Pub Date : 2013-12-09DOI: 10.1080/17442508.2014.995659
Dejun Luo
We consider stochastic differential equations on the group of volume-preserving homeomorphisms of the sphere . The diffusion part is given by the divergence-free eigenvector fields of the Laplacian acting on -vector fields, while the drift is some other divergence-free vector field. We show that the equation generates a unique flow of measure-preserving homeomorphisms when the drift has first-order Sobolev regularity, and derive a formula for the distance between two Lagrangian flows. We also compute the rotation process of two particles on the sphere when they are close to each other.
{"title":"Stochastic Lagrangian flows on the group of volume-preserving homeomorphisms of the spheres","authors":"Dejun Luo","doi":"10.1080/17442508.2014.995659","DOIUrl":"https://doi.org/10.1080/17442508.2014.995659","url":null,"abstract":"We consider stochastic differential equations on the group of volume-preserving homeomorphisms of the sphere . The diffusion part is given by the divergence-free eigenvector fields of the Laplacian acting on -vector fields, while the drift is some other divergence-free vector field. We show that the equation generates a unique flow of measure-preserving homeomorphisms when the drift has first-order Sobolev regularity, and derive a formula for the distance between two Lagrangian flows. We also compute the rotation process of two particles on the sphere when they are close to each other.","PeriodicalId":49269,"journal":{"name":"Stochastics-An International Journal of Probability and Stochastic Processes","volume":"24 1","pages":"680 - 701"},"PeriodicalIF":0.9,"publicationDate":"2013-12-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75829788","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2013-11-27DOI: 10.1080/17442508.2014.895358
Claudio Fontana
We provide a critical analysis of the proof of the fundamental theorem of asset pricing given in the paper Arbitrage and approximate arbitrage: the fundamental theorem of asset pricing by B. Wong and C.C. Heyde [Stochastics 82 (2010), pp. 189–200] in the context of incomplete Itô-process models. We show that their approach can only work in the known case of a complete financial market model and give an explicit counter example.
本文对B. Wong和C.C. Heyde [Stochastics 82 (2010)， pp. 189-200]论文《套利和近似套利:资产定价基本定理》中资产定价基本定理在不完全Itô-process模型背景下的证明进行了批判性分析。我们证明了他们的方法只能在一个完整的金融市场模型的已知情况下起作用，并给出了一个明确的反例。
{"title":"A note on arbitrage, approximate arbitrage and the fundamental theorem of asset pricing","authors":"Claudio Fontana","doi":"10.1080/17442508.2014.895358","DOIUrl":"https://doi.org/10.1080/17442508.2014.895358","url":null,"abstract":"We provide a critical analysis of the proof of the fundamental theorem of asset pricing given in the paper Arbitrage and approximate arbitrage: the fundamental theorem of asset pricing by B. Wong and C.C. Heyde [Stochastics 82 (2010), pp. 189–200] in the context of incomplete Itô-process models. We show that their approach can only work in the known case of a complete financial market model and give an explicit counter example.","PeriodicalId":49269,"journal":{"name":"Stochastics-An International Journal of Probability and Stochastic Processes","volume":"18 1","pages":"922 - 931"},"PeriodicalIF":0.9,"publicationDate":"2013-11-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75373933","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2013-11-07DOI: 10.1080/17442508.2013.865131
Charles Curry, K. Ebrahimi-Fard, S. Malham, Anke Wiese
We investigate the algebra of repeated integrals of semimartingales. We prove that a minimal family of semimartingales generates a quasi-shuffle algebra. In essence, to fulfil the minimality criterion, first, the family must be a minimal generator of the algebra of repeated integrals generated by its elements and by quadratic covariation processes recursively constructed from the elements of the family. Second, recursively constructed quadratic covariation processes may lie in the linear span of previously constructed quadratic covariation processes and of the family, but may not lie in the linear span of repeated integrals of these. We prove that a finite family of independent Lévy processes that have finite moments generates a minimal family. Key to the proof are the Teugels martingales and a strong orthogonalization of them. We conclude that a finite family of independent Lévy processes forms a quasi-shuffle algebra. We discuss important potential applications to constructing efficient numerical methods for the strong approximation of stochastic differential equations driven by Lévy processes.
