Motivated by recent advances made in the study of dividend control and risk management problems involving the U.S.bankruptcy code,in this paper we follow[44]to revisit the De Finetti dividend control problem under the...Motivated by recent advances made in the study of dividend control and risk management problems involving the U.S.bankruptcy code,in this paper we follow[44]to revisit the De Finetti dividend control problem under the reorganization process and the regulator's intervention documented in U.S.Chapter 11 bankruptcy.We do this by further accommodating the fixed transaction costs on dividends to imitate the real-world procedure of dividend payments.Incorporating the fixed transaction costs transforms the targeting optimal dividend problem into an impulse control problem rather than a singular control problem,and hence computations and proofs that are distinct from[44]are needed.To account for the financial stress that is due to the more subtle concept of Chapter 11 bankruptcy,the surplus process after dividends is driven by a piece-wise spectrally negative Lévy process with endogenous regime switching.Some explicit expressions of the expected net present values under a double barrier dividend strategy,new to the literature,are established in terms of scale functions.With the help of these expressions,we are able to characterize the optimal strategy among the set of admissible double barrier dividend strategies.When the tail of the Lévy measure is log-convex,this optimal double barrier dividend strategy is then verified as the optimal dividend strategy,solving our optimal impulse control problem.展开更多
We introduce a super-Lévy process and study maximal speed of all particles in the range and the support of the super-Lévy process. The state of historical super-Lévy process is a measure on the set of p...We introduce a super-Lévy process and study maximal speed of all particles in the range and the support of the super-Lévy process. The state of historical super-Lévy process is a measure on the set of paths. We study the maximal speed of all particles during a given time period, which turns out to be a function of the packing dimension of the time period. We calculate the Hausdorff dimension of the set of a-fast paths in the support and the range of the historical super-Lévy process.展开更多
In this paper, we introduce the definition of a multi-parameter fractional Lévy process and its local time, and show its decomposition. Using the decomposition, we prove existence and joint continuity of its loca...In this paper, we introduce the definition of a multi-parameter fractional Lévy process and its local time, and show its decomposition. Using the decomposition, we prove existence and joint continuity of its local time.展开更多
A new class of reflected backward stochastic differential equations (RBSDEs) driven by Teugels martingales associated with Lévy process and Countable Brownian Motions are investigated. Via approximation, the exis...A new class of reflected backward stochastic differential equations (RBSDEs) driven by Teugels martingales associated with Lévy process and Countable Brownian Motions are investigated. Via approximation, the existence and uniqueness of solution to this kind of RBSDEs are obtained.展开更多
In this article,we first prove the existence and uniqueness of the solution to the stochastic generalized porous medium equation perturbed by Lévy process,and then show the exponential convergence of(pt)t≥0 to...In this article,we first prove the existence and uniqueness of the solution to the stochastic generalized porous medium equation perturbed by Lévy process,and then show the exponential convergence of(pt)t≥0 to equilibrium uniform on any bounded subset in H.展开更多
With the development of information technology,rumors propagate faster and more widely than in the past.In this paper,a stochastic rumor propagation model incorporating media coverage and driven by Lévy noise is ...With the development of information technology,rumors propagate faster and more widely than in the past.In this paper,a stochastic rumor propagation model incorporating media coverage and driven by Lévy noise is proposed.The global positivity of the solution process is proved,and further the basic reproductive number R_(0) is obtained.When R_(0)<1,the dynamical process of system with Lévy jump tends to the rumor-free equilibrium point of the deterministic system,and the rumor tends to extinction;when R_(0)>1,the rumor will keep spreading and the system will oscillate randomly near the rumor equilibrium point of the deterministic system.The results show that the oscillation amplitude is related to the disturbance of the system.In addition,increasing media coverage can effectively reduce the final spread of rumors.Finally,the above results are verified by numerical simulation.展开更多
In this paper, we consider a Markov switching Lévy process model in which the underlying risky assets are driven by the stochastic exponential of Markov switching Lévy process and then apply the model to opt...In this paper, we consider a Markov switching Lévy process model in which the underlying risky assets are driven by the stochastic exponential of Markov switching Lévy process and then apply the model to option pricing and hedging. In this model, the market interest rate, the volatility of the underlying risky assets and the N-state compensator,depend on unobservable states of the economy which are modeled by a continuous-time Hidden Markov process. We use the MEMM(minimal entropy martingale measure) as the equivalent martingale measure. The option price using this model is obtained by the Fourier transform method. We obtain a closed-form solution for the hedge ratio by applying the local risk minimizing hedging.展开更多
