. In this paper we investigate consequences of covariance of a uni- formly Quantum Markov Semigroup, under a group action, on the structure of its minimal invariant projections. We obtain that, under suitable hypotheses, minimal invariant projections correspond to irreducible sub-representations in which the initial covariant representation is decomposed. We apply this results in the study circulant Quantum Markov Semigroups.
{"title":"Invariant Projections for Covariant Quantum Markov Semigroups","authors":"F. Fagnola, E. Sasso, V. Umanità","doi":"10.31390/JOSA.1.4.03","DOIUrl":"https://doi.org/10.31390/JOSA.1.4.03","url":null,"abstract":". In this paper we investigate consequences of covariance of a uni- formly Quantum Markov Semigroup, under a group action, on the structure of its minimal invariant projections. We obtain that, under suitable hypotheses, minimal invariant projections correspond to irreducible sub-representations in which the initial covariant representation is decomposed. We apply this results in the study circulant Quantum Markov Semigroups.","PeriodicalId":263604,"journal":{"name":"Journal of Stochastic Analysis","volume":"4 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125979076","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 paper investigates the value of information to an investor under the partial information setting for exponential utility. The only information available to the investor is the one generated by the asset price processes and, in particular, the underlying appreciation rate of the risky asset cannot be observed directly. Filtering theory is used to find a filtered estimate of the underlying appreciation rate. This brings about two maximisation problems from which we determine the optimal expected utilities of wealth under partial and full information, via Hamilton-Jacobi-Bellman equations. The value of information is, therefore, calculated as the di↵erence between the two optimal expected utilities. The e↵ect of parameter changes on the value of information is determined by carrying out numerical simulations.
{"title":"The Value of Information under Partial Information for Exponential Utility","authors":"F. J. Mhlanga, M. Dube","doi":"10.31390/JOSA.1.3.01","DOIUrl":"https://doi.org/10.31390/JOSA.1.3.01","url":null,"abstract":"The paper investigates the value of information to an investor under the partial information setting for exponential utility. The only information available to the investor is the one generated by the asset price processes and, in particular, the underlying appreciation rate of the risky asset cannot be observed directly. Filtering theory is used to find a filtered estimate of the underlying appreciation rate. This brings about two maximisation problems from which we determine the optimal expected utilities of wealth under partial and full information, via Hamilton-Jacobi-Bellman equations. The value of information is, therefore, calculated as the di↵erence between the two optimal expected utilities. The e↵ect of parameter changes on the value of information is determined by carrying out numerical simulations.","PeriodicalId":263604,"journal":{"name":"Journal of Stochastic Analysis","volume":"9 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-08-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127873023","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 consider a stochastic version of a system of coupled two equations formulated by Burgers with the aim to describe the laminar and turbulent motions of a fluid in a channel. The existence and uniqueness of the solution as well as the irreducibility property of such system were given by Twardowska and Zabczyk. In the paper the existence of a unique invariant measure is investigated. The paper generalizes the results of Da Prato, Debussche and Temam, and Da Prato and Gatarek, dealing with one equation describing the turbulent motion only.
{"title":"Ergodicity of Burgers' System","authors":"S. Peszat, K. Twardowska, J. Zabczyk","doi":"10.31390/josa.2.3.10","DOIUrl":"https://doi.org/10.31390/josa.2.3.10","url":null,"abstract":"We consider a stochastic version of a system of coupled two equations formulated by Burgers with the aim to describe the laminar and turbulent motions of a fluid in a channel. The existence and uniqueness of the solution as well as the irreducibility property of such system were given by Twardowska and Zabczyk. In the paper the existence of a unique invariant measure is investigated. The paper generalizes the results of Da Prato, Debussche and Temam, and Da Prato and Gatarek, dealing with one equation describing the turbulent motion only.","PeriodicalId":263604,"journal":{"name":"Journal of Stochastic Analysis","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-06-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115521952","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, an approximate version of the Barndorff-Nielsen and Shephard model, driven by a Brownian motion and a L'evy subordinator, is formulated. The first-exit time of the log-return process for this model is analyzed. It is shown that with certain probability, the first-exit time process of the log-return is decomposable into the sum of the first exit time of the Brownian motion with drift, and the first exit time of a L'evy subordinator with drift. Subsequently, the probability density functions of the first exit time of some specific L'evy subordinators, connected to stationary, self-decomposable variance processes, are studied. Analytical expressions of the probability density function of the first-exit time of three such L'evy subordinators are obtained in terms of various special functions. The results are implemented to empirical S&P 500 dataset.
