This is the second part(also the main part) of the papers with the same title.Here we will discuss the existence and uniqueness theorem for the quasi-linear SEE(Stochastic equation of evolution).We will also discuss a...This is the second part(also the main part) of the papers with the same title.Here we will discuss the existence and uniqueness theorem for the quasi-linear SEE(Stochastic equation of evolution).We will also discuss an aproximation theorem展开更多
In this paper we prove the pathwise uniqueness of a kind of two-parameter Volterra type stochastic differential equations under the coefficients satisfy the non-Lipschitz conditions. We use a martingale formula in ste...In this paper we prove the pathwise uniqueness of a kind of two-parameter Volterra type stochastic differential equations under the coefficients satisfy the non-Lipschitz conditions. We use a martingale formula in stead of Ito formula, which leads to simplicity the process of proof and extends the result to unbounded coefficients case.展开更多
This is the third part of the papers with the same title. We will discuss the problem of convergence of the semi-implicit difference scheme for a class of quasilinear SEE, which generalize the Crandall's work to t...This is the third part of the papers with the same title. We will discuss the problem of convergence of the semi-implicit difference scheme for a class of quasilinear SEE, which generalize the Crandall's work to the stochastic case.展开更多
In this paper, we are concerned with the problem of the pathwise uniqueness of one-dimensional reflected stochastic differential equations with jumps under the assumption of non-Lipschitz continuous coefficients whose...In this paper, we are concerned with the problem of the pathwise uniqueness of one-dimensional reflected stochastic differential equations with jumps under the assumption of non-Lipschitz continuous coefficients whose proof are based on the technique of local time.展开更多
We consider a strictly pathwise setting for Delta hedging exotic options,based on Follmer’s pathwise It¨o calculus.Price trajectories areˆd-dimensional continuous functions whose pathwise quadratic variations an...We consider a strictly pathwise setting for Delta hedging exotic options,based on Follmer’s pathwise It¨o calculus.Price trajectories areˆd-dimensional continuous functions whose pathwise quadratic variations and covariations are determined by a given local volatility matrix.The existence of Delta hedging strategies in this pathwise setting is established via existence results for recursive schemes of parabolic Cauchy problems and via the existence of functional Cauchy problems on path space.Our main results establish the nonexistence of pathwise arbitrage opportunities in classes of strategies containing these Delta hedging strategies and under relatively mild conditions on the local volatility matrix.展开更多
Without the linear growth condition, by the use of Lyapunov function, this paper estab- lishes the existence^and-uniqueness theorem of global solutions to a class of neutral stochastic differen- tim equations with unb...Without the linear growth condition, by the use of Lyapunov function, this paper estab- lishes the existence^and-uniqueness theorem of global solutions to a class of neutral stochastic differen- tim equations with unbounded delay, and examines the pathwise stability of this solution with general decay rate. As an application of our results, this paper also considers in detail a two-dimensional unbounded delay neutral stochastic differential equation with polynomial coefficients.展开更多
Let (Ω,(?), P) be a complete probability space with a family of sub-σ-fields {(?)_z}_z∈R_+~2 which satisfies the usual conditions. Yeh considered the existence and uniqueness of strong solutions of the following no...Let (Ω,(?), P) be a complete probability space with a family of sub-σ-fields {(?)_z}_z∈R_+~2 which satisfies the usual conditions. Yeh considered the existence and uniqueness of strong solutions of the following non-Markovian stochastic differential equations (SDE)展开更多
The authors integrate two well-known systems, the Rssler and Lorentz systems,to introduce a new chaotic system, called the Lorentz-Rssler system. Then, taking into account the effect of environmental noise, the author...The authors integrate two well-known systems, the Rssler and Lorentz systems,to introduce a new chaotic system, called the Lorentz-Rssler system. Then, taking into account the effect of environmental noise, the authors incorporate white noise in both Rssler and Lorentz systems to have a corresponding stochastic system. By deriving the uniform a priori estimates for an approximate system and then taking them to the limit,the authors prove the global existence, uniqueness and the pathwise property of solutions to the Lorentz-Rssler system. Moreover, the authors carried out a number of numerical experiments, and the numerical results demonstrate their theoretic analysis and show some new qualitative properties of solutions which reveal that the Lorentz-Ro¨ssler system could be used to design more complex and more secure nonlinear hop-frequence time series.展开更多
