It is well known that the Girsanov transform (or, Girsanov's theorem) plays an important role in the stochastic analysis and this transform is closely related with the uniform integrability of local martingales. The ̄rst aim of this article is to give concrete, necessary and su±cient conditions of uniform integrability of positive local martingales with jumps. Then we shall apply Girsanov transform to Zakai equation (Zakai SDE) arisen from the ̄ltering problem of stochastic processes with jumps. Using Girsanov transform for L¶evy processes, Malliavin calculus could be applied to show the existence of smooth density of the ̄ltering measure. The second aim of this article is to show the uniqueness of solutions of Zakai equation. This is worthwhile from the fact that the solution of Zakai equation can be obtained from the ̄ltering measure by using Girsanov transform.
{"title":"Generalized Girsanov Transform of Processes and Zakai Equation with Jumps","authors":"M. Fujisaki, T. Komatsu","doi":"10.31390/josa.2.3.17","DOIUrl":"https://doi.org/10.31390/josa.2.3.17","url":null,"abstract":"It is well known that the Girsanov transform (or, Girsanov's theorem) plays an important role in the stochastic analysis and this transform is closely related with the uniform integrability of local martingales. The ̄rst aim of this article is to give concrete, necessary and su±cient conditions of uniform integrability of positive local martingales with jumps. Then we shall apply Girsanov transform to Zakai equation (Zakai SDE) arisen from the ̄ltering problem of stochastic processes with jumps. Using Girsanov transform for L¶evy processes, Malliavin calculus could be applied to show the existence of smooth density of the ̄ltering measure. The second aim of this article is to show the uniqueness of solutions of Zakai equation. This is worthwhile from the fact that the solution of Zakai equation can be obtained from the ̄ltering measure by using Girsanov transform.","PeriodicalId":263604,"journal":{"name":"Journal of Stochastic Analysis","volume":"27 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126008776","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 uniqueness of solutions to martingale problems for diffusion operators with progressively measurable coefficients is studied and a uniqueness result is obtained: the uniqueness holds under the conditions of the boundedness and uniform ellipticity for the coefficients of the diffusion operators and under an additional condition for the diffusion coefficients. Construction of appropriate approximation consisting of simple functions to the diffusion coefficients plays a key role; the additional condition is used to ensure the simpleness and then the uniqueness follows from the result in the case of diffusion operators with simple type coefficients, which is due to Stroock and Varadhan.
{"title":"On the Uniqueness of Solutions to Martingale Problems for Diffusion Operators with Progressively Measurable Random Coefficients","authors":"M. Tsuchiya","doi":"10.31390/josa.2.3.16","DOIUrl":"https://doi.org/10.31390/josa.2.3.16","url":null,"abstract":"The uniqueness of solutions to martingale problems for diffusion operators with progressively measurable coefficients is studied and a uniqueness result is obtained: the uniqueness holds under the conditions of the boundedness and uniform ellipticity for the coefficients of the diffusion operators and under an additional condition for the diffusion coefficients. Construction of appropriate approximation consisting of simple functions to the diffusion coefficients plays a key role; the additional condition is used to ensure the simpleness and then the uniqueness follows from the result in the case of diffusion operators with simple type coefficients, which is due to Stroock and Varadhan.","PeriodicalId":263604,"journal":{"name":"Journal of Stochastic Analysis","volume":"73 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-09-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127330767","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 the present paper we will introduce an anti-symmetric version of Malliavin calculus which consists of operators with anti-commuting relations, which actually form an in(cid:12)nite-dimensional Clifford algebra.
{"title":"An Anti-Symmetric Version of Malliavin Calculus","authors":"J. Akahori, T. Matsusita, Yasufumi Nitta","doi":"10.31390/josa.2.3.14","DOIUrl":"https://doi.org/10.31390/josa.2.3.14","url":null,"abstract":". In the present paper we will introduce an anti-symmetric version of Malliavin calculus which consists of operators with anti-commuting relations, which actually form an in(cid:12)nite-dimensional Clifford algebra.","PeriodicalId":263604,"journal":{"name":"Journal of Stochastic Analysis","volume":"74 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123805920","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 study the regularity of a Markov semigroup (Pt)t>0, that is, when Pt(x, dy) = pt(x, y)dy for a suitable smooth function pt(x, y). This is done by transferring the regularity from an approximating Markov semigroup sequence (Pn t )t>0, n ∈ N, whose associated densities pt (x, y) are smooth and can blow up as n → ∞. We use an interpolation type result and we show that if there exists a good equilibrium between the blow-up and the speed of convergence, then Pt(x, dy) = pt(x, y)dy and pt has some regularity properties.
