By using large deviation theory that deals with the decay of probabilities of rare events on an exponential scale, we study the longtime behaviors and establish action functionals for scaled Brownian motion and Levy processes with existing finite exponential moments. Based on extended contraction principle, Legendre transform and Levy symbols, we derive the action functionals for stochastic differential equations driven by Levy processes.
{"title":"Action Functionals for Stochastic Differential Equations with Lévy Noise","authors":"S. Yuan, Jinqiao Duan","doi":"10.31390/cosa.13.3.10","DOIUrl":"https://doi.org/10.31390/cosa.13.3.10","url":null,"abstract":"By using large deviation theory that deals with the decay of probabilities of rare events on an exponential scale, we study the longtime behaviors and establish action functionals for scaled Brownian motion and Levy processes with existing finite exponential moments. Based on extended contraction principle, Legendre transform and Levy symbols, we derive the action functionals for stochastic differential equations driven by Levy processes.","PeriodicalId":53434,"journal":{"name":"Communications on Stochastic Analysis","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-08-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"69689406","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}
In this work we discuss a random Tug-of-War game in graphs where one of the players has the power to decide at each turn whether to play a round of classical random Tug-of-War, or let the other player choose the new game position in exchange of a fixed payoff. We prove that this game has a value using a discrete comparison principle and viscosity tools, as well as probabilistic arguments. This game is related to Jensen's extremal equations, which have a key role in Jensen's celebrated proof of uniqueness of infinity harmonic functions.
{"title":"Totalitarian Random Tug-of-War Games in Graphs","authors":"Marcos Ant'on, Fernando Charro, Peiyong Wang","doi":"10.31390/cosa.13.3.06","DOIUrl":"https://doi.org/10.31390/cosa.13.3.06","url":null,"abstract":"In this work we discuss a random Tug-of-War game in graphs where one of the players has the power to decide at each turn whether to play a round of classical random Tug-of-War, or let the other player choose the new game position in exchange of a fixed payoff. We prove that this game has a value using a discrete comparison principle and viscosity tools, as well as probabilistic arguments. This game is related to Jensen's extremal equations, which have a key role in Jensen's celebrated proof of uniqueness of infinity harmonic functions.","PeriodicalId":53434,"journal":{"name":"Communications on Stochastic Analysis","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41721291","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}
In this paper, we are concerned with the nonparametric estimation of an unknown density under censoring. Firstly, we propose a recursive kernel density estimators under censoring, based on a stochastic approximation algorithm. Then, we showed that our recursive estimator is consistent and asymptotically normally distributed. Moreover, we describe and investigate a data-driven bandwidth selection procedure based on normal pilot bandwidth reference distributions. We showed that the proposed recursive estimators can be better than the non-recursive in terms of estimation error and much better in terms of computational costs. We corroborated these theoretical results through a simulation study and on Malaria in Senegalese children dataset.
{"title":"Smoothing Parameters for Recursive Kernel Density Estimators under Censoring","authors":"Y. Slaoui","doi":"10.31390/COSA.13.2.02","DOIUrl":"https://doi.org/10.31390/COSA.13.2.02","url":null,"abstract":"In this paper, we are concerned with the nonparametric estimation of an unknown density under censoring. Firstly, we propose a recursive kernel density estimators under censoring, based on a stochastic approximation algorithm. Then, we showed that our recursive estimator is consistent and asymptotically normally distributed. Moreover, we describe and investigate a data-driven bandwidth selection procedure based on normal pilot bandwidth reference distributions. We showed that the proposed recursive estimators can be better than the non-recursive in terms of estimation error and much better in terms of computational costs. We corroborated these theoretical results through a simulation study and on Malaria in Senegalese children dataset.","PeriodicalId":53434,"journal":{"name":"Communications on Stochastic Analysis","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46078974","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}
Using the spectral theorem we compute the Quantum Fourier Transform (or Vacuum Characteristic Function) $langle Phi, e^{itH}Phirangle$ of an observable $H$ defined as a self-adjoint sum of the generators of a finite-dimensional Lie algebra, where $Phi$ is a unit vector in a Hilbert space $mathcal{H}$. We show how Stone's formula for computing the spectral resolution of a Hilbert space self-adjoint operator, can serve as an alternative to the traditional reliance on splitting (or disentanglement) formulas for the operator exponential.
{"title":"Spectral Theorem Approach to the Characteristic Function of Quantum Observables","authors":"A. Boukas, P. Feinsilver","doi":"10.31390/cosa.13.2.03.","DOIUrl":"https://doi.org/10.31390/cosa.13.2.03.","url":null,"abstract":"Using the spectral theorem we compute the Quantum Fourier Transform (or Vacuum Characteristic Function) $langle Phi, e^{itH}Phirangle$ of an observable $H$ defined as a self-adjoint sum of the generators of a finite-dimensional Lie algebra, where $Phi$ is a unit vector in a Hilbert space $mathcal{H}$. We show how Stone's formula for computing the spectral resolution of a Hilbert space self-adjoint operator, can serve as an alternative to the traditional reliance on splitting (or disentanglement) formulas for the operator exponential.","PeriodicalId":53434,"journal":{"name":"Communications on Stochastic Analysis","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43099706","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}
It is shown that any infinitely divisible distribution μ on R+ gives rise to a class of increasing additive processes we call C-additive processes, where C is a continuous semigroup of cumulant generating functions. The marginal and increment distributions of these pocesses are characterized in terms of their Lévy measure and their drift coefficient. Integral representations of C-additive processes in terms of a Poisson random measure are obtained. The limiting behavior (as t → ∞) of two subclasses of C-additive processes leads to new characterizations of C-selfdecomposable and C-stable distributions on R+.