This paper addresses the estimation problem of an unknown drift parameter matrix for a fractional Ornstein-Uhlenbeck process in a multi-dimensional setting.To tackle this problem,we propose a novel approach based on r...This paper addresses the estimation problem of an unknown drift parameter matrix for a fractional Ornstein-Uhlenbeck process in a multi-dimensional setting.To tackle this problem,we propose a novel approach based on rough path theory that allows us to construct pathwise rough path estimators from both continuous and discrete observations of a single path.Our approach is particularly suitable for high-frequency data.To formulate the parameter estimators,we introduce a theory of pathwise Itôintegrals with respect to fractional Brownian motion.By establishing the regularity of fractional Ornstein-Uhlenbeck processes and analyzing the long-term behavior of the associated Lévy area processes,we demonstrate that our estimators are strongly consistent and pathwise stable.Our findings offer a new perspective on estimating the drift parameter matrix for fractional Ornstein-Uhlenbeck processes in multi-dimensional settings,and may have practical implications for fields including finance,economics,and engineering.展开更多
In this note, we consider stochastic heat equation with general additive Gaussian noise. Our aim is to derive some necessary and sufficient conditions on the Gaussian noise in order to solve the corresponding heat equ...In this note, we consider stochastic heat equation with general additive Gaussian noise. Our aim is to derive some necessary and sufficient conditions on the Gaussian noise in order to solve the corresponding heat equation. We investigate this problem invoking two differen t met hods, respectively, based on variance compu tations and on pat h-wise considerations in Besov spaces. We are going to see that, as anticipated, both approaches lead to the same necessary and sufficient condition on the noise. In addition, the path-wise approach brings out regularity results for the solution.展开更多
This is a survey on the strong uniqueness of the solutions to stochastic partial differential equations(SPDEs) related to two measure-valued processes: superprocess and Fleming-Viot process which are given as rescalin...This is a survey on the strong uniqueness of the solutions to stochastic partial differential equations(SPDEs) related to two measure-valued processes: superprocess and Fleming-Viot process which are given as rescaling limits of population biology models. We summarize recent results for Konno-Shiga-Reimers’ and Mytnik’s SPDEs, and their related distribution-function-valued SPDEs.展开更多
In this paper,a novel stochastic two-species competitive system with saturation effect is formulated,in which there exist two noise resources and their coupling mode is relatively complex and every noise source has el...In this paper,a novel stochastic two-species competitive system with saturation effect is formulated,in which there exist two noise resources and their coupling mode is relatively complex and every noise source has elfect on the intrinsic growth rates of both species.With the help of some suitable Lyapunov functions,sufficient conditions for stochastic permanence are established as exponential extinction,extinction,permanence in time average and asymptotic pathwise estimation of system.The effect of coupling noise on the asymptotic behaviors of the populations is shown.展开更多
Let M = {M<sub>z</sub>, z∈R<sub>+</sub><sup>2</sup>} be a continuous square integrable martingale and A = {A<sub>z</sub>, z∈ R<sub>+</sub><sup>2</...Let M = {M<sub>z</sub>, z∈R<sub>+</sub><sup>2</sup>} be a continuous square integrable martingale and A = {A<sub>z</sub>, z∈ R<sub>+</sub><sup>2</sup>} be a continuous adapted increasing process. Consider the following stochastic partial differential equations in the plane: dX<sub>z</sub>=α（z, X<sub>z</sub>）dM<sub>2</sub>+β（z,X<sub>z</sub>）dA<sub>z</sub>, z∈R<sub>+</sub><sup>2</sup>, X<sub>z</sub>=Z<sub>z</sub>, z∈R<sub>+</sub><sup>2</sup>, where R<sub>+</sub><sup>2</sup>=[0,+∞)×[0,+∞) and R<sub>+</sub><sup>2</sup> is its boundary, Z is a continuous stochastic process on R<sub>+</sub><sup>2</sup>. We establish a new theorem on the pathwise uniqueness of solutions for the equation under a weaker condition than the Lipschitz one. The result concerning the one-parameter analogue of the problem we consider here is immediate （see [1, Theorem 3.2]）. Unfortunately, the situation is much more complicated for two-parameter process and we believe that our result is the first one of its kind and is interesting in itself. We have proved the existence theorem for the equation in.展开更多
In this paper, a stochastic predator-prey （PP） model with mutual interference is considered. Some sufficient conditions for the existence of globally positive solution, non- persistence in the mean, weak persistence...In this paper, a stochastic predator-prey （PP） model with mutual interference is considered. Some sufficient conditions for the existence of globally positive solution, non- persistence in the mean, weak persistence in the mean, strong persistence in the mean and almost surely extinction of the the model are established. Moreover, the thresh- old between weak persistence in the mean and almost surely extinction of the prey is obtained. Some examples are given to show the feasibility of the results by numeri- cal simulation. It is significant that such a model is firstly proposed with stochastic perturbation.展开更多
文摘This is the second part(also the main part) of the papers with the same title.Here we will discuss the existence and uniqueness theorem for the quasi-linear SEE(Stochastic equation of evolution).We will also discuss an aproximation theorem
基金Foundation item: Hubei University Youngth Foundations (099206).
