations numerically. Convergence analysis for both presented methods have been investigated. Also, the results obtained by proposed methods have been compared. The accuracy and reliability of the presented methods have been proved by some numerical instances.
{"title":"Two reliable methods for numerical solution of nonlinear stochastic Itô–Volterra integral equation","authors":"Priya Singh, S. Saha Ray","doi":"10.1080/07362994.2021.1967761","DOIUrl":"https://doi.org/10.1080/07362994.2021.1967761","url":null,"abstract":"Abstract This article proposes two efficient methods to solve nonlinear stochastic Itô–Volterra integral equations. The shifted Jacobi spectral Galerkin method and shifted Jacobi operational matrix method have been applied to solve these equations. The presented methods convert this equation to a system of nonlinear algebraic equations and then Newton’s method have been implemented to solve the obtained algebraic equations numerically. Convergence analysis for both presented methods have been investigated. Also, the results obtained by proposed methods have been compared. The accuracy and reliability of the presented methods have been proved by some numerical instances.","PeriodicalId":49474,"journal":{"name":"Stochastic Analysis and Applications","volume":"40 1","pages":"891 - 913"},"PeriodicalIF":1.3,"publicationDate":"2021-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47342775","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 : 2021-09-03DOI: 10.1080/07362994.2020.1844022
Fulgence Eyi Obiang
Abstract Contributions of the present paper consist of two parts. In the first one, we contribute to the theory of stochastic calculus for signed measures. For instance, we provide some results permitting to characterize martingales and Brownian motion both defined under a signed measure. We also prove that the uniformly integrable martingales (defined with respect to a signed measure) can be expressed as relative martingales and we provide some new results to the study of the class . The second part is devoted to the construction of solutions for the homogeneous skew Brownian motion equation and for the inhomogeneous skew Brownian motion equation. To do this, our ingredients are the techniques and results developed in the first part that we apply on some stochastic processes borrowed from the theory of stochastic calculus for signed measures. Our methods are inspired by those used by Bouhadou and Ouknine in [2013]. Moreover, their solution of the inhomogeneous skew Brownian motion equation is a particular case of those we propose in this paper.
{"title":"Resolution of the skew Brownian motion equations with stochastic calculus for signed measures","authors":"Fulgence Eyi Obiang","doi":"10.1080/07362994.2020.1844022","DOIUrl":"https://doi.org/10.1080/07362994.2020.1844022","url":null,"abstract":"Abstract Contributions of the present paper consist of two parts. In the first one, we contribute to the theory of stochastic calculus for signed measures. For instance, we provide some results permitting to characterize martingales and Brownian motion both defined under a signed measure. We also prove that the uniformly integrable martingales (defined with respect to a signed measure) can be expressed as relative martingales and we provide some new results to the study of the class . The second part is devoted to the construction of solutions for the homogeneous skew Brownian motion equation and for the inhomogeneous skew Brownian motion equation. To do this, our ingredients are the techniques and results developed in the first part that we apply on some stochastic processes borrowed from the theory of stochastic calculus for signed measures. Our methods are inspired by those used by Bouhadou and Ouknine in [2013]. Moreover, their solution of the inhomogeneous skew Brownian motion equation is a particular case of those we propose in this paper.","PeriodicalId":49474,"journal":{"name":"Stochastic Analysis and Applications","volume":"39 1","pages":"775 - 803"},"PeriodicalIF":1.3,"publicationDate":"2021-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/07362994.2020.1844022","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"59595499","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 : 2021-09-02DOI: 10.1080/07362994.2021.1967171
Ling Qin, Dandan Ma, J. Shu
Abstract In this paper, we study the Wong-Zakai approximations and long term behavior of the stochastic FitzHugh-Nagumo system driven by a white noise. We first prove the existence and uniqueness of tempered pullback attractors for the Wong-Zakai approximation system. Then we show that the attractors of Wong-Zakai approximations converges to the attractor of the stochastic FitzHugh-Nagumo system for both additive and multiplicative noise on unbounded domains. The tail estimates and decomposition method are employed to derive the pullback asymptotic compactness of solutions in order to overcome the obstacles caused by the non-compactness of Sobolev embeddings on unbounded domains as well as the absence of regularity of one component of the solutions.