{"title":"Increasing C-Additive Processes","authors":"N. Bouzar","doi":"10.31390/cosa.13.2.05","DOIUrl":"https://doi.org/10.31390/cosa.13.2.05","url":null,"abstract":"It is shown that any infinitely divisible distribution μ on R+ gives rise to a class of increasing additive processes we call C-additive processes, where C is a continuous semigroup of cumulant generating functions. The marginal and increment distributions of these pocesses are characterized in terms of their Lévy measure and their drift coefficient. Integral representations of C-additive processes in terms of a Poisson random measure are obtained. The limiting behavior (as t → ∞) of two subclasses of C-additive processes leads to new characterizations of C-selfdecomposable and C-stable distributions on R+.","PeriodicalId":53434,"journal":{"name":"Communications on Stochastic Analysis","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41784816","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":"Strong Convergence Rate in Averaging Principle for the Heat Equation Driven by a General Stochastic Measure","authors":"V. Radchenko","doi":"10.31390/COSA.13.2.01","DOIUrl":"https://doi.org/10.31390/COSA.13.2.01","url":null,"abstract":"","PeriodicalId":53434,"journal":{"name":"Communications on Stochastic Analysis","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49044002","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}
In this work we extend the finite dimensional Radon transform [23] to the Gaussian measure. We develop an inversion formula for this GaussRadon transform by way of Fourier inversion formula. We then proceed to extend these results to the infinite dimensional setting.
{"title":"A Limiting Process to Invert the Gauss-Radon Transform","authors":"Jeremy J. Becnel","doi":"10.31390/cosa.13.2.04","DOIUrl":"https://doi.org/10.31390/cosa.13.2.04","url":null,"abstract":"In this work we extend the finite dimensional Radon transform [23] to the Gaussian measure. We develop an inversion formula for this GaussRadon transform by way of Fourier inversion formula. We then proceed to extend these results to the infinite dimensional setting.","PeriodicalId":53434,"journal":{"name":"Communications on Stochastic Analysis","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49258539","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 classical processes (Poisson, Bernoulli, negative binomial) are the most popular discrete counting processes; however, these rely on strict assumptions. We studied an inhomogeneous counting process (which is known as the inhomogeneous Panjer process IPP) that not only includes the classical processes as special cases, but also allows to describe counting processes to approximate data with overor under-dispersion. We present the most relevant properties of this process and establish the probability mass function and cumulative distribution function using intensity rates. This counting process will allow risk analysts who work modeling the counting processes where data dispersion exists in a more flexible and efficient way.
{"title":"Some Properties of the Inhomogeneous Panjer Process","authors":"Ana María Beltrán Cortés, J. A. Jiménez-Moscoso","doi":"10.31390/COSA.13.1.07","DOIUrl":"https://doi.org/10.31390/COSA.13.1.07","url":null,"abstract":"The classical processes (Poisson, Bernoulli, negative binomial) are the most popular discrete counting processes; however, these rely on strict assumptions. We studied an inhomogeneous counting process (which is known as the inhomogeneous Panjer process IPP) that not only includes the classical processes as special cases, but also allows to describe counting processes to approximate data with overor under-dispersion. We present the most relevant properties of this process and establish the probability mass function and cumulative distribution function using intensity rates. This counting process will allow risk analysts who work modeling the counting processes where data dispersion exists in a more flexible and efficient way.","PeriodicalId":53434,"journal":{"name":"Communications on Stochastic Analysis","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48392909","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}
We investigate the spectrum of the generator of a self-adjoint transition semigroup of a (symmetric) Levy process taking values in $d$--dimensional space.
研究了d维空间中取值的(对称)Levy过程的自伴随跃迁半群的谱。
{"title":"On the Spectrum of Self-Adjoint Lévy Generators","authors":"D. Applebaum","doi":"10.31390/COSA.13.1.04","DOIUrl":"https://doi.org/10.31390/COSA.13.1.04","url":null,"abstract":"We investigate the spectrum of the generator of a self-adjoint transition semigroup of a (symmetric) Levy process taking values in $d$--dimensional space.","PeriodicalId":53434,"journal":{"name":"Communications on Stochastic Analysis","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45599934","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":"Second Order Stochastic Partial Integro Differential Equations with Delay and Impulses","authors":"M.V.S.S.B.B.K. Sastry, G.V.S.R. Deekshitulu","doi":"10.31390/COSA.13.1.06","DOIUrl":"https://doi.org/10.31390/COSA.13.1.06","url":null,"abstract":"","PeriodicalId":53434,"journal":{"name":"Communications on Stochastic Analysis","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44286437","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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