In this paper, we consider the Lévy Laplacian acting on multiple Wiener integrals by the stochastic process given as a difference of two independent Lévy processes, and give a necessary and sufficient condition for eigenfunctions of the Lévy Laplacian. Moreover we give a decomposition of the L2-space on Lévy noise probability space by eigenspaces consisting of multiple Wiener integrals by the above process in terms of the Lévy Laplacian. By this decomposition, we obtain an expression of the semigroup generated by the Lévy Laplacian related to the semigroup generated by the number operator.
{"title":"A Decomposition of a Space of Multiple Wiener Integrals by the Difference of Two Independent Lévy Processes in Terms of the Lévy Laplacian","authors":"Atsushi Ishikawa","doi":"10.31390/COSA.12.2.05","DOIUrl":"https://doi.org/10.31390/COSA.12.2.05","url":null,"abstract":"In this paper, we consider the Lévy Laplacian acting on multiple Wiener integrals by the stochastic process given as a difference of two independent Lévy processes, and give a necessary and sufficient condition for eigenfunctions of the Lévy Laplacian. Moreover we give a decomposition of the L2-space on Lévy noise probability space by eigenspaces consisting of multiple Wiener integrals by the above process in terms of the Lévy Laplacian. By this decomposition, we obtain an expression of the semigroup generated by the Lévy Laplacian related to the semigroup generated by the number operator.","PeriodicalId":53434,"journal":{"name":"Communications on Stochastic Analysis","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2018-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48244204","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}
Both Wick-Ito-Skorokhod and Stratonovich interpretations of the parabolic Anderson model (PAM) lead to solutions that are real analytic as functions of the noise intensity e, and, in the limit e->0, the difference between the two solutions is of order e^2 and is non-random.
{"title":"An Asymptotic Comparison of Two Time-Homogeneous PAM Models","authors":"Hyun-Jung Kim, S. Lototsky","doi":"10.31390/COSA.12.2.06","DOIUrl":"https://doi.org/10.31390/COSA.12.2.06","url":null,"abstract":"Both Wick-Ito-Skorokhod and Stratonovich interpretations of the parabolic Anderson model (PAM) lead to solutions that are real analytic as functions of the noise intensity e, and, in the limit e->0, the difference between the two solutions is of order e^2 and is non-random.","PeriodicalId":53434,"journal":{"name":"Communications on Stochastic Analysis","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2018-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47990103","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 are interested in a nonlinear partial differential equation: the granular media one. Thanks to some of our previous results [10, 11], we know that under easily checked assumptions, there is a unique steady state. We point out that we consider a case in which the con(cid:12)ning potential is not globally convex. According to recent articles [8, 9], we know that there is weak convergence towards this steady state. However, we do not know anything about the rate of convergence. In this paper, we make a (cid:12)rst step to this direction by proving a deterministic Kramers’type law concerning the (cid:12)rst time that the solution of the granular media equation leaves a local well. In other words, we show that the solution of the granular media equation is trapped around a local minimum during a time exponentially equivalent to exp { 2 (cid:27) 2 H } , H being the so-called exit-cost.
{"title":"Exit-Time of Granular Media Equation Starting in a Local Minimum","authors":"J. Tugaut","doi":"10.31390/COSA.12.1.03","DOIUrl":"https://doi.org/10.31390/COSA.12.1.03","url":null,"abstract":". We are interested in a nonlinear partial differential equation: the granular media one. Thanks to some of our previous results [10, 11], we know that under easily checked assumptions, there is a unique steady state. We point out that we consider a case in which the con(cid:12)ning potential is not globally convex. According to recent articles [8, 9], we know that there is weak convergence towards this steady state. However, we do not know anything about the rate of convergence. In this paper, we make a (cid:12)rst step to this direction by proving a deterministic Kramers’type law concerning the (cid:12)rst time that the solution of the granular media equation leaves a local well. In other words, we show that the solution of the granular media equation is trapped around a local minimum during a time exponentially equivalent to exp { 2 (cid:27) 2 H } , H being the so-called exit-cost.","PeriodicalId":53434,"journal":{"name":"Communications on Stochastic Analysis","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2018-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45473936","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 express the tau functions considered by Pöppe in [23] for the Korteweg de Vries (KdV) equation, as the Laplace transforms of iterated Skorohod integrals. Our main tool is the notion of Fredholm determinant of an integral operator. A stochastic representation of tau functions for the N -soliton solutions of KdV has been proved by Ikeda and Taniguchi in [14]. They express the N -soliton solutions as the Laplace transform of a quadratic functional of N independent Ornstein-Uhlenbeck processes. Our first step is to provide the Wiener chaos decomposition of the underlying functional and to identify the Fredholm determinant of an integral operator in their representation. Our general result goes beyond the N -soliton case and enables us to consider a non soliton solution of KdV associated to a Gaussian process with Cauchy covariance function.