{"title":"Lévy processes and quasi-shuffle algebras","authors":"Charles Curry, K. Ebrahimi-Fard, S. Malham, Anke Wiese","doi":"10.1080/17442508.2013.865131","DOIUrl":"https://doi.org/10.1080/17442508.2013.865131","url":null,"abstract":"We investigate the algebra of repeated integrals of semimartingales. We prove that a minimal family of semimartingales generates a quasi-shuffle algebra. In essence, to fulfil the minimality criterion, first, the family must be a minimal generator of the algebra of repeated integrals generated by its elements and by quadratic covariation processes recursively constructed from the elements of the family. Second, recursively constructed quadratic covariation processes may lie in the linear span of previously constructed quadratic covariation processes and of the family, but may not lie in the linear span of repeated integrals of these. We prove that a finite family of independent Lévy processes that have finite moments generates a minimal family. Key to the proof are the Teugels martingales and a strong orthogonalization of them. We conclude that a finite family of independent Lévy processes forms a quasi-shuffle algebra. We discuss important potential applications to constructing efficient numerical methods for the strong approximation of stochastic differential equations driven by Lévy processes.","PeriodicalId":49269,"journal":{"name":"Stochastics-An International Journal of Probability and Stochastic Processes","volume":"54 1","pages":"632 - 642"},"PeriodicalIF":0.9,"publicationDate":"2013-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91095763","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2013-09-29DOI: 10.1080/17442508.2015.1019882
E. Hashorva, Y. Mishura, O. Seleznjev
In this paper, we investigate the boundary non-crossing probabilities of a fractional Brownian motion considering some general deterministic trend function. We derive bounds for non-crossing probabilities and discuss the case of a large trend function. As a by-product, we solve a minimization problem related to the norm of the trend function.
{"title":"Boundary non-crossing probabilities for fractional Brownian motion with trend","authors":"E. Hashorva, Y. Mishura, O. Seleznjev","doi":"10.1080/17442508.2015.1019882","DOIUrl":"https://doi.org/10.1080/17442508.2015.1019882","url":null,"abstract":"In this paper, we investigate the boundary non-crossing probabilities of a fractional Brownian motion considering some general deterministic trend function. We derive bounds for non-crossing probabilities and discuss the case of a large trend function. As a by-product, we solve a minimization problem related to the norm of the trend function.","PeriodicalId":49269,"journal":{"name":"Stochastics-An International Journal of Probability and Stochastic Processes","volume":"34 1","pages":"946 - 965"},"PeriodicalIF":0.9,"publicationDate":"2013-09-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75635057","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2013-09-26DOI: 10.1080/17442508.2015.1013959
Patrick Cheridito, Kihun Nam
We study multidimensional BSDEs of the formwith bounded terminal conditions and drivers that grow at most quadratically in . We consider three different cases. In the first case, the BSDE is Markovian, and a solution can be obtained from a solution to a related FBSDE. In the second case, the BSDE becomes a one-dimensional quadratic BSDE when projected to a one-dimensional subspace, and a solution can be derived from a solution of the one-dimensional equation. In the third case, the growth of the driver in is strictly subquadratic, and the existence and uniqueness of a solution can be shown by first solving the BSDE on a short time interval and then extending it recursively.
