Pub Date : 2014-08-05DOI: 10.1080/17442508.2013.879141
N. D. Cong, S. Siegmund, N. The
We introduce a concept of adjoint equation and Lyapunov regularity of a stochastic differential algebraic Equation (SDAE) of index 1. The notion of adjoint SDAE is introduced in a similar way as in the deterministic differential algebraic equation case. We prove a multiplicative ergodic theorem for the adjoint SDAE and the adjoint Lyapunov spectrum. Employing the notion of adjoint equation and Lyapunov spectrum of an SDAE, we are able to define Lyapunov regularity of SDAEs. Some properties and an example of a metal oxide semiconductor field-effect transistor ring oscillator under thermal noise are discussed.
{"title":"Adjoint equation and Lyapunov regularity for linear stochastic differential algebraic equations of index 1","authors":"N. D. Cong, S. Siegmund, N. The","doi":"10.1080/17442508.2013.879141","DOIUrl":"https://doi.org/10.1080/17442508.2013.879141","url":null,"abstract":"We introduce a concept of adjoint equation and Lyapunov regularity of a stochastic differential algebraic Equation (SDAE) of index 1. The notion of adjoint SDAE is introduced in a similar way as in the deterministic differential algebraic equation case. We prove a multiplicative ergodic theorem for the adjoint SDAE and the adjoint Lyapunov spectrum. Employing the notion of adjoint equation and Lyapunov spectrum of an SDAE, we are able to define Lyapunov regularity of SDAEs. Some properties and an example of a metal oxide semiconductor field-effect transistor ring oscillator under thermal noise are discussed.","PeriodicalId":49269,"journal":{"name":"Stochastics-An International Journal of Probability and Stochastic Processes","volume":"28 1","pages":"776 - 802"},"PeriodicalIF":0.9,"publicationDate":"2014-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90822368","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 : 2014-08-05DOI: 10.1080/17442508.2013.879145
M. Barski
The problem of existence of arbitrage-free and monotone collateralized debt obligations term structure models is studied. Conditions for positivity and monotonicity of the corresponding Heath–Jarrow–Morton–Musiela equation for the -forward rates with the use of the Milian-type result are formulated. Two state spaces are taken into account – of square integrable functions and a Sobolev space. For the first the regularity results concerning pointwise monotonicity are proven. Arbitrage-free and monotone models are characterized in terms of the volatility of the model and characteristics of the driving Lévy process.
{"title":"Monotonicity of the collateralized debt obligations term structure model","authors":"M. Barski","doi":"10.1080/17442508.2013.879145","DOIUrl":"https://doi.org/10.1080/17442508.2013.879145","url":null,"abstract":"The problem of existence of arbitrage-free and monotone collateralized debt obligations term structure models is studied. Conditions for positivity and monotonicity of the corresponding Heath–Jarrow–Morton–Musiela equation for the -forward rates with the use of the Milian-type result are formulated. Two state spaces are taken into account – of square integrable functions and a Sobolev space. For the first the regularity results concerning pointwise monotonicity are proven. Arbitrage-free and monotone models are characterized in terms of the volatility of the model and characteristics of the driving Lévy process.","PeriodicalId":49269,"journal":{"name":"Stochastics-An International Journal of Probability and Stochastic Processes","volume":"170 1","pages":"835 - 864"},"PeriodicalIF":0.9,"publicationDate":"2014-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76612862","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 : 2014-08-05DOI: 10.1080/17442508.2013.878345
Litan Yan, Zhi Wang, Huiting Jing
In this paper we consider the weighted-fractional Brownian motion with indexes a, b () and narrow the focus to obtain some properties of sample paths. Motivated by the asymptotic propertyfor all s>0, we consider the -strong variation of the principal value type defined by the limitwith for all t>0, where the limits are uniform in probability on each compact interval. We show that is strongly locally -non-deterministic with , and by applying this property we study Chung's law of the iterated logarithm for and intersection local time on .