This article establishes a universal robust limit theorem under a sublinear expectation framework.Under moment and consistency conditions,we show that,forα∈(1,2),the i.i.d.sequence{(1/√∑_(i=1)^(n)X_(i),1/n∑_(i=1)...This article establishes a universal robust limit theorem under a sublinear expectation framework.Under moment and consistency conditions,we show that,forα∈(1,2),the i.i.d.sequence{(1/√∑_(i=1)^(n)X_(i),1/n∑_(i=1)^(n)X_(i)Y_(i),1/α√n∑_(i=1)^(n)X_(i))}_(n=1)^(∞)converges in distribution to L_(1),where L_(t=(ε_(t),η_(t),ζ_(t))),t∈[0,1],is a multidimensional nonlinear Lévy process with an uncertainty■set as a set of Lévy triplets.This nonlinear Lévy process is characterized by a fully nonlinear and possibly degenerate partial integro-differential equation(PIDE){δ_(t)u(t,x,y,z)-sup_(F_(μ),q,Q)∈■{∫_(R^(d)δλu(t,x,y,z)(dλ)with.To construct the limit process,we develop a novel weak convergence approach based on the notions of tightness and weak compactness on a sublinear expectation space.We further prove a new type of Lévy-Khintchine representation formula to characterize.As a byproduct,we also provide a probabilistic approach to prove the existence of the above fully nonlinear degenerate PIDE.展开更多
In this paper,we use a unified framework to study Poisson stable(including stationary,periodic,quasi-periodic,almost periodic,almost automorphic,Birkhoff recurrent,almost recurrent in the sense of Bebutov,Levitan almo...In this paper,we use a unified framework to study Poisson stable(including stationary,periodic,quasi-periodic,almost periodic,almost automorphic,Birkhoff recurrent,almost recurrent in the sense of Bebutov,Levitan almost periodic,pseudo-periodic,pseudo-recurrent and Poisson stable)solutions for semilinear stochastic differential equations driven by infinite dimensional L′evy noise with large jumps.Under suitable conditions on drift,diffusion and jump coefficients,we prove that there exist solutions which inherit the Poisson stability of coefficients.Further we show that these solutions are globally asymptotically stable in square-mean sense.Finally,we illustrate our theoretical results by several examples.展开更多
We introduce G-Lévy processes which develop the theory of processes with independent and stationary increments under the framework of sublinear expectations.We then obtain the Lévy-Khintchine formula and the...We introduce G-Lévy processes which develop the theory of processes with independent and stationary increments under the framework of sublinear expectations.We then obtain the Lévy-Khintchine formula and the existence for G-Lévy processes.We also introduce G-Poisson processes.展开更多
We consider the spectrally negative L@vy processes and determine the joint laws for the quantities such as the first and last passage times over a fixed level, the overshoots and undershoots at first passage, the mini...We consider the spectrally negative L@vy processes and determine the joint laws for the quantities such as the first and last passage times over a fixed level, the overshoots and undershoots at first passage, the minimum, the maximum, and the duration of negative values. We apply our results to insurance risk theory to find an explicit expression for the generalized expected discounted penalty function in terms of scale functions. Furthermore, a new expression for the generalized Dickson's formula is provided.展开更多
This paper is concerned with a class of mean-field type stochastic optimal control systems,which are governed by fully coupled mean-field forward-backward stochastic differential equations with Teugels martingales ass...This paper is concerned with a class of mean-field type stochastic optimal control systems,which are governed by fully coupled mean-field forward-backward stochastic differential equations with Teugels martingales associated to Lévy processes.In these systems,the coefficients contain not only the state processes but also their marginal distribution,and the cost function is of mean-field type as well.The necessary and sufficient conditions for such optimal problems are obtained.Furthermore,the applications to the linear quadratic stochastic optimization control problem are investigated.展开更多
A new class of generalized backward doubly stochastic differential equations (GBDSDEs in short) driven by Teugels martingales associated with Levy process are investigated. We establish a comparison theorem which al...A new class of generalized backward doubly stochastic differential equations (GBDSDEs in short) driven by Teugels martingales associated with Levy process are investigated. We establish a comparison theorem which allows us to derive an existence result of solutions under continuous and linear growth conditions.展开更多
Let {X(t), t ≥ 0} be a Lévy process with EX(1) = 0 and EX^2(1) 〈 ∞. In this paper, we shall give two precise asymptotic theorems for {X(t), t 〉 0}. By the way, we prove the corresponding conclusions f...Let {X(t), t ≥ 0} be a Lévy process with EX(1) = 0 and EX^2(1) 〈 ∞. In this paper, we shall give two precise asymptotic theorems for {X(t), t 〉 0}. By the way, we prove the corresponding conclusions for strictly stable processes and a general precise asymptotic proposition for sums of i.i.d. random variables.展开更多
For spectrally negative Lévy process (SNLP), we find an expression, in terms of scale functions, for a potential measure involving the maximum and the last time of reaching the maximum up to a draw-down time. As ...For spectrally negative Lévy process (SNLP), we find an expression, in terms of scale functions, for a potential measure involving the maximum and the last time of reaching the maximum up to a draw-down time. As applications, we obtain a potential measure for the reflected SNLP and recover a joint Laplace transform for the Wiener-Hopf factorization for SNLP.展开更多
