The paper is devoted to a stochastic heat equation with a mixed fractional Brownian noise. We investigate the covariance structure, stationarity, upper bounds and asymptotic behavior of the solution. Based on its discrete-time observations, we construct a strongly consistent estimator for the Hurst index H and prove the asymptotic normality for $H<3/4$. Then assuming the parameter H to be known, we deal with joint estimation of the coefficients at the Wiener process and at the fractional Brownian motion. The quality of estimators is illustrated by simulation experiments.
{"title":"Parameter estimation in mixed fractional stochastic heat equation","authors":"D. Avetisian, K. Ralchenko","doi":"10.15559/23-vmsta221","DOIUrl":"https://doi.org/10.15559/23-vmsta221","url":null,"abstract":"The paper is devoted to a stochastic heat equation with a mixed fractional Brownian noise. We investigate the covariance structure, stationarity, upper bounds and asymptotic behavior of the solution. Based on its discrete-time observations, we construct a strongly consistent estimator for the Hurst index H and prove the asymptotic normality for $H<3/4$. Then assuming the parameter H to be known, we deal with joint estimation of the coefficients at the Wiener process and at the fractional Brownian motion. The quality of estimators is illustrated by simulation experiments.","PeriodicalId":42685,"journal":{"name":"Modern Stochastics-Theory and Applications","volume":"60 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78816377","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}
A multivariate trigonometric regression model is considered. In the paper strong consistency of the least squares estimator for amplitudes and angular frequencies is obtained for such a multivariate model on the assumption that the random noise is a homogeneous or homogeneous and isotropic Gaussian, specifically, strongly dependent random field on ${mathbb{R}^{M}},Mge 3$.
{"title":"Consistency of LSE for the many-dimensional symmetric textured surface parameters","authors":"O. Dykyi, Alexander Ivanov","doi":"10.15559/23-vmsta225","DOIUrl":"https://doi.org/10.15559/23-vmsta225","url":null,"abstract":"A multivariate trigonometric regression model is considered. In the paper strong consistency of the least squares estimator for amplitudes and angular frequencies is obtained for such a multivariate model on the assumption that the random noise is a homogeneous or homogeneous and isotropic Gaussian, specifically, strongly dependent random field on ${mathbb{R}^{M}},Mge 3$.","PeriodicalId":42685,"journal":{"name":"Modern Stochastics-Theory and Applications","volume":"3 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89216207","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}
The so-called multi-mixed fractional Brownian motions (mmfBm) and multi-mixed fractional Ornstein–Uhlenbeck (mmfOU) processes are studied. These processes are constructed by mixing by superimposing or mixing (infinitely many) independent fractional Brownian motions (fBm) and fractional Ornstein–Uhlenbeck processes (fOU), respectively. Their existence as ${L^{2}}$ processes is proved, and their path properties, viz. long-range and short-range dependence, Hölder continuity, p-variation, and conditional full support, are studied.
{"title":"Multi-mixed fractional Brownian motions and Ornstein–Uhlenbeck processes","authors":"Hamidreza Maleki Almani, T. Sottinen","doi":"10.15559/23-vmsta229","DOIUrl":"https://doi.org/10.15559/23-vmsta229","url":null,"abstract":"The so-called multi-mixed fractional Brownian motions (mmfBm) and multi-mixed fractional Ornstein–Uhlenbeck (mmfOU) processes are studied. These processes are constructed by mixing by superimposing or mixing (infinitely many) independent fractional Brownian motions (fBm) and fractional Ornstein–Uhlenbeck processes (fOU), respectively. Their existence as ${L^{2}}$ processes is proved, and their path properties, viz. long-range and short-range dependence, Hölder continuity, p-variation, and conditional full support, are studied.","PeriodicalId":42685,"journal":{"name":"Modern Stochastics-Theory and Applications","volume":"42 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74007110","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}
The stochastic transport equation is considered where the randomness is given by a symmetric integral with respect to a stochastic measure. For a stochastic measure, only σ-additivity in probability and continuity of paths is assumed. Existence and uniqueness of a weak solution to the equation are proved.
