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A Cameron-Martin type formula for general Gaussian processes--a filtering approach 一般高斯过程的Cameron-Martin型公式——一种滤波方法
Stochastics and Stochastic Reports Pub Date : 2002-01-01 DOI: 10.1080/10451120290019203
M. Kleptsyna, A. Breton
{"title":"A Cameron-Martin type formula for general Gaussian processes--a filtering approach","authors":"M. Kleptsyna, A. Breton","doi":"10.1080/10451120290019203","DOIUrl":"https://doi.org/10.1080/10451120290019203","url":null,"abstract":"A Cameron-Martin type formula is derived for the Laplace transform of some integrals of the square of a general continuous Gaussian process. The formula involves in particular the variance of the filtering error in some auxiliary optimal filtering problem which is used in the proof. This variance is expressed in terms of the solution of a Riccati-Volterra type integral equation containing the covariance function of the process. In various specific cases this equation is solved and then the formula becomes completely explicit.","PeriodicalId":289792,"journal":{"name":"Stochastics and Stochastic Reports","volume":"43 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2002-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115857771","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}
引用次数: 20
Pseudo-moderate deviations in the euler method for real diffusion process 真实扩散过程中欧拉方法的拟中等偏差
Stochastics and Stochastic Reports Pub Date : 2002-01-01 DOI: 10.1080/10451120290008575
E. Clement
{"title":"Pseudo-moderate deviations in the euler method for real diffusion process","authors":"E. Clement","doi":"10.1080/10451120290008575","DOIUrl":"https://doi.org/10.1080/10451120290008575","url":null,"abstract":"Consider a one-dimensional process (X t ) solution of a stochastic differential equation driven by a Brownian motion. In order to simulate the process (X t ), it is usual in practice to approximate it by an Euler scheme denoted by (X n t ). We are interested in the study of the asymptotic behaviour of the error process (X-X) and we prove that it satisfies a pseudo-moderate deviation principle.","PeriodicalId":289792,"journal":{"name":"Stochastics and Stochastic Reports","volume":"65 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2002-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130787755","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}
引用次数: 2
Backward stochastic differential equations with Azéma's martingale 带azsamma鞅的倒向随机微分方程
Stochastics and Stochastic Reports Pub Date : 2002-01-01 DOI: 10.1080/1045112021000004216
C. Rainer
{"title":"Backward stochastic differential equations with Azéma's martingale","authors":"C. Rainer","doi":"10.1080/1045112021000004216","DOIUrl":"https://doi.org/10.1080/1045112021000004216","url":null,"abstract":"The aim of this paper is to study backward stochastic differential equations (BSDE) driven by Azéma's martingale and the associated deterministic functional equations. More precisely, we introduce BSDE's vs. Azéma's martingale in a general frame, then we prove that the existence of a solution to a Markovian BSDE implies the existence of a solution to a deterministic functional equation of a new type. Uniqueness for the functional equation is proved in a particular case. Then we discuss BSDE's vs. an asymmetric martingale: half Brownian motion/half Azéma's martingale, which leads to an asymmetric deterministic functional equation.","PeriodicalId":289792,"journal":{"name":"Stochastics and Stochastic Reports","volume":"35 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2002-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125038145","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}
引用次数: 5
On a variational inequality associated with a stopping game combined with a control 关于与停止博弈结合控制相关联的变分不等式
Stochastics and Stochastic Reports Pub Date : 2002-01-01 DOI: 10.1080/10451120212867
Kenji Kamizono, H. Morimoto
{"title":"On a variational inequality associated with a stopping game combined with a control","authors":"Kenji Kamizono, H. Morimoto","doi":"10.1080/10451120212867","DOIUrl":"https://doi.org/10.1080/10451120212867","url":null,"abstract":"We study a non-linear elliptic variational inequality which corresponds to a zero-sum stopping game (Dynkin game) combined with a control. Our result is a generalization of the existing works by Bensoussan [ Stochastic Control by Functional Analysis Methods (North-Holland, Amsterdam), 1982], Bensoussan and Lions [ Applications des Inéquations Variationnelles en Contrôle Stochastique (Dunod, Paris), 1978] and Friedman [ Stochastic Differential Equations and Applications (Academic Press, New York), 1976] in the sense that a non-linear term appears in the variational inequality, or equivalently, that the underlying process for the corresponding stopping game is subject to a control. By using the dynamic programming principle and the method of penalization, we show the existence and uniqueness of a viscosity solution of the variational inequality and describe it as the value function of the corresponding combined-stochastic game problem.","PeriodicalId":289792,"journal":{"name":"Stochastics and Stochastic Reports","volume":"313 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2002-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121293259","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}
引用次数: 13
Discrete chaotic calculus and covariance identities 离散混沌演算与协方差恒等式
Stochastics and Stochastic Reports Pub Date : 2002-01-01 DOI: 10.1080/10451120290019230
Nicolas Privault, W. Schoutens
{"title":"Discrete chaotic calculus and covariance identities","authors":"Nicolas Privault, W. Schoutens","doi":"10.1080/10451120290019230","DOIUrl":"https://doi.org/10.1080/10451120290019230","url":null,"abstract":"We show that for the binomial process (or Bernoulli random walk) the orthogonal functionals constructed in Kroeker, J.P. (1980) \"Wiener analysis of functionals of a Markov chain: application to neural transformations of random signals\", Biol. Cybernetics 36 , 243-248, [14] for Markov chains can be expressed using the Krawtchouk polynomials, and by iterated stochastic integrals. This allows to construct a chaotic calculus based on gradient and divergence operators and structure equations, and to establish a Clark representation formula. As an application we obtain simple infinite dimensional proofs of covariance identities on the discrete cube.","PeriodicalId":289792,"journal":{"name":"Stochastics and Stochastic Reports","volume":"110 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2002-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124223121","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}
引用次数: 24
On the stochastic versions of Neumann and oblique derivative problems 关于Neumann和斜导数问题的随机版本