{"title":"Multidimensional quadratic and subquadratic BSDEs with special structure","authors":"Patrick Cheridito, Kihun Nam","doi":"10.1080/17442508.2015.1013959","DOIUrl":"https://doi.org/10.1080/17442508.2015.1013959","url":null,"abstract":"We study multidimensional BSDEs of the formwith bounded terminal conditions and drivers that grow at most quadratically in . We consider three different cases. In the first case, the BSDE is Markovian, and a solution can be obtained from a solution to a related FBSDE. In the second case, the BSDE becomes a one-dimensional quadratic BSDE when projected to a one-dimensional subspace, and a solution can be derived from a solution of the one-dimensional equation. In the third case, the growth of the driver in is strictly subquadratic, and the existence and uniqueness of a solution can be shown by first solving the BSDE on a short time interval and then extending it recursively.","PeriodicalId":49269,"journal":{"name":"Stochastics-An International Journal of Probability and Stochastic Processes","volume":"33 1","pages":"871 - 884"},"PeriodicalIF":0.9,"publicationDate":"2013-09-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80825067","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2013-09-01DOI: 10.1080/17442508.2013.879142
J. Choi
We revisit the optimal investment and consumption problem with proportional transaction costs. We prove that both the value function and the slopes of the lines demarcating the no-trading region are analytic functions of cube root of the transaction cost parameter. Also, we can explicitly calculate the coefficients of the fractional power series expansions of the value function and the no-trading region.
{"title":"Asymptotic analysis for Merton's problem with transaction costs in power utility case","authors":"J. Choi","doi":"10.1080/17442508.2013.879142","DOIUrl":"https://doi.org/10.1080/17442508.2013.879142","url":null,"abstract":"We revisit the optimal investment and consumption problem with proportional transaction costs. We prove that both the value function and the slopes of the lines demarcating the no-trading region are analytic functions of cube root of the transaction cost parameter. Also, we can explicitly calculate the coefficients of the fractional power series expansions of the value function and the no-trading region.","PeriodicalId":49269,"journal":{"name":"Stochastics-An International Journal of Probability and Stochastic Processes","volume":"7 1","pages":"803 - 816"},"PeriodicalIF":0.9,"publicationDate":"2013-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87237990","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2013-08-27DOI: 10.1080/17442508.2014.915973
S. Hamadène, Rui Mu
This paper is related to non-zero-sum stochastic differential games in the Markovian framework. We show existence of a Nash equilibrium point for the game when the drift is no longer bounded and only satisfies a linear growth condition. The main tool is the notion of backward stochastic differential equations which, in our case, are multidimensional with continuous coefficient and stochastic linear growth.
{"title":"Existence of Nash equilibrium points for Markovian non-zero-sum stochastic differential games with unbounded coefficients","authors":"S. Hamadène, Rui Mu","doi":"10.1080/17442508.2014.915973","DOIUrl":"https://doi.org/10.1080/17442508.2014.915973","url":null,"abstract":"This paper is related to non-zero-sum stochastic differential games in the Markovian framework. We show existence of a Nash equilibrium point for the game when the drift is no longer bounded and only satisfies a linear growth condition. The main tool is the notion of backward stochastic differential equations which, in our case, are multidimensional with continuous coefficient and stochastic linear growth.","PeriodicalId":49269,"journal":{"name":"Stochastics-An International Journal of Probability and Stochastic Processes","volume":"42 5 1","pages":"111 - 85"},"PeriodicalIF":0.9,"publicationDate":"2013-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72861377","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2013-08-16DOI: 10.1080/17442508.2014.882924
G. Blower, C. Brett, I. Doust
The nonlinear Schrödinger equation , , arises from a Hamiltonian on infinite-dimensional phase space . For , Bourgain (Comm. Math. Phys. 166 (1994), 1–26) has shown that there exists a Gibbs measure on balls in phase space such that the Cauchy problem for is well posed on the support of , and that is invariant under the flow. This paper shows that satisfies a logarithmic Sobolev inequality (LSI) for the focusing case and on for all N>0; also satisfies a restricted LSI for on compact subsets of determined by Hölder norms. Hence for p = 4, the spectral data of the periodic Dirac operator in with random potential subject to are concentrated near to their mean values. The paper concludes with a similar result for the spectral data of Hill's equation when the potential is random and subject to the Gibbs measure of Korteweg–de Vries.
