Pub Date : 2022-06-17DOI: 10.1080/17442508.2022.2088236
Jingyu Li, Yong Zhang
Consider the parabolic Anderson model of the form , where for t>0 and with , and η is a centered Gaussian noise that is white in time and has a spatially homogeneous covariance given by a nonnegative-definite measure f that satisfies Dalang's condition. Let denote the standard Gaussian heat kernel on and set for all t>0 and . In this paper, we present an almost sure central limit theorem (ASCLT) and a functional ASCLT for spatial averages of the form as for fixed t>0 based on the quantitative analysis of f. In particular, when f is given by a Riesz kernel, that is, for some , we can also obtain the ASCLT.
{"title":"An almost sure central limit theorem for the parabolic Anderson model with delta initial condition","authors":"Jingyu Li, Yong Zhang","doi":"10.1080/17442508.2022.2088236","DOIUrl":"https://doi.org/10.1080/17442508.2022.2088236","url":null,"abstract":"Consider the parabolic Anderson model of the form , where for t>0 and with , and η is a centered Gaussian noise that is white in time and has a spatially homogeneous covariance given by a nonnegative-definite measure f that satisfies Dalang's condition. Let denote the standard Gaussian heat kernel on and set for all t>0 and . In this paper, we present an almost sure central limit theorem (ASCLT) and a functional ASCLT for spatial averages of the form as for fixed t>0 based on the quantitative analysis of f. In particular, when f is given by a Riesz kernel, that is, for some , we can also obtain the ASCLT.","PeriodicalId":50447,"journal":{"name":"Finance and Stochastics","volume":"1 1","pages":"483 - 500"},"PeriodicalIF":1.7,"publicationDate":"2022-06-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88667565","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"经济学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-06-11DOI: 10.1080/17442508.2022.2085503
Yufeng Shi, Jiaqiang Wen, J. Xiong
In this paper, a class of non-Markovian forward-backward doubly stochastic systems is studied. By using the technique of functional Itô (or path-dependent) calculus, the relationship between the systems and related path-dependent quasi-linear stochastic partial differential equations (SPDEs in short) is established, and the well-known nonlinear stochastic Feynman-Kac formula of Pardoux and Peng [Backward doubly stochastic differential equations and systems of quasilinear SPDEs, Probab. Theory Relat. Fields 98 (1994), pp. 209–227.] is developed to the non-Markovian situation. Moreover, we obtain the differentiability of the solution to the forward-backward doubly stochastic systems and some properties of solutions to the path-dependent SPDEs.
{"title":"Forward-backward doubly stochastic systems and classical solutions of path-dependent stochastic PDEs","authors":"Yufeng Shi, Jiaqiang Wen, J. Xiong","doi":"10.1080/17442508.2022.2085503","DOIUrl":"https://doi.org/10.1080/17442508.2022.2085503","url":null,"abstract":"In this paper, a class of non-Markovian forward-backward doubly stochastic systems is studied. By using the technique of functional Itô (or path-dependent) calculus, the relationship between the systems and related path-dependent quasi-linear stochastic partial differential equations (SPDEs in short) is established, and the well-known nonlinear stochastic Feynman-Kac formula of Pardoux and Peng [Backward doubly stochastic differential equations and systems of quasilinear SPDEs, Probab. Theory Relat. Fields 98 (1994), pp. 209–227.] is developed to the non-Markovian situation. Moreover, we obtain the differentiability of the solution to the forward-backward doubly stochastic systems and some properties of solutions to the path-dependent SPDEs.","PeriodicalId":50447,"journal":{"name":"Finance and Stochastics","volume":"10 1","pages":"396 - 422"},"PeriodicalIF":1.7,"publicationDate":"2022-06-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87970437","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"经济学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-06-08DOI: 10.1007/s00780-023-00501-5
Donghan Kim
{"title":"Market-to-book ratio in stochastic portfolio theory","authors":"Donghan Kim","doi":"10.1007/s00780-023-00501-5","DOIUrl":"https://doi.org/10.1007/s00780-023-00501-5","url":null,"abstract":"","PeriodicalId":50447,"journal":{"name":"Finance and Stochastics","volume":"1 1","pages":"1 - 34"},"PeriodicalIF":1.7,"publicationDate":"2022-06-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46978331","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"经济学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-05-27DOI: 10.1080/17442508.2023.2238954
Sefika Kuzgun, D. Nualart
In this paper, we present a rate of convergence in the uniform norm for the densities of spatial averages of the solution to the d-dimensional parabolic Anderson model driven by a Gaussian multiplicative noise, which is white in time and has a spatial covariance given by the Riesz kernel. The proof is based on the combination of Malliavin calculus techniques and the Stein's method for normal approximations.