Stochastics and Stochastic Reports Pub Date : 2002-01-01 DOI: 10.1080/10451120290028104
Y. Rozanov, F. Sansò
{"title":"On the stochastic versions of Neumann and oblique derivative problems","authors":"Y. Rozanov, F. Sansò","doi":"10.1080/10451120290028104","DOIUrl":"https://doi.org/10.1080/10451120290028104","url":null,"abstract":"In some areas, for instance geodesy, one finds the need of suitably defining the solution of certain boundary value problems (BVPs), for instance, the Laplace equation, where boundary data are very irregular and can be described as fields of random variables, with suitable regularity constraints (Rummel and Sansò, Lecture Notes on Earth Sciences, Vol. 65, 1997). This item has been attacked in the literature, although mainly for the case of the Dirichlet problem while much less material is available, for instance, for the Neumann and the Oblique Derivative Problem. In studying these stochastic problems in detail, the authors have found fairly general criteria which provide an automatic translation of a deterministic result into the corresponding stochastic one.","PeriodicalId":289792,"journal":{"name":"Stochastics and Stochastic Reports","volume":"76 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2002-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127695686","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}
引用次数: 4
Monte Carlo construction of hedging strategies against multi-asset European claims 针对多资产欧洲债权的蒙特卡洛对冲策略构建
Stochastics and Stochastic Reports Pub Date : 2002-01-01 DOI: 10.1080/10451120212868
G. Milstein, J. Schoenmakers
{"title":"Monte Carlo construction of hedging strategies against multi-asset European claims","authors":"G. Milstein, J. Schoenmakers","doi":"10.1080/10451120212868","DOIUrl":"https://doi.org/10.1080/10451120212868","url":null,"abstract":"For evaluating a hedging strategy we have to know at every moment the solution of the Cauchy problem for a corresponding parabolic equation (the value of the hedging portfolio) and its derivatives (the deltas). We suggest to find these quantities by Monte Carlo simulation of the corresponding system of stochastic differential equations using weak solution schemes. It turns out that with one and the same control function a variance reduction can be achieved simultaneously for the claim value as well as for the deltas. As illustrations we consider a Markovian multi-asset model with an instantaneously riskless saving bond and also some applications to the LIBOR rate model of Brace, Gatarck, Musiela and Jamshidian.","PeriodicalId":289792,"journal":{"name":"Stochastics and Stochastic Reports","volume":"27 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2002-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125279872","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}
引用次数: 31
Stochastic integral representations, stochastic derivatives and minimal variance hedging 随机积分表示，随机导数和最小方差对冲
Stochastics and Stochastic Reports Pub Date : 2002-01-01 DOI: 10.1080/10451120212869
G. di Nunno
{"title":"Stochastic integral representations, stochastic derivatives and minimal variance hedging","authors":"G. di Nunno","doi":"10.1080/10451120212869","DOIUrl":"https://doi.org/10.1080/10451120212869","url":null,"abstract":"The stochastic integral representation for an arbitrary random variable in a standard L 2 -space is considered in the case of the integrator as a martingale. In relation to this, a certain stochastic derivative is defined. It is shown that this derivative determines the integrand in the stochastic integral which serves as the best L 2 - approximation to the random variable considered. For a general Lévy process as integrator some specification of the suggested stochastic derivative is given. In the case of the Wiener process the considered specification reduces to the well-known Clark-Haussmann-Ocone formula. This result provides a general solution to the problem of minimal variance hedging in incomplete markets.","PeriodicalId":289792,"journal":{"name":"Stochastics and Stochastic Reports","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2002-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124258468","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}
引用次数: 30
Random times at which insiders can have free lunches 内部人员可以免费享用午餐的随机时间
Stochastics and Stochastic Reports Pub Date : 2002-01-01 DOI: 10.1080/10451120290024892
P. Imkeller
{"title":"Random times at which insiders can have free lunches","authors":"P. Imkeller","doi":"10.1080/10451120290024892","DOIUrl":"https://doi.org/10.1080/10451120290024892","url":null,"abstract":"We consider models of time continuous financial markets with a regular trader and an insider who are able to invest into one risky asset. The insider's additional knowledge consists in his ability to stop at a random time which is inaccessible to the regular trader, such as the last passage of a certain level before maturity by some stock price process, or the time at which the stock price reaches its maximum during the trading interval. We show that under very mild assumptions on the coefficients of the diffusion process describing these price processes the information drift caused by the additional knowledge of the insider cannot be eliminated by an equivalent change of probability measure. As a consequence, all our models allow the insider to have free lunches with vanishing risk, or even to exercise arbitrage.","PeriodicalId":289792,"journal":{"name":"Stochastics and Stochastic Reports","volume":"35 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2002-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131605737","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}
引用次数: 68
Reflected BSDE's with discontinuous barrier and application 反映了BSDE的不连续屏障及其应用
Stochastics and Stochastic Reports Pub Date : 2002-01-01 DOI: 10.1080/1045112021000036545
S. Hamadène
{"title":"Reflected BSDE's with discontinuous barrier and application","authors":"S. Hamadène","doi":"10.1080/1045112021000036545","DOIUrl":"https://doi.org/10.1080/1045112021000036545","url":null,"abstract":"We deal with reflected backward stochastic differential equations with right continuous and left limited barrier. We show the existence and uniqueness of the solution and we give a comparison theorem. As an application, we study the link between such an equations with stochastic mixed control problems.","PeriodicalId":289792,"journal":{"name":"Stochastics and Stochastic Reports","volume":"102 4","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2002-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"120941678","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}
引用次数: 141
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