{"title":"Some path properties of weighted-fractional Brownian motion","authors":"Litan Yan, Zhi Wang, Huiting Jing","doi":"10.1080/17442508.2013.878345","DOIUrl":"https://doi.org/10.1080/17442508.2013.878345","url":null,"abstract":"In this paper we consider the weighted-fractional Brownian motion with indexes a, b () and narrow the focus to obtain some properties of sample paths. Motivated by the asymptotic propertyfor all s>0, we consider the -strong variation of the principal value type defined by the limitwith for all t>0, where the limits are uniform in probability on each compact interval. We show that is strongly locally -non-deterministic with , and by applying this property we study Chung's law of the iterated logarithm for and intersection local time on .","PeriodicalId":49269,"journal":{"name":"Stochastics-An International Journal of Probability and Stochastic Processes","volume":"4 1","pages":"721 - 758"},"PeriodicalIF":0.9,"publicationDate":"2014-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84745128","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 : 2014-08-05DOI: 10.1080/17442508.2013.879144
M. Claude
The wavelet transform is defined for Wiener functionals. We characterize global and local regularities of Wiener functionals and we give a criterion for the existence and regularity of densities. Such a criterion is applied to diffusion processes and to the solutions to backward stochastic differential equations.
{"title":"The wavelet transform for Wiener functionals and some applications","authors":"M. Claude","doi":"10.1080/17442508.2013.879144","DOIUrl":"https://doi.org/10.1080/17442508.2013.879144","url":null,"abstract":"The wavelet transform is defined for Wiener functionals. We characterize global and local regularities of Wiener functionals and we give a criterion for the existence and regularity of densities. Such a criterion is applied to diffusion processes and to the solutions to backward stochastic differential equations.","PeriodicalId":49269,"journal":{"name":"Stochastics-An International Journal of Probability and Stochastic Processes","volume":"15 1","pages":"817 - 834"},"PeriodicalIF":0.9,"publicationDate":"2014-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87262811","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 : 2014-07-04DOI: 10.1080/17442508.2013.848864
Satoshi Yokoyama
We prove the existence of weak solutions of stochastic Navier–Stokes equation on a two-dimensional torus, which appears in a certain variational problem. Our equation does not satisfy the coercivity condition. We construct its weak solutions due to an approximation by a sequence of solutions of equations with enlarged viscosity terms and then by showing an a priori estimate for them.
{"title":"Construction of weak solutions of a certain stochastic Navier–Stokes equation","authors":"Satoshi Yokoyama","doi":"10.1080/17442508.2013.848864","DOIUrl":"https://doi.org/10.1080/17442508.2013.848864","url":null,"abstract":"We prove the existence of weak solutions of stochastic Navier–Stokes equation on a two-dimensional torus, which appears in a certain variational problem. Our equation does not satisfy the coercivity condition. We construct its weak solutions due to an approximation by a sequence of solutions of equations with enlarged viscosity terms and then by showing an a priori estimate for them.","PeriodicalId":49269,"journal":{"name":"Stochastics-An International Journal of Probability and Stochastic Processes","volume":"1 1","pages":"573 - 593"},"PeriodicalIF":0.9,"publicationDate":"2014-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82858764","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 : 2014-07-04DOI: 10.1080/17442508.2013.859388
Xiaoming Liu, R. Mamon, Huan Gao
Annuity-contingent derivatives involve both mortality and interest risks, which could have a correlation. In this article, we propose a generalized pricing framework in which the dependence between the two risks can be explicitly modelled. We also utilize the change of measure technique to simplify the valuation expressions. We illustrate our methodology in the valuation of a guaranteed annuity option (GAO). Using both forward measure associated with the bond price as numéraire and the newly introduced concept of endowment-risk-adjusted measure, we derive a simplified formula for the GAO price under the generalized framework. Numerical results show that the methodology proposed in this article is highly efficient and accurate.