{"title":"First Exit-Time Analysis for an Approximate Barndorff-Nielsen and Shephard Model with Stationary Self-Decomposable Variance Process","authors":"Shantanu Awasthi, I. Sengupta","doi":"10.31390/josa.2.1.05","DOIUrl":"https://doi.org/10.31390/josa.2.1.05","url":null,"abstract":"In this paper, an approximate version of the Barndorff-Nielsen and Shephard model, driven by a Brownian motion and a L'evy subordinator, is formulated. The first-exit time of the log-return process for this model is analyzed. It is shown that with certain probability, the first-exit time process of the log-return is decomposable into the sum of the first exit time of the Brownian motion with drift, and the first exit time of a L'evy subordinator with drift. Subsequently, the probability density functions of the first exit time of some specific L'evy subordinators, connected to stationary, self-decomposable variance processes, are studied. Analytical expressions of the probability density function of the first-exit time of three such L'evy subordinators are obtained in terms of various special functions. The results are implemented to empirical S&P 500 dataset.","PeriodicalId":263604,"journal":{"name":"Journal of Stochastic Analysis","volume":"60 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-06-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126699650","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 objective is to provide an Al`os type decomposition formula of call option prices for the Barndorff-Nielsen and Shephard model: an Ornstein-Uhlenbeck type stochastic volatility model driven by a subordinator without drift. Al`os (2012) introduced a decomposition expression for the Heston model by using Ito's formula. In this paper, we extend it to the Barndorff-Nielsen and Shephard model. As far as we know, this is the first result on the Al`os type decomposition formula for models with infinite active jumps.
{"title":"Alòs Type Decomposition Formula for Barndorff-Nielsen and Shephard Model","authors":"Takuji Arai","doi":"10.31390/josa.2.2.03","DOIUrl":"https://doi.org/10.31390/josa.2.2.03","url":null,"abstract":"The objective is to provide an Al`os type decomposition formula of call option prices for the Barndorff-Nielsen and Shephard model: an Ornstein-Uhlenbeck type stochastic volatility model driven by a subordinator without drift. Al`os (2012) introduced a decomposition expression for the Heston model by using Ito's formula. In this paper, we extend it to the Barndorff-Nielsen and Shephard model. As far as we know, this is the first result on the Al`os type decomposition formula for models with infinite active jumps.","PeriodicalId":263604,"journal":{"name":"Journal of Stochastic Analysis","volume":"41 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132250978","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 consider a finite state, continuous time homogeneous semiMarkov chain X = {Xt, t ≥ 0}. Without loss of generality the state space of the chain can be identified with the set of unit vectors S = {e1, e2, . . . , eN} where ei = (0, . . . , 0, 1, 0, . . . , 0) ′ ∈ RN . The probabilistic and dynamic properties of X can be described by either a rate matrix A or a matrix which gives the occupation times in the various states together with the probabilities of jumping to a different state. For a continuous time Markov chain the occupation times are memoryless, implying the distributions are exponential. For semi-Markov chains the occupation times can have more general distributions. The relation between these two descriptions is first investigated and the semimartingale dynamics of a semi-Markov chain obtained in contrast to the traditional description of a semi-Markov chain in terms of a renewal process. An equation giving the dynamics of the occupation times is derived together with an equation for the density of the conditional occupation time and state. Some approximations for these dynamics are then obtained.
{"title":"The Semimartingale Dynamics and Generator of a Continuous Time Semi-Markov Chain","authors":"R. Elliott","doi":"10.31390/josa.1.1.01","DOIUrl":"https://doi.org/10.31390/josa.1.1.01","url":null,"abstract":"We consider a finite state, continuous time homogeneous semiMarkov chain X = {Xt, t ≥ 0}. Without loss of generality the state space of the chain can be identified with the set of unit vectors S = {e1, e2, . . . , eN} where ei = (0, . . . , 0, 1, 0, . . . , 0) ′ ∈ RN . The probabilistic and dynamic properties of X can be described by either a rate matrix A or a matrix which gives the occupation times in the various states together with the probabilities of jumping to a different state. For a continuous time Markov chain the occupation times are memoryless, implying the distributions are exponential. For semi-Markov chains the occupation times can have more general distributions. The relation between these two descriptions is first investigated and the semimartingale dynamics of a semi-Markov chain obtained in contrast to the traditional description of a semi-Markov chain in terms of a renewal process. An equation giving the dynamics of the occupation times is derived together with an equation for the density of the conditional occupation time and state. Some approximations for these dynamics are then obtained.","PeriodicalId":263604,"journal":{"name":"Journal of Stochastic Analysis","volume":"34 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125595176","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 give a nonstandard analytic proof of de Finetti's theorem for an exchangeable sequence of Bernoulli random variables. The theorem postulates that such a sequence is uniquely representable as a mixture of iid sequences of Bernoulli random variables. We use combinatorial arguments to show that this probability distribution is induced by a hyperfinite sample mean.