{"title":"Stochastic Representation of Tau Functions With an Application to the Korteweg-De Vries Equation","authors":"M. Thieullen, A. Vigot","doi":"10.31390/COSA.12.1.01","DOIUrl":"https://doi.org/10.31390/COSA.12.1.01","url":null,"abstract":"In this paper we express the tau functions considered by Pöppe in [23] for the Korteweg de Vries (KdV) equation, as the Laplace transforms of iterated Skorohod integrals. Our main tool is the notion of Fredholm determinant of an integral operator. A stochastic representation of tau functions for the N -soliton solutions of KdV has been proved by Ikeda and Taniguchi in [14]. They express the N -soliton solutions as the Laplace transform of a quadratic functional of N independent Ornstein-Uhlenbeck processes. Our first step is to provide the Wiener chaos decomposition of the underlying functional and to identify the Fredholm determinant of an integral operator in their representation. Our general result goes beyond the N -soliton case and enables us to consider a non soliton solution of KdV associated to a Gaussian process with Cauchy covariance function.","PeriodicalId":53434,"journal":{"name":"Communications on Stochastic Analysis","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2018-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42997535","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":"A Discrete Time Approximations for Certain Class of One-Dimensional Backward Stochastic Differential Equations via Girsanov's Theorem","authors":"A. Sghir, D. Seghir, S. Hadiri","doi":"10.31390/cosa.12.1.02","DOIUrl":"https://doi.org/10.31390/cosa.12.1.02","url":null,"abstract":"","PeriodicalId":53434,"journal":{"name":"Communications on Stochastic Analysis","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2018-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"69689586","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 present reversibility preserving operations on Markov chain transition matrices. Simple row and column operations allow us to create new reversible transition matrices and yield an easy method for checking a Markov chain for reversibility.
{"title":"Reversibility Checking for Markov Chains","authors":"Q. Jiang, M. Hlynka, P. Brill, C. H. Cheung","doi":"10.31390/COSA.12.2.02","DOIUrl":"https://doi.org/10.31390/COSA.12.2.02","url":null,"abstract":"In this paper, we present reversibility preserving operations on Markov chain transition matrices. Simple row and column operations allow us to create new reversible transition matrices and yield an easy method for checking a Markov chain for reversibility.","PeriodicalId":53434,"journal":{"name":"Communications on Stochastic Analysis","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2018-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41882237","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 present a Hilbert space approach to the limit joint *-distributions of complex independent Gaussian random matrices. For that purpose, we use a suitably defined family of creation and annihilation operators living in some direct integral of Hilbert spaces. These operators are decomposed in terms of continuous circular systems of operators acting between the fibers of the considered Hilbert space direct integral. In the case of square matrices with i.i.d. entries, we obtain the circular operators of Voiculescu, whereas in the case of upper-triangular matrices with i.i.d. entries, we obtain the triangular operators of Dykema and Haagerup. We apply this approach to give a bijective proof of a formula for *-moments of the triangular operator, using the enumeration formula of Chauve, Dulucq and Rechnizter for alternating ordered rooted trees.