{"title":"A generalized pricing framework addressing correlated mortality and interest risks: a change of probability measure approach","authors":"Xiaoming Liu, R. Mamon, Huan Gao","doi":"10.1080/17442508.2013.859388","DOIUrl":"https://doi.org/10.1080/17442508.2013.859388","url":null,"abstract":"Annuity-contingent derivatives involve both mortality and interest risks, which could have a correlation. In this article, we propose a generalized pricing framework in which the dependence between the two risks can be explicitly modelled. We also utilize the change of measure technique to simplify the valuation expressions. We illustrate our methodology in the valuation of a guaranteed annuity option (GAO). Using both forward measure associated with the bond price as numéraire and the newly introduced concept of endowment-risk-adjusted measure, we derive a simplified formula for the GAO price under the generalized framework. Numerical results show that the methodology proposed in this article is highly efficient and accurate.","PeriodicalId":49269,"journal":{"name":"Stochastics-An International Journal of Probability and Stochastic Processes","volume":"4 1","pages":"594 - 608"},"PeriodicalIF":0.9,"publicationDate":"2014-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87446731","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 : 2014-07-04DOI: 10.1080/17442508.2013.872645
C. Lefèvre, P. Picard
Appell polynomials are known to play a key role in certain first-crossing problems. The present paper considers a rather general insurance risk model where the claim interarrival times are independent and exponentially distributed with different parameters, the successive claim amounts may be dependent and the premium income is an arbitrary deterministic function. It is shown that the non-ruin (or survival) probability over a finite horizon may be expressed in terms of a remarkable family of functions, named pseudopolynomials, that generalize the classical Appell polynomials. The presence of that underlying algebraic structure is exploited to provide a closed formula, almost explicit, for the non-ruin probability.
{"title":"Appell pseudopolynomials and Erlang-type risk models","authors":"C. Lefèvre, P. Picard","doi":"10.1080/17442508.2013.872645","DOIUrl":"https://doi.org/10.1080/17442508.2013.872645","url":null,"abstract":"Appell polynomials are known to play a key role in certain first-crossing problems. The present paper considers a rather general insurance risk model where the claim interarrival times are independent and exponentially distributed with different parameters, the successive claim amounts may be dependent and the premium income is an arbitrary deterministic function. It is shown that the non-ruin (or survival) probability over a finite horizon may be expressed in terms of a remarkable family of functions, named pseudopolynomials, that generalize the classical Appell polynomials. The presence of that underlying algebraic structure is exploited to provide a closed formula, almost explicit, for the non-ruin probability.","PeriodicalId":49269,"journal":{"name":"Stochastics-An International Journal of Probability and Stochastic Processes","volume":"203 1","pages":"676 - 695"},"PeriodicalIF":0.9,"publicationDate":"2014-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77022357","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 : 2014-07-04DOI: 10.1080/17442508.2013.872644
M. K. Ghosh, Subhamay Saha
We study risk-sensitive control of continuous time Markov chains taking values in discrete state space. We study both finite and infinite horizon problems. In the finite horizon problem we characterize the value function via Hamilton Jacobi Bellman equation and obtain an optimal Markov control. We do the same for infinite horizon discounted cost case. In the infinite horizon average cost case we establish the existence of an optimal stationary control under certain Lyapunov condition. We also develop a policy iteration algorithm for finding an optimal control.