{"title":"Transfer of Regularity for Markov Semigroups by Using an Interpolation Technique","authors":"V. Bally, L. Caramellino","doi":"10.31390/josa.2.3.13","DOIUrl":"https://doi.org/10.31390/josa.2.3.13","url":null,"abstract":"We study the regularity of a Markov semigroup (Pt)t>0, that is, when Pt(x, dy) = pt(x, y)dy for a suitable smooth function pt(x, y). This is done by transferring the regularity from an approximating Markov semigroup sequence (Pn t )t>0, n ∈ N, whose associated densities pt (x, y) are smooth and can blow up as n → ∞. We use an interpolation type result and we show that if there exists a good equilibrium between the blow-up and the speed of convergence, then Pt(x, dy) = pt(x, y)dy and pt has some regularity properties.","PeriodicalId":263604,"journal":{"name":"Journal of Stochastic Analysis","volume":"11 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-08-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115448067","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}
Consider solutions to Marcus-type stochastic differential equations with jumps on the bundle of orthonormal frames O(M) over a Riemannian manifold M , and define the M -valued process by its canonical projection, which is parallel to the Eells-Elworthy-Malliavin construction of Brownian motions on M . In the present paper, the integration by parts formula for such jump processes is studied, and the strategy is based upon the calculus on Brownian motions via the Kolmogorov backward equations. The celebrated Bismut formula can be also obtained in our setting.
{"title":"Integration by Parts Formula on Solutions to Stochastic Differential Equations with Jumps on Riemannian Manifolds","authors":"Hirotaka Kai, Atsushi Takeuchi","doi":"10.31390/josa.2.3.12","DOIUrl":"https://doi.org/10.31390/josa.2.3.12","url":null,"abstract":"Consider solutions to Marcus-type stochastic differential equations with jumps on the bundle of orthonormal frames O(M) over a Riemannian manifold M , and define the M -valued process by its canonical projection, which is parallel to the Eells-Elworthy-Malliavin construction of Brownian motions on M . In the present paper, the integration by parts formula for such jump processes is studied, and the strategy is based upon the calculus on Brownian motions via the Kolmogorov backward equations. The celebrated Bismut formula can be also obtained in our setting.","PeriodicalId":263604,"journal":{"name":"Journal of Stochastic Analysis","volume":"66 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-08-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122882499","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 theorem on the exponential moments of general R -valued additive processes will be established. A condition that implies the integrability of the exponential of additive processes will be proposed and furthermore the representation of their exponential moments by their characteristics will be shown. In the previous paper [1], the same problem as above has been investigated in the case when the underlying additive processes have the structure of semimartingales. In this paper, another proof for this case will be presented. It will be more inherent and simpler than the previous one. Moreover, the result will be generalized to the case when the underlying additive processes do not necessarily have the structure of semimartingales.
{"title":"On the Exponential Moments of Additive Processes","authors":"Tsukasa Fujiwara","doi":"10.31390/josa.2.3.11","DOIUrl":"https://doi.org/10.31390/josa.2.3.11","url":null,"abstract":". A theorem on the exponential moments of general R -valued additive processes will be established. A condition that implies the integrability of the exponential of additive processes will be proposed and furthermore the representation of their exponential moments by their characteristics will be shown. In the previous paper [1], the same problem as above has been investigated in the case when the underlying additive processes have the structure of semimartingales. In this paper, another proof for this case will be presented. It will be more inherent and simpler than the previous one. Moreover, the result will be generalized to the case when the underlying additive processes do not necessarily have the structure of semimartingales.","PeriodicalId":263604,"journal":{"name":"Journal of Stochastic Analysis","volume":"150 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-08-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125435257","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}
. Two of Kunita’s papers in early 1980s on diffeomorphic property of stochastic (cid:13)ows are revisited, and corresponding results by the author are presented.
{"title":"Two of Kunita's Papers on Stochastic Flows in Early 1980s","authors":"S. Taniguchi","doi":"10.31390/josa.2.3.09","DOIUrl":"https://doi.org/10.31390/josa.2.3.09","url":null,"abstract":". Two of Kunita’s papers in early 1980s on diffeomorphic property of stochastic (cid:13)ows are revisited, and corresponding results by the author are presented.","PeriodicalId":263604,"journal":{"name":"Journal of Stochastic Analysis","volume":"47 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-08-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127111967","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 describe the life and mathematical work of Hiroshi Kunita and add a few personal recollections which show how admirable person he was.
我们描述了国田浩的生活和数学工作，并添加了一些个人回忆，表明他是一个多么令人钦佩的人。
{"title":"The Life and Scientific Work of Hiroshi Kunita","authors":"Y. Ishikawa","doi":"10.31390/josa.2.3.05","DOIUrl":"https://doi.org/10.31390/josa.2.3.05","url":null,"abstract":"We describe the life and mathematical work of Hiroshi Kunita and add a few personal recollections which show how admirable person he was.","PeriodicalId":263604,"journal":{"name":"Journal of Stochastic Analysis","volume":"17 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125574930","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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