{"title":"Wong-Zakai approximations and attractors for non-autonomous stochastic FitzHugh-Nagumo system on unbounded domains","authors":"Ling Qin, Dandan Ma, J. Shu","doi":"10.1080/07362994.2021.1967171","DOIUrl":"https://doi.org/10.1080/07362994.2021.1967171","url":null,"abstract":"Abstract In this paper, we study the Wong-Zakai approximations and long term behavior of the stochastic FitzHugh-Nagumo system driven by a white noise. We first prove the existence and uniqueness of tempered pullback attractors for the Wong-Zakai approximation system. Then we show that the attractors of Wong-Zakai approximations converges to the attractor of the stochastic FitzHugh-Nagumo system for both additive and multiplicative noise on unbounded domains. The tail estimates and decomposition method are employed to derive the pullback asymptotic compactness of solutions in order to overcome the obstacles caused by the non-compactness of Sobolev embeddings on unbounded domains as well as the absence of regularity of one component of the solutions.","PeriodicalId":49474,"journal":{"name":"Stochastic Analysis and Applications","volume":"40 1","pages":"854 - 890"},"PeriodicalIF":1.3,"publicationDate":"2021-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46866527","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 : 2021-08-24DOI: 10.1080/07362994.2021.1947855
U. Manna, D. Mukherjee
Abstract We investigate the existence of a weak martingale solution for a two-dimensional critical viscoelastic flow of the Oldroyd type driven by pure jump Lévy noise. Due to the viscoelastic nature, noise in the equation modeling stress tensor is considered in the Marcus canonical form. Owing to the lack of dissipation and taking into account of the structure of the non-linear terms, the proof requires higher order estimates.
{"title":"Weak martingale solution of stochastic critical Oldroyd-B type models perturbed by pure jump noise","authors":"U. Manna, D. Mukherjee","doi":"10.1080/07362994.2021.1947855","DOIUrl":"https://doi.org/10.1080/07362994.2021.1947855","url":null,"abstract":"Abstract We investigate the existence of a weak martingale solution for a two-dimensional critical viscoelastic flow of the Oldroyd type driven by pure jump Lévy noise. Due to the viscoelastic nature, noise in the equation modeling stress tensor is considered in the Marcus canonical form. Owing to the lack of dissipation and taking into account of the structure of the non-linear terms, the proof requires higher order estimates.","PeriodicalId":49474,"journal":{"name":"Stochastic Analysis and Applications","volume":"40 1","pages":"657 - 690"},"PeriodicalIF":1.3,"publicationDate":"2021-08-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45127486","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 : 2021-08-20DOI: 10.1080/07362994.2021.1962343
Shraddha Ramdas Bandekar, M. Ghosh
Abstract India, unlike several other countries, has witnessed a greater challenge in overcoming the pandemic during the second wave crisis. The strategies on lockdown imposition during the first wave of the pandemic were well implemented due to which the year-long wave did not witness uncontrollable surges in the infections. In this study, we present a detailed study on the disease spread during the first and second wave of COVID-19, by performing numerical simulation in a phase-wise manner as per the lockdown and unlock phases implemented in India. The inclusion of a piecewise treatment function in the framed epidemiological model is a noteworthy aspect of the study since this function takes into consideration the availability of medical equipment, based upon the threshold number of infections. This function framed in this model first grows linearly reaching a peak, then it declines due to shortage of medical resources and finally gets saturated. Analysis of the impact of lockdown is presented for each phase of the pandemic in India. Though the data considered for the study is for a period that marked the beginning of the pandemic, major analysis and predictions are presented based on the second wave data in terms of sensitivity analysis and time series behavior. A comparison of deterministic and stochastic differential equations is presented with simulation results on certain parameter set to examine variation in treatment and recovery. The simulations are performed using MATLAB and R softwares. The work is validated with the real data and model fitting is done applying the Maximum Likelihood method. The study implies that, under accurate lockdown strategies and sufficient medical care, the peak in cases would be attained by 16 May 2021, after which a decline in the cases could be observed.