{"title":"Logarithmic Sobolev inequalities and spectral concentration for the cubic Schrödinger equation","authors":"G. Blower, C. Brett, I. Doust","doi":"10.1080/17442508.2014.882924","DOIUrl":"https://doi.org/10.1080/17442508.2014.882924","url":null,"abstract":"The nonlinear Schrödinger equation , , arises from a Hamiltonian on infinite-dimensional phase space . For , Bourgain (Comm. Math. Phys. 166 (1994), 1–26) has shown that there exists a Gibbs measure on balls in phase space such that the Cauchy problem for is well posed on the support of , and that is invariant under the flow. This paper shows that satisfies a logarithmic Sobolev inequality (LSI) for the focusing case and on for all N>0; also satisfies a restricted LSI for on compact subsets of determined by Hölder norms. Hence for p = 4, the spectral data of the periodic Dirac operator in with random potential subject to are concentrated near to their mean values. The paper concludes with a similar result for the spectral data of Hill's equation when the potential is random and subject to the Gibbs measure of Korteweg–de Vries.","PeriodicalId":49269,"journal":{"name":"Stochastics-An International Journal of Probability and Stochastic Processes","volume":"18 1","pages":"870 - 881"},"PeriodicalIF":0.9,"publicationDate":"2013-08-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81687268","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2013-06-17DOI: 10.1080/17442508.2014.945451
J. D. da Silva, M. Erraoui
In this paper we investigate the class of generalized grey Brownian motions (ggBms) (, ). We show that ggBm admits different representations in terms of certain known processes, such as fractional Brownian motion, multivariate elliptical distribution or as a subordination. We establish almost-sure weak convergence of the increments of in the measure space . We also obtain weak convergence of the weighted power variation of process . Using the Berman criterion we show that admits a -square integrable local time almost surely ( denoting Lebesgue measure). Moreover, we prove that this local time can be weak-approximated by the number of crossings , of level x, of the convolution approximation of ggBm.
{"title":"Generalized grey Brownian motion local time: existence and weak approximation","authors":"J. D. da Silva, M. Erraoui","doi":"10.1080/17442508.2014.945451","DOIUrl":"https://doi.org/10.1080/17442508.2014.945451","url":null,"abstract":"In this paper we investigate the class of generalized grey Brownian motions (ggBms) (, ). We show that ggBm admits different representations in terms of certain known processes, such as fractional Brownian motion, multivariate elliptical distribution or as a subordination. We establish almost-sure weak convergence of the increments of in the measure space . We also obtain weak convergence of the weighted power variation of process . Using the Berman criterion we show that admits a -square integrable local time almost surely ( denoting Lebesgue measure). Moreover, we prove that this local time can be weak-approximated by the number of crossings , of level x, of the convolution approximation of ggBm.","PeriodicalId":49269,"journal":{"name":"Stochastics-An International Journal of Probability and Stochastic Processes","volume":"57 1","pages":"347 - 361"},"PeriodicalIF":0.9,"publicationDate":"2013-06-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90824377","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2013-02-08DOI: 10.1080/17442508.2013.774402
Stéphane Goutte, N. Oudjane, F. Russo
For a large class of vanilla contingent claims, we establish an explicit Föllmer–Schweizer decomposition when the underlying is an exponential of an additive process. This allows to provide an efficient algorithm for solving the mean variance hedging problem. Applications to models derived from the electricity market are performed.
{"title":"Variance optimal hedging for continuous time additive processes and applications","authors":"Stéphane Goutte, N. Oudjane, F. Russo","doi":"10.1080/17442508.2013.774402","DOIUrl":"https://doi.org/10.1080/17442508.2013.774402","url":null,"abstract":"For a large class of vanilla contingent claims, we establish an explicit Föllmer–Schweizer decomposition when the underlying is an exponential of an additive process. This allows to provide an efficient algorithm for solving the mean variance hedging problem. Applications to models derived from the electricity market are performed.","PeriodicalId":49269,"journal":{"name":"Stochastics-An International Journal of Probability and Stochastic Processes","volume":"24 1","pages":"147 - 185"},"PeriodicalIF":0.9,"publicationDate":"2013-02-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87689908","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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