{"title":"Transport equation driven by a stochastic measure","authors":"V. Radchenko","doi":"10.15559/23-vmsta222","DOIUrl":"https://doi.org/10.15559/23-vmsta222","url":null,"abstract":"The stochastic transport equation is considered where the randomness is given by a symmetric integral with respect to a stochastic measure. For a stochastic measure, only σ-additivity in probability and continuity of paths is assumed. Existence and uniqueness of a weak solution to the equation are proved.","PeriodicalId":42685,"journal":{"name":"Modern Stochastics-Theory and Applications","volume":"7 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86562629","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}
The major characteristic of the cancellable American options is the existing writer’s right to cancel the contract prematurely paying some penalty amount. The main purpose of this paper is to introduce and examine a new subclass of such options for which the penalty which the writer owes for this right consists of three parts – a fixed amount, shares of the underlying asset, and a proportion of the usual option payment. We examine the asymptotic case in which the maturity is set to be infinity. We determine the optimal exercise regions for the option’s holder and writer and derive the fair option price.
{"title":"Perpetual cancellable American options with convertible features","authors":"Tsvetelin S. Zaevski","doi":"10.15559/23-vmsta230","DOIUrl":"https://doi.org/10.15559/23-vmsta230","url":null,"abstract":"The major characteristic of the cancellable American options is the existing writer’s right to cancel the contract prematurely paying some penalty amount. The main purpose of this paper is to introduce and examine a new subclass of such options for which the penalty which the writer owes for this right consists of three parts – a fixed amount, shares of the underlying asset, and a proportion of the usual option payment. We examine the asymptotic case in which the maturity is set to be infinity. We determine the optimal exercise regions for the option’s holder and writer and derive the fair option price.","PeriodicalId":42685,"journal":{"name":"Modern Stochastics-Theory and Applications","volume":"23 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75324211","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}
The minimax identity for a nondecreasing upper-semicontinuous utility function satisfying mild growth assumption is studied. In contrast to the classical setting, concavity of the utility function is not asumed. By considering the concave envelope of the utility function, equalities and inequalities between the robust utility functionals of an initial utility function and its concavification are obtained. Furthermore, similar equalities and inequalities are proved in the case of implementing an upper bound on the final endowment of the initial model.
{"title":"Minimax identity with robust utility functional for a nonconcave utility","authors":"O. Bahchedjioglou, G. Shevchenko","doi":"10.15559/22-vmsta215","DOIUrl":"https://doi.org/10.15559/22-vmsta215","url":null,"abstract":"The minimax identity for a nondecreasing upper-semicontinuous utility function satisfying mild growth assumption is studied. In contrast to the classical setting, concavity of the utility function is not asumed. By considering the concave envelope of the utility function, equalities and inequalities between the robust utility functionals of an initial utility function and its concavification are obtained. Furthermore, similar equalities and inequalities are proved in the case of implementing an upper bound on the final endowment of the initial model.","PeriodicalId":42685,"journal":{"name":"Modern Stochastics-Theory and Applications","volume":"27 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2022-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82647520","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}
A new class of multidimensional locally perturbed random walks called random walks with sticky barriers is introduced and analyzed. The laws of large numbers and functional limit theorems are proved for hitting times of successive barriers.