{"title":"Convergence of densities of spatial averages of the parabolic Anderson model driven by colored noise","authors":"Sefika Kuzgun, D. Nualart","doi":"10.1080/17442508.2023.2238954","DOIUrl":"https://doi.org/10.1080/17442508.2023.2238954","url":null,"abstract":"In this paper, we present a rate of convergence in the uniform norm for the densities of spatial averages of the solution to the d-dimensional parabolic Anderson model driven by a Gaussian multiplicative noise, which is white in time and has a spatial covariance given by the Riesz kernel. The proof is based on the combination of Malliavin calculus techniques and the Stein's method for normal approximations.","PeriodicalId":50447,"journal":{"name":"Finance and Stochastics","volume":"1 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2022-05-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81454475","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"经济学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-05-13DOI: 10.1007/s00780-022-00478-7
Hongjun Ha, Daniel Bauer
{"title":"A least-squares Monte Carlo approach to the estimation of enterprise risk","authors":"Hongjun Ha, Daniel Bauer","doi":"10.1007/s00780-022-00478-7","DOIUrl":"https://doi.org/10.1007/s00780-022-00478-7","url":null,"abstract":"","PeriodicalId":50447,"journal":{"name":"Finance and Stochastics","volume":"26 1","pages":"417 - 459"},"PeriodicalIF":1.7,"publicationDate":"2022-05-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44268806","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"经济学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-05-12DOI: 10.1080/17442508.2022.2071107
Kaoutar Nasroallah, Y. Ouknine
In this paper, we consider a stochastic differential equation with jumps for which pathwise uniqueness hold. We establish a fundamental mean square convergence theorem for the Euler approximation scheme. We provide some results on strong stability with respect to small perturbations of the initial conditions, and we study the convergence of Picard approximations.
{"title":"Euler approximation and stability of the solution to stochastic differential equations with jumps under pathwise uniqueness","authors":"Kaoutar Nasroallah, Y. Ouknine","doi":"10.1080/17442508.2022.2071107","DOIUrl":"https://doi.org/10.1080/17442508.2022.2071107","url":null,"abstract":"In this paper, we consider a stochastic differential equation with jumps for which pathwise uniqueness hold. We establish a fundamental mean square convergence theorem for the Euler approximation scheme. We provide some results on strong stability with respect to small perturbations of the initial conditions, and we study the convergence of Picard approximations.","PeriodicalId":50447,"journal":{"name":"Finance and Stochastics","volume":"77 1","pages":"266 - 302"},"PeriodicalIF":1.7,"publicationDate":"2022-05-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74267237","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"经济学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-05-06DOI: 10.1007/s00780-022-00477-8
Tahir Choulli, Sina Yansori
{"title":"Log-optimal and numéraire portfolios for market models stopped at a random time","authors":"Tahir Choulli, Sina Yansori","doi":"10.1007/s00780-022-00477-8","DOIUrl":"https://doi.org/10.1007/s00780-022-00477-8","url":null,"abstract":"","PeriodicalId":50447,"journal":{"name":"Finance and Stochastics","volume":"26 1","pages":"535 - 585"},"PeriodicalIF":1.7,"publicationDate":"2022-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47966658","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"经济学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-05-04DOI: 10.1080/17442508.2022.2070019
Kenneth Uda
We present the existence and uniqueness of random periodic path for stochastic dynamical systems generated by random switching stochastic differential equations (SDEs). These classes of SDEs arise as concrete models in molecular dynamics, biochemistry, climatology, wireless communications, financial mathematics, biological and artificial neural networks, etc. Random periodic processes are inevitable in these classes of stochastic dynamical systems due to the nonlinearity of their processing and the presence of time-dependent applied current. In our investigation, we employed Lyapunov second method and the theory of M-matrices, which are verifiable in terms of the coefficients of the SDE and the switching rates.