文摘In this paper we prove the pathwise uniqueness of a kind of two-parameter Volterra type stochastic differential equations under the coefficients satisfy the non-Lipschitz conditions. We use a martingale formula in stead of Ito formula, which leads to simplicity the process of proof and extends the result to unbounded coefficients case.
基金Work supported by National Natural Science Foundation of China.
文摘This is the third part of the papers with the same title. We will discuss the problem of convergence of the semi-implicit difference scheme for a class of quasilinear SEE, which generalize the Crandall's work to the stochastic case.
基金supported by the National Natural Science Foundation of China (No.12261038, 11671408 and11871484)Natural Science Foundation of Jiangxi Province (No.20232BAB201004, 20212BAB201009)Training Program of Young Talents for academic and technical leaders of major disciplines in Jiangxi Province(No.20204BCJL23057)。
文摘In this paper, we are concerned with the problem of the pathwise uniqueness of one-dimensional reflected stochastic differential equations with jumps under the assumption of non-Lipschitz continuous coefficients whose proof are based on the technique of local time.
基金support by Deutsche Forschungsgemeinschaft through the Research Training Group RTG 1953.
文摘We consider a strictly pathwise setting for Delta hedging exotic options,based on Follmer’s pathwise It¨o calculus.Price trajectories areˆd-dimensional continuous functions whose pathwise quadratic variations and covariations are determined by a given local volatility matrix.The existence of Delta hedging strategies in this pathwise setting is established via existence results for recursive schemes of parabolic Cauchy problems and via the existence of functional Cauchy problems on path space.Our main results establish the nonexistence of pathwise arbitrage opportunities in classes of strategies containing these Delta hedging strategies and under relatively mild conditions on the local volatility matrix.
基金Supported by National Natural Science Foundation of China （Grant No. 11001091） and Chinese University Research Foundation （Grant No. 2010MS129）
文摘Without the linear growth condition, by the use of Lyapunov function, this paper estab- lishes the existence^and-uniqueness theorem of global solutions to a class of neutral stochastic differen- tim equations with unbounded delay, and examines the pathwise stability of this solution with general decay rate. As an application of our results, this paper also considers in detail a two-dimensional unbounded delay neutral stochastic differential equation with polynomial coefficients.
文摘Let (Ω,(?), P) be a complete probability space with a family of sub-σ-fields {(?)_z}_z∈R_+~2 which satisfies the usual conditions. Yeh considered the existence and uniqueness of strong solutions of the following non-Markovian stochastic differential equations (SDE)
基金supported by the National Natural Science Foundation of China(Nos.11101044,11371065)the Beijing Center for Mathematics and Information Interdisciplinary Sciences
文摘The authors integrate two well-known systems, the Rssler and Lorentz systems,to introduce a new chaotic system, called the Lorentz-Rssler system. Then, taking into account the effect of environmental noise, the authors incorporate white noise in both Rssler and Lorentz systems to have a corresponding stochastic system. By deriving the uniform a priori estimates for an approximate system and then taking them to the limit,the authors prove the global existence, uniqueness and the pathwise property of solutions to the Lorentz-Rssler system. Moreover, the authors carried out a number of numerical experiments, and the numerical results demonstrate their theoretic analysis and show some new qualitative properties of solutions which reveal that the Lorentz-Ro¨ssler system could be used to design more complex and more secure nonlinear hop-frequence time series.
基金supported by Shanghai Artificial Intelligence Laboratory.