{"title":"A Nonstandard Proof of De Finetti’s Theorem for Bernoulli Random Variables","authors":"Irfan Alam","doi":"10.31390/JOSA.1.4.15","DOIUrl":"https://doi.org/10.31390/JOSA.1.4.15","url":null,"abstract":"We give a nonstandard analytic proof of de Finetti's theorem for an exchangeable sequence of Bernoulli random variables. The theorem postulates that such a sequence is uniquely representable as a mixture of iid sequences of Bernoulli random variables. We use combinatorial arguments to show that this probability distribution is induced by a hyperfinite sample mean.","PeriodicalId":263604,"journal":{"name":"Journal of Stochastic Analysis","volume":"1976 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-12-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130157062","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 derive covariant Weyl operators for light-like fields, with the massless Weyl fermion as an illustrative example, in such a way that they correspond to quantum white noises in vacuum state of a symmetric Fock space. First, we build a representation of a light-like little group in terms of Weyl operators. We then use this construction to induce a representation of Poincar'e group to construct relativistic quantum white noises from the fields via Mackey's systems of imprimitivity (SI) machinery. Our construction proceeds by fashioning the fermionic processes on a symmetric Fock space using re ection and identifying the corresponding processes on the isomorphic white noise space.
{"title":"Covariant Quantum White Noise from Light-like Quantum Fields","authors":"R. Balu","doi":"10.31390/JOSA.1.4.07","DOIUrl":"https://doi.org/10.31390/JOSA.1.4.07","url":null,"abstract":"We derive covariant Weyl operators for light-like fields, with the massless Weyl fermion as an illustrative example, in such a way that they correspond to quantum white noises in vacuum state of a symmetric Fock space. First, we build a representation of a light-like little group in terms of Weyl operators. We then use this construction to induce a representation of Poincar'e group to construct relativistic quantum white noises from the fields via Mackey's systems of imprimitivity (SI) machinery. Our construction proceeds by fashioning the fermionic processes on a symmetric Fock space using re ection and identifying the corresponding processes on the isomorphic white noise space.","PeriodicalId":263604,"journal":{"name":"Journal of Stochastic Analysis","volume":"5 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-11-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122737023","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 definition due to Accardi of a pair of complementary observables is adapted to the context of the Lie algebra $ su(2) $. We show that the pair of Pauli matrices $ A,B $ associated to the unit directions $ alpha $ and $ beta $ in $ mathbb{R}^{3} $ are Accardi complementary if and only if $ alpha $ and $ beta $ are orthogonal if and only if $ A $ and $ B $ are orthogonal. In particular, any pair of the standard triple of Pauli matrices is complementary.
一对互补可观测量的Accardi定义适用于李代数$ su(2) $。我们证明了$ mathbb{R}^{3} $中与单位方向$ alpha $和$ beta $相关的泡利矩阵对$ A,B $当且仅当$ alpha $和$ beta $正交当且仅当$ A $和$ B $正交时为Accardi互补。特别地，泡利矩阵的标准三元组中的任何一对都是互补的。
{"title":"Pauli Matrices: A Triple of Accardi Complementary Observables","authors":"S. B. Sontz","doi":"10.31390/JOSA.1.4.02","DOIUrl":"https://doi.org/10.31390/JOSA.1.4.02","url":null,"abstract":"The definition due to Accardi of a pair of complementary observables is adapted to the context of the Lie algebra $ su(2) $. We show that the pair of Pauli matrices $ A,B $ associated to the unit directions $ alpha $ and $ beta $ in $ mathbb{R}^{3} $ are Accardi complementary if and only if $ alpha $ and $ beta $ are orthogonal if and only if $ A $ and $ B $ are orthogonal. In particular, any pair of the standard triple of Pauli matrices is complementary.","PeriodicalId":263604,"journal":{"name":"Journal of Stochastic Analysis","volume":"392 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-08-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116523168","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 define and discuss $mathcal{R}(p,q)$- deformations of basic univariate discrete distributions of the probability theory. We mainly focus on binomial, Euler, P'olya and inverse P'olya distributions. We discuss relevant $mathcal{R}(p,q)-$ deformed factorial moments of a random variable, and establish associated expressions of mean and variance. Futhermore, we derive a recursion relation for the probability distributions. Then, we apply the same approach to build main distributional properties characterizing the generalized $q-$ Quesne quantum algebra, used in physics. Other known results in the literature are also recovered as particular cases.
{"title":"R(p,q) Analogs of Discrete Distributions: General Formalism and Applications","authors":"M. N. Hounkonnou, Fridolin Melong","doi":"10.31390/JOSA.1.4.11","DOIUrl":"https://doi.org/10.31390/JOSA.1.4.11","url":null,"abstract":"In this paper, we define and discuss $mathcal{R}(p,q)$- deformations of basic univariate discrete distributions of the probability theory. We mainly focus on binomial, Euler, P'olya and inverse P'olya distributions. We discuss relevant $mathcal{R}(p,q)-$ deformed factorial moments of a random variable, and establish associated expressions of mean and variance. Futhermore, we derive a recursion relation for the probability distributions. Then, we apply the same approach to build main distributional properties characterizing the generalized $q-$ Quesne quantum algebra, used in physics. Other known results in the literature are also recovered as particular cases.","PeriodicalId":263604,"journal":{"name":"Journal of Stochastic Analysis","volume":"103 1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-12-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116520264","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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