{"title":"Modeling and analysis of COVID-19 in India with treatment function through different phases of lockdown and unlock","authors":"Shraddha Ramdas Bandekar, M. Ghosh","doi":"10.1080/07362994.2021.1962343","DOIUrl":"https://doi.org/10.1080/07362994.2021.1962343","url":null,"abstract":"Abstract India, unlike several other countries, has witnessed a greater challenge in overcoming the pandemic during the second wave crisis. The strategies on lockdown imposition during the first wave of the pandemic were well implemented due to which the year-long wave did not witness uncontrollable surges in the infections. In this study, we present a detailed study on the disease spread during the first and second wave of COVID-19, by performing numerical simulation in a phase-wise manner as per the lockdown and unlock phases implemented in India. The inclusion of a piecewise treatment function in the framed epidemiological model is a noteworthy aspect of the study since this function takes into consideration the availability of medical equipment, based upon the threshold number of infections. This function framed in this model first grows linearly reaching a peak, then it declines due to shortage of medical resources and finally gets saturated. Analysis of the impact of lockdown is presented for each phase of the pandemic in India. Though the data considered for the study is for a period that marked the beginning of the pandemic, major analysis and predictions are presented based on the second wave data in terms of sensitivity analysis and time series behavior. A comparison of deterministic and stochastic differential equations is presented with simulation results on certain parameter set to examine variation in treatment and recovery. The simulations are performed using MATLAB and R softwares. The work is validated with the real data and model fitting is done applying the Maximum Likelihood method. The study implies that, under accurate lockdown strategies and sufficient medical care, the peak in cases would be attained by 16 May 2021, after which a decline in the cases could be observed.","PeriodicalId":49474,"journal":{"name":"Stochastic Analysis and Applications","volume":"40 1","pages":"812 - 829"},"PeriodicalIF":1.3,"publicationDate":"2021-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42224742","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 : 2021-08-20DOI: 10.1080/07362994.2021.1963776
Qun Liu, D. Jiang
Abstract In this paper, we analyze a stochastic multigroup DS-DI-A model for the transmission of HIV. We establish sufficient conditions for the existence and uniqueness of an ergodic stationary distribution of positive solutions to the system by constructing a series of suitable Lyapunov functions. Moreover, we make up adequate conditions for complete eradication and wiping out of the diseases. In a biological viewpoint, the existence of a stationary distribution shows that the diseases will be prevalent and persistent in the long term.
{"title":"Stationary distribution and extinction of a stochastic multigroup DS-DI-a model for the transmission of HIV","authors":"Qun Liu, D. Jiang","doi":"10.1080/07362994.2021.1963776","DOIUrl":"https://doi.org/10.1080/07362994.2021.1963776","url":null,"abstract":"Abstract In this paper, we analyze a stochastic multigroup DS-DI-A model for the transmission of HIV. We establish sufficient conditions for the existence and uniqueness of an ergodic stationary distribution of positive solutions to the system by constructing a series of suitable Lyapunov functions. Moreover, we make up adequate conditions for complete eradication and wiping out of the diseases. In a biological viewpoint, the existence of a stationary distribution shows that the diseases will be prevalent and persistent in the long term.","PeriodicalId":49474,"journal":{"name":"Stochastic Analysis and Applications","volume":"40 1","pages":"830 - 853"},"PeriodicalIF":1.3,"publicationDate":"2021-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46894874","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 : 2021-08-17DOI: 10.1080/07362994.2021.1960565
Hua Zhang
ABSTRACT In this paper, the problem of the large deviations for the invariant measures of the multivalued stochastic differential equations is considered. Under the assumptions of diffusion coefficient being non-Lipschitz and elliptic, we establish the large deviation principle for the invariant measures of the solutions to the multivalued stochastic differential equations. The proof is based on the work of large deviations and invariant measures for the solutions to the multivalued stochastic differential equations.
{"title":"Large deviations for invariant measures of multivalued stochastic differential equations","authors":"Hua Zhang","doi":"10.1080/07362994.2021.1960565","DOIUrl":"https://doi.org/10.1080/07362994.2021.1960565","url":null,"abstract":"ABSTRACT In this paper, the problem of the large deviations for the invariant measures of the multivalued stochastic differential equations is considered. Under the assumptions of diffusion coefficient being non-Lipschitz and elliptic, we establish the large deviation principle for the invariant measures of the solutions to the multivalued stochastic differential equations. The proof is based on the work of large deviations and invariant measures for the solutions to the multivalued stochastic differential equations.","PeriodicalId":49474,"journal":{"name":"Stochastic Analysis and Applications","volume":"40 1","pages":"798 - 811"},"PeriodicalIF":1.3,"publicationDate":"2021-08-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45615163","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 : 2021-08-17DOI: 10.1080/07362994.2021.1959349
N. Medhin, C. Xu
Abstract An optimal asset allocation problem involving restrictions on liquidity is studied in this article. The portfolio consists of liquid and illiquid asset. The portfolio is only allowed to rebalance at particular times. An investor tries to maximize the total utility of a hyperbolic absolute risk aversion function depending on the consumption, which is sourced only from the liquid asset. The optimal policies of the consumption, investment, and allocation are derived. A numerical approximation scheme is developed to show the optimal allocation policy in our model is path-dependent. Paths of the value function and other optimal controls are illustrated to validate our results.