{"title":"Risk-sensitive control of continuous time Markov chains","authors":"M. K. Ghosh, Subhamay Saha","doi":"10.1080/17442508.2013.872644","DOIUrl":"https://doi.org/10.1080/17442508.2013.872644","url":null,"abstract":"We study risk-sensitive control of continuous time Markov chains taking values in discrete state space. We study both finite and infinite horizon problems. In the finite horizon problem we characterize the value function via Hamilton Jacobi Bellman equation and obtain an optimal Markov control. We do the same for infinite horizon discounted cost case. In the infinite horizon average cost case we establish the existence of an optimal stationary control under certain Lyapunov condition. We also develop a policy iteration algorithm for finding an optimal control.","PeriodicalId":49269,"journal":{"name":"Stochastics-An International Journal of Probability and Stochastic Processes","volume":"244 1","pages":"655 - 675"},"PeriodicalIF":0.9,"publicationDate":"2014-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78826815","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 : 2014-07-04DOI: 10.1080/17442508.2013.879143
M. Diop, K. Ezzinbi, Modou Lo
In this paper, we study the existence and asymptotic stability in the p-th moment of mild solutions of nonlinear impulsive stochastic partial functional integrodifferential equations with delays. We suppose that the linear part possesses a resolvent operator in the sense given in Grimmer [R. Grimmer, Resolvent operators for integral equations in a Banach space, Trans. Am. Math. Soc. 273(1) (1982), 333–349] and the nonlinear terms are assumed to be Lipschitz continuous. A fixed point approach is employed for achieving the required result. An example is provided to illustrate the results of this work.
{"title":"Asymptotic stability of impulsive stochastic partial integrodifferential equations with delays","authors":"M. Diop, K. Ezzinbi, Modou Lo","doi":"10.1080/17442508.2013.879143","DOIUrl":"https://doi.org/10.1080/17442508.2013.879143","url":null,"abstract":"In this paper, we study the existence and asymptotic stability in the p-th moment of mild solutions of nonlinear impulsive stochastic partial functional integrodifferential equations with delays. We suppose that the linear part possesses a resolvent operator in the sense given in Grimmer [R. Grimmer, Resolvent operators for integral equations in a Banach space, Trans. Am. Math. Soc. 273(1) (1982), 333–349] and the nonlinear terms are assumed to be Lipschitz continuous. A fixed point approach is employed for achieving the required result. An example is provided to illustrate the results of this work.","PeriodicalId":49269,"journal":{"name":"Stochastics-An International Journal of Probability and Stochastic Processes","volume":"35 1","pages":"696 - 706"},"PeriodicalIF":0.9,"publicationDate":"2014-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88994946","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 : 2014-05-04DOI: 10.1080/17442508.2013.819510
D. Crisan, J. Xiong
Parabolic stochastic partial differential Equations (SPDEs) with multiplicative noise play a central rôle in nonlinear filtering. More precisely, the conditional distribution of a partially observed diffusion solves the normalized version of an equation of this type. We show that one can approximate the solution of the SPDE by the (unweighted) empirical measure of a finite system of interacting particle for the case when the diffusion evolves in a compact state space with reflecting boundary. This approximation differs from existing approximations where the particles are weighted and the particle interaction arises through the choice of the weights and not at the level of the particles' motion as it is the case in this work. The system of stochastic differential equations modelling the trajectories of the particles is approximated by the recursive projection scheme introduced by Pettersson [Stoch. Process. Appl. 59(2) (1995), pp. 295–308].
{"title":"Numerical solution for a class of SPDEs over bounded domains","authors":"D. Crisan, J. Xiong","doi":"10.1080/17442508.2013.819510","DOIUrl":"https://doi.org/10.1080/17442508.2013.819510","url":null,"abstract":"Parabolic stochastic partial differential Equations (SPDEs) with multiplicative noise play a central rôle in nonlinear filtering. More precisely, the conditional distribution of a partially observed diffusion solves the normalized version of an equation of this type. We show that one can approximate the solution of the SPDE by the (unweighted) empirical measure of a finite system of interacting particle for the case when the diffusion evolves in a compact state space with reflecting boundary. This approximation differs from existing approximations where the particles are weighted and the particle interaction arises through the choice of the weights and not at the level of the particles' motion as it is the case in this work. The system of stochastic differential equations modelling the trajectories of the particles is approximated by the recursive projection scheme introduced by Pettersson [Stoch. Process. Appl. 59(2) (1995), pp. 295–308].","PeriodicalId":49269,"journal":{"name":"Stochastics-An International Journal of Probability and Stochastic Processes","volume":"35 1","pages":"450 - 472"},"PeriodicalIF":0.9,"publicationDate":"2014-05-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75649232","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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