This paper focuses on optimal control of nonlinear stochastic delay system constructed through Teugels martingales associated with Lévy processes and standard Brownian motion,in which finite horizon is extended t...This paper focuses on optimal control of nonlinear stochastic delay system constructed through Teugels martingales associated with Lévy processes and standard Brownian motion,in which finite horizon is extended to infinite horizon.In order to describe the interacting many-body system,the expectation values of state processes are added to the concerned system.Further,sufficient and necessary conditions are established under convexity assumptions of the control domain.Finally,an example is given to demonstrate the application of the theory.展开更多
基金the financial support from the National Natural Science Foundation of China(12171405 and 11661074)the Program for New Century Excellent Talents in Fujian Province University+2 种基金the financial support from the Characteristic&Preponderant Discipline of Key Construction Universities in Zhejiang Province(Zhejiang Gongshang University-Statistics)Collaborative Innovation Center of Statistical Data Engineering Technology&ApplicationDigital+Discipline Construction Project(SZJ2022B004)。
文摘Motivated by recent advances made in the study of dividend control and risk management problems involving the U.S.bankruptcy code,in this paper we follow[44]to revisit the De Finetti dividend control problem under the reorganization process and the regulator's intervention documented in U.S.Chapter 11 bankruptcy.We do this by further accommodating the fixed transaction costs on dividends to imitate the real-world procedure of dividend payments.Incorporating the fixed transaction costs transforms the targeting optimal dividend problem into an impulse control problem rather than a singular control problem,and hence computations and proofs that are distinct from[44]are needed.To account for the financial stress that is due to the more subtle concept of Chapter 11 bankruptcy,the surplus process after dividends is driven by a piece-wise spectrally negative Lévy process with endogenous regime switching.Some explicit expressions of the expected net present values under a double barrier dividend strategy,new to the literature,are established in terms of scale functions.With the help of these expressions,we are able to characterize the optimal strategy among the set of admissible double barrier dividend strategies.When the tail of the Lévy measure is log-convex,this optimal double barrier dividend strategy is then verified as the optimal dividend strategy,solving our optimal impulse control problem.
基金Project supported by the National Natural Science Foundation of China(No.10571159)the Ph.D.Programs Foundation of Ministry of Education of China(No.20060335032)and the Foundation of Hangzhou Dianzi University(No.KYS091506042)
文摘We introduce a super-Lévy process and study maximal speed of all particles in the range and the support of the super-Lévy process. The state of historical super-Lévy process is a measure on the set of paths. We study the maximal speed of all particles during a given time period, which turns out to be a function of the packing dimension of the time period. We calculate the Hausdorff dimension of the set of a-fast paths in the support and the range of the historical super-Lévy process.
基金supported by the National Natural Science Foundation of China (No. 10871177)the Ph. D.Programs Foundation of Ministry of Education of China (No. 20060335032)the Natural Science Foundation of Zhejiang Province of China (No. Y7080044)
文摘In this paper, we introduce the definition of a multi-parameter fractional Lévy process and its local time, and show its decomposition. Using the decomposition, we prove existence and joint continuity of its local time.
文摘A new class of reflected backward stochastic differential equations (RBSDEs) driven by Teugels martingales associated with Lévy process and Countable Brownian Motions are investigated. Via approximation, the existence and uniqueness of solution to this kind of RBSDEs are obtained.
基金the National Science Foundations of China(10971180,11271169)the Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions
基金supported by the National Science Foundation of China(1067121290820302)the National Science Foundation of Hunan Province
文摘In this article,we first prove the existence and uniqueness of the solution to the stochastic generalized porous medium equation perturbed by Lévy process,and then show the exponential convergence of(pt)t≥0 to equilibrium uniform on any bounded subset in H.
基金Program for Professor of Special Appointment(Eastern Scholar)at Shanghai Institutions of Higher Learning,and the Project for the Natural Science Foundation of Shanghai(Grant No.21ZR1444100)the Project for the National Natural Science Foundation of China(Grant Nos.71774111,61702331,71871144).
文摘With the development of information technology,rumors propagate faster and more widely than in the past.In this paper,a stochastic rumor propagation model incorporating media coverage and driven by Lévy noise is proposed.The global positivity of the solution process is proved,and further the basic reproductive number R_(0) is obtained.When R_(0)<1,the dynamical process of system with Lévy jump tends to the rumor-free equilibrium point of the deterministic system,and the rumor tends to extinction;when R_(0)>1,the rumor will keep spreading and the system will oscillate randomly near the rumor equilibrium point of the deterministic system.The results show that the oscillation amplitude is related to the disturbance of the system.In addition,increasing media coverage