{"title":"Random walks with sticky barriers","authors":"V. Bohun, A. Marynych","doi":"10.15559/22-vmsta202","DOIUrl":"https://doi.org/10.15559/22-vmsta202","url":null,"abstract":"A new class of multidimensional locally perturbed random walks called random walks with sticky barriers is introduced and analyzed. The laws of large numbers and functional limit theorems are proved for hitting times of successive barriers.","PeriodicalId":42685,"journal":{"name":"Modern Stochastics-Theory and Applications","volume":"8 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89385685","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}
{"title":"On the denseness of the subset of discrete distributions in a certain set of two-dimensional distributions","authors":"D. Borzykh, A. Gushchin","doi":"10.15559/22-vmsta204","DOIUrl":"https://doi.org/10.15559/22-vmsta204","url":null,"abstract":"<jats:p />","PeriodicalId":42685,"journal":{"name":"Modern Stochastics-Theory and Applications","volume":"69 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74604963","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}
This article provides survival probability calculation formulas for bi-risk discrete time risk model with income rate two. More precisely, the possibility for the stochastic process $u+2t-{textstylesum _{i=1}^{t}}{X_{i}}-{textstylesum _{j=1}^{lfloor t/2rfloor }}{Y_{j}}$, $uin mathbb{N}cup {0}$, to stay positive for all $tin {1,hspace{0.1667em}2,hspace{0.1667em}dots ,hspace{0.1667em}T}$, when $Tin mathbb{N}$ or $Tto infty $, is considered, where the subtracted random part consists of the sum of random variables, which occur in time in the following order: ${X_{1}},hspace{0.1667em}{X_{2}}+{Y_{1}},hspace{0.1667em}{X_{3}},hspace{0.1667em}{X_{4}}+{Y_{2}},hspace{0.1667em}dots $ Here ${X_{i}},hspace{0.1667em}iin mathbb{N}$, and ${Y_{j}},hspace{0.1667em}jin mathbb{N}$, are independent copies of two independent, but not necessarily identically distributed, nonnegative and integer-valued random variables X and Y. Following the known survival probability formulas of the similar bi-seasonal model with income rate two, $u+2t-{textstylesum _{i=1}^{t}}{X_{i}}{mathbb{1}_{{ihspace{2.5pt}text{is odd}}}}-{textstylesum _{j=1}^{t}}{Y_{i}}{mathbb{1}_{{jhspace{2.5pt}text{is even}}}}$, it is demonstrated how the bi-seasonal model is used to express survival probability calculation formulas in the bi-risk case. Several numerical examples are given where the derived theoretical statements are applied.
{"title":"Note on the bi-risk discrete time risk model with income rate two","authors":"A. Grigutis, Artur Nakliuda","doi":"10.15559/22-vmsta209","DOIUrl":"https://doi.org/10.15559/22-vmsta209","url":null,"abstract":"This article provides survival probability calculation formulas for bi-risk discrete time risk model with income rate two. More precisely, the possibility for the stochastic process $u+2t-{textstylesum _{i=1}^{t}}{X_{i}}-{textstylesum _{j=1}^{lfloor t/2rfloor }}{Y_{j}}$, $uin mathbb{N}cup {0}$, to stay positive for all $tin {1,hspace{0.1667em}2,hspace{0.1667em}dots ,hspace{0.1667em}T}$, when $Tin mathbb{N}$ or $Tto infty $, is considered, where the subtracted random part consists of the sum of random variables, which occur in time in the following order: ${X_{1}},hspace{0.1667em}{X_{2}}+{Y_{1}},hspace{0.1667em}{X_{3}},hspace{0.1667em}{X_{4}}+{Y_{2}},hspace{0.1667em}dots $ Here ${X_{i}},hspace{0.1667em}iin mathbb{N}$, and ${Y_{j}},hspace{0.1667em}jin mathbb{N}$, are independent copies of two independent, but not necessarily identically distributed, nonnegative and integer-valued random variables X and Y. Following the known survival probability formulas of the similar bi-seasonal model with income rate two, $u+2t-{textstylesum _{i=1}^{t}}{X_{i}}{mathbb{1}_{{ihspace{2.5pt}text{is odd}}}}-{textstylesum _{j=1}^{t}}{Y_{i}}{mathbb{1}_{{jhspace{2.5pt}text{is even}}}}$, it is demonstrated how the bi-seasonal model is used to express survival probability calculation formulas in the bi-risk case. Several numerical examples are given where the derived theoretical statements are applied.","PeriodicalId":42685,"journal":{"name":"Modern Stochastics-Theory and Applications","volume":"13 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81417938","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}
A stochastic parabolic equation on $[0,T]times mathbb{R}$ driven by a general stochastic measure, for which we assume only σ-additivity in probability, is considered. The asymptotic behavior of its solution as $tto infty $ is studied.
{"title":"Asymptotic properties of the parabolic equation driven by stochastic measure","authors":"B. Manikin","doi":"10.15559/22-vmsta213","DOIUrl":"https://doi.org/10.15559/22-vmsta213","url":null,"abstract":"A stochastic parabolic equation on $[0,T]times mathbb{R}$ driven by a general stochastic measure, for which we assume only σ-additivity in probability, is considered. The asymptotic behavior of its solution as $tto infty $ is studied.","PeriodicalId":42685,"journal":{"name":"Modern Stochastics-Theory and Applications","volume":"29 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88213917","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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