{"title":"Random Matrices, Continuous Circular Systems and the Triangular Operator","authors":"R. Lenczewski","doi":"10.31390/cosa.12.3.02","DOIUrl":"https://doi.org/10.31390/cosa.12.3.02","url":null,"abstract":"We present a Hilbert space approach to the limit joint *-distributions of complex independent Gaussian random matrices. For that purpose, we use a suitably defined family of creation and annihilation operators living in some direct integral of Hilbert spaces. These operators are decomposed in terms of continuous circular systems of operators acting between the fibers of the considered Hilbert space direct integral. In the case of square matrices with i.i.d. entries, we obtain the circular operators of Voiculescu, whereas in the case of upper-triangular matrices with i.i.d. entries, we obtain the triangular operators of Dykema and Haagerup. We apply this approach to give a bijective proof of a formula for *-moments of the triangular operator, using the enumeration formula of Chauve, Dulucq and Rechnizter for alternating ordered rooted trees.","PeriodicalId":53434,"journal":{"name":"Communications on Stochastic Analysis","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2018-04-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44716223","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 article we study the existence and uniqueness of strong solutions of a class of parameterized family of SDEs driven by L'evy noise. These SDEs occurs in connection with a class of stochastic PDEs, which take values in the space of tempered distributions $mathcal{S}^prime$. This correspondence for diffusion processes was proved in [Rajeev, Translation invariant diffusion in the space of tempered distributions, Indian J. Pure Appl. Math. 44 (2013), no.~2, 231--258].
{"title":"Parametric Family of SDEs Driven by Lévy Noise","authors":"Suprio Bhar, Barun Sarkar","doi":"10.31390/COSA.12.2.04","DOIUrl":"https://doi.org/10.31390/COSA.12.2.04","url":null,"abstract":"In this article we study the existence and uniqueness of strong solutions of a class of parameterized family of SDEs driven by L'evy noise. These SDEs occurs in connection with a class of stochastic PDEs, which take values in the space of tempered distributions $mathcal{S}^prime$. This correspondence for diffusion processes was proved in [Rajeev, Translation invariant diffusion in the space of tempered distributions, Indian J. Pure Appl. Math. 44 (2013), no.~2, 231--258].","PeriodicalId":53434,"journal":{"name":"Communications on Stochastic Analysis","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2018-01-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42046540","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 simplified version of the problem of insider trading in a financial market. We approach it by means of anticipating stochastic calculus and compare the use of the Hitsuda-Skorokhod, the Ayed-Kuo, and the Russo-Vallois forward integrals within this context. We conclude that, while the forward integral yields results with a suitable financial meaning, the Hitsuda-Skorokhod and the Ayed-Kuo integrals do not provide an appropriate formulation of this problem. Further results regarding the use of the Ayed-Kuo integral in this context are also provided, including the proof of the fact that the expectation of a Russo-Vallois solution is strictly greater than that of an Ayed-Kuo solution. Finally, we conjecture the explicit solution of an Ayed-Kuo stochastic differential equation that possesses discontinuous sample paths with finite probability.
{"title":"A Triple Comparison between Anticipating Stochastic Integrals in Financial Modeling","authors":"Joan C. Bastons, C. Escudero","doi":"10.31390/COSA.12.1.06","DOIUrl":"https://doi.org/10.31390/COSA.12.1.06","url":null,"abstract":"We consider a simplified version of the problem of insider trading in a financial market. We approach it by means of anticipating stochastic calculus and compare the use of the Hitsuda-Skorokhod, the Ayed-Kuo, and the Russo-Vallois forward integrals within this context. We conclude that, while the forward integral yields results with a suitable financial meaning, the Hitsuda-Skorokhod and the Ayed-Kuo integrals do not provide an appropriate formulation of this problem. Further results regarding the use of the Ayed-Kuo integral in this context are also provided, including the proof of the fact that the expectation of a Russo-Vallois solution is strictly greater than that of an Ayed-Kuo solution. Finally, we conjecture the explicit solution of an Ayed-Kuo stochastic differential equation that possesses discontinuous sample paths with finite probability.","PeriodicalId":53434,"journal":{"name":"Communications on Stochastic Analysis","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2018-01-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46854356","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 prove a non-central limit theorem for the symmetric weighted odd-power variations of the fractional Brownian motion with Hurst parameter H = 0, where X is a fractional Brownian motion and Y is an independent Brownian motion.
{"title":"Symmetric Weighted Odd-Power Variations of Fractional Brownian Motion and Applications","authors":"D. Nualart, Raghid Zeineddine","doi":"10.31390/COSA.12.1.04","DOIUrl":"https://doi.org/10.31390/COSA.12.1.04","url":null,"abstract":"We prove a non-central limit theorem for the symmetric weighted odd-power variations of the fractional Brownian motion with Hurst parameter H = 0, where X is a fractional Brownian motion and Y is an independent Brownian motion.","PeriodicalId":53434,"journal":{"name":"Communications on Stochastic Analysis","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2017-11-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49271156","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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