{"title":"Random periodic solutions for a class of hybrid stochastic differential equations","authors":"Kenneth Uda","doi":"10.1080/17442508.2022.2070019","DOIUrl":"https://doi.org/10.1080/17442508.2022.2070019","url":null,"abstract":"We present the existence and uniqueness of random periodic path for stochastic dynamical systems generated by random switching stochastic differential equations (SDEs). These classes of SDEs arise as concrete models in molecular dynamics, biochemistry, climatology, wireless communications, financial mathematics, biological and artificial neural networks, etc. Random periodic processes are inevitable in these classes of stochastic dynamical systems due to the nonlinearity of their processing and the presence of time-dependent applied current. In our investigation, we employed Lyapunov second method and the theory of M-matrices, which are verifiable in terms of the coefficients of the SDE and the switching rates.","PeriodicalId":50447,"journal":{"name":"Finance and Stochastics","volume":"1 1","pages":"211 - 234"},"PeriodicalIF":1.7,"publicationDate":"2022-05-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82392399","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"经济学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-05-04DOI: 10.1080/17442508.2022.2070433
Yike Wang, Jingzhen Liu, Jiaqin Wei
This paper is devoted to consumption-portfolio control problems with regime-switching-modulated habit formation. The formation of a habit depends on the current regime, which leads to time inconsistency in optimal control. General utility functions are used to evaluate the preference for the consumption net of the habit. To derive an analytical solution, we consider an enlarged financial market with Markov jump assets. Then, the time-inconsistent problem for pre-commitment controls is reduced to solving a regime-switching Hamilton–Jacobi–Bellman (HJB) equation. In seeking a time-consistent control, we investigate an alternate problem by means of multi-person sequential games with an essentially cooperative approach. The analytical expression of the time-consistent control is derived via a straightforward application of the pre-commitment control results. Moreover, the limit case in which the mesh size of time approaches zero is studied.
{"title":"Time-consistent consumption-portfolio control problems with regime-switching-modulated habit formation: an essentially cooperative approach","authors":"Yike Wang, Jingzhen Liu, Jiaqin Wei","doi":"10.1080/17442508.2022.2070433","DOIUrl":"https://doi.org/10.1080/17442508.2022.2070433","url":null,"abstract":"This paper is devoted to consumption-portfolio control problems with regime-switching-modulated habit formation. The formation of a habit depends on the current regime, which leads to time inconsistency in optimal control. General utility functions are used to evaluate the preference for the consumption net of the habit. To derive an analytical solution, we consider an enlarged financial market with Markov jump assets. Then, the time-inconsistent problem for pre-commitment controls is reduced to solving a regime-switching Hamilton–Jacobi–Bellman (HJB) equation. In seeking a time-consistent control, we investigate an alternate problem by means of multi-person sequential games with an essentially cooperative approach. The analytical expression of the time-consistent control is derived via a straightforward application of the pre-commitment control results. Moreover, the limit case in which the mesh size of time approaches zero is studied.","PeriodicalId":50447,"journal":{"name":"Finance and Stochastics","volume":"62 1","pages":"235 - 265"},"PeriodicalIF":1.7,"publicationDate":"2022-05-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85764071","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"经济学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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