{"title":"Optimal asset allocation with restrictions on liquidity","authors":"N. Medhin, C. Xu","doi":"10.1080/07362994.2021.1959349","DOIUrl":"https://doi.org/10.1080/07362994.2021.1959349","url":null,"abstract":"Abstract An optimal asset allocation problem involving restrictions on liquidity is studied in this article. The portfolio consists of liquid and illiquid asset. The portfolio is only allowed to rebalance at particular times. An investor tries to maximize the total utility of a hyperbolic absolute risk aversion function depending on the consumption, which is sourced only from the liquid asset. The optimal policies of the consumption, investment, and allocation are derived. A numerical approximation scheme is developed to show the optimal allocation policy in our model is path-dependent. Paths of the value function and other optimal controls are illustrated to validate our results.","PeriodicalId":49474,"journal":{"name":"Stochastic Analysis and Applications","volume":"40 1","pages":"776 - 797"},"PeriodicalIF":1.3,"publicationDate":"2021-08-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44361087","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 : 2021-08-12DOI: 10.1080/07362994.2021.1958687
Xing Huang, Fen-Fen Yang
Abstract Sufficient and necessary conditions are presented for the comparison theorem of path dependent G-SDEs. Different from the corresponding study in path independent G-SDEs, a probability method is applied to prove these results. Moreover, the results extend the ones in the linear expectation case.
{"title":"Comparison theorem for path dependent SDEs driven by G-Brownian motion","authors":"Xing Huang, Fen-Fen Yang","doi":"10.1080/07362994.2021.1958687","DOIUrl":"https://doi.org/10.1080/07362994.2021.1958687","url":null,"abstract":"Abstract Sufficient and necessary conditions are presented for the comparison theorem of path dependent G-SDEs. Different from the corresponding study in path independent G-SDEs, a probability method is applied to prove these results. Moreover, the results extend the ones in the linear expectation case.","PeriodicalId":49474,"journal":{"name":"Stochastic Analysis and Applications","volume":"40 1","pages":"765 - 775"},"PeriodicalIF":1.3,"publicationDate":"2021-08-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"59595708","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 : 2021-08-12DOI: 10.1080/07362994.2021.1950551
Trung-Thuy Kieu, D. Luong, H. Ngo
Abstract We propose a tamed-adaptive Euler-Maruyama approximation scheme for stochastic differential equations with locally Lipschitz continuous, polynomial growth drift, and locally Hölder continuous, polynomial growth diffusion coefficients. We consider the strong convergence and the stability of the new scheme. In particular, we show that under some sufficient conditions for the stability of the exact solution, the tamed-adaptive scheme converges strongly in both finite and infinite time intervals.
{"title":"Tamed-adaptive Euler-Maruyama approximation for SDEs with locally Lipschitz continuous drift and locally Hölder continuous diffusion coefficients","authors":"Trung-Thuy Kieu, D. Luong, H. Ngo","doi":"10.1080/07362994.2021.1950551","DOIUrl":"https://doi.org/10.1080/07362994.2021.1950551","url":null,"abstract":"Abstract We propose a tamed-adaptive Euler-Maruyama approximation scheme for stochastic differential equations with locally Lipschitz continuous, polynomial growth drift, and locally Hölder continuous, polynomial growth diffusion coefficients. We consider the strong convergence and the stability of the new scheme. In particular, we show that under some sufficient conditions for the stability of the exact solution, the tamed-adaptive scheme converges strongly in both finite and infinite time intervals.","PeriodicalId":49474,"journal":{"name":"Stochastic Analysis and Applications","volume":"40 1","pages":"714 - 734"},"PeriodicalIF":1.3,"publicationDate":"2021-08-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49119070","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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