文摘This paper addresses the estimation problem of an unknown drift parameter matrix for a fractional Ornstein-Uhlenbeck process in a multi-dimensional setting.To tackle this problem,we propose a novel approach based on rough path theory that allows us to construct pathwise rough path estimators from both continuous and discrete observations of a single path.Our approach is particularly suitable for high-frequency data.To formulate the parameter estimators,we introduce a theory of pathwise Itôintegrals with respect to fractional Brownian motion.By establishing the regularity of fractional Ornstein-Uhlenbeck processes and analyzing the long-term behavior of the associated Lévy area processes,we demonstrate that our estimators are strongly consistent and pathwise stable.Our findings offer a new perspective on estimating the drift parameter matrix for fractional Ornstein-Uhlenbeck processes in multi-dimensional settings,and may have practical implications for fields including finance,economics,and engineering.
基金supported by an NSERC granta startup fund of University of Albertasupported by the NSF grant DMS1613163
文摘In this note, we consider stochastic heat equation with general additive Gaussian noise. Our aim is to derive some necessary and sufficient conditions on the Gaussian noise in order to solve the corresponding heat equation. We investigate this problem invoking two differen t met hods, respectively, based on variance compu tations and on pat h-wise considerations in Besov spaces. We are going to see that, as anticipated, both approaches lead to the same necessary and sufficient condition on the noise. In addition, the path-wise approach brings out regularity results for the solution.
基金Supported partially by SUST startup fund 28/Y01286120NSF of Ningxia(2018AAC03245)+1 种基金NSFC(11771018)First-Class Disciplines Foundation Ningxia(NXYLXK2017B09)
文摘This is a survey on the strong uniqueness of the solutions to stochastic partial differential equations(SPDEs) related to two measure-valued processes: superprocess and Fleming-Viot process which are given as rescaling limits of population biology models. We summarize recent results for Konno-Shiga-Reimers’ and Mytnik’s SPDEs, and their related distribution-function-valued SPDEs.
基金the National Natural Science Foundation of China(Nos.11871201 and 11261017)Natural Science Foundation of Hubei Province(Nos.2019CFB241 and 2019CFB773).
文摘In this paper,a novel stochastic two-species competitive system with saturation effect is formulated,in which there exist two noise resources and their coupling mode is relatively complex and every noise source has elfect on the intrinsic growth rates of both species.With the help of some suitable Lyapunov functions,sufficient conditions for stochastic permanence are established as exponential extinction,extinction,permanence in time average and asymptotic pathwise estimation of system.The effect of coupling noise on the asymptotic behaviors of the populations is shown.
基金Supported by the National Science Foundationthe Postdoctoral Science Foundation of China
文摘Let M = {M<sub>z</sub>, z∈R<sub>+</sub><sup>2</sup>} be a continuous square integrable martingale and A = {A<sub>z</sub>, z∈ R<sub>+</sub><sup>2</sup>} be a continuous adapted increasing process. Consider the following stochastic partial differential equations in the plane: dX<sub>z</sub>=α（z, X<sub>z</sub>）dM<sub>2</sub>+β（z,X<sub>z</sub>）dA<sub>z</sub>, z∈R<sub>+</sub><sup>2</sup>, X<sub>z</sub>=Z<sub>z</sub>, z∈R<sub>+</sub><sup>2</sup>, where R<sub>+</sub><sup>2</sup>=[0,+∞)×[0,+∞) and R<sub>+</sub><sup>2</sup> is its boundary, Z is a continuous stochastic process on R<sub>+</sub><sup>2</sup>. We establish a new theorem on the pathwise uniqueness of solutions for the equation under a weaker condition than the Lipschitz one. The result concerning the one-parameter analogue of the problem we consider here is immediate （see [1, Theorem 3.2]）. Unfortunately, the situation is much more complicated for two-parameter process and we believe that our result is the first one of its kind and is interesting in itself. We have proved the existence theorem for the equation in.
文摘In this paper, a stochastic predator-prey （PP） model with mutual interference is considered. Some sufficient conditions for the existence of globally positive solution, non- persistence in the mean, weak persistence in the mean, strong persistence in the mean and almost surely extinction of the the model are established. Moreover, the thresh- old between weak persistence in the mean and almost surely extinction of the prey is obtained. Some examples are given to show the feasibility of the results by numeri- cal simulation. It is significant that such a model is firstly proposed with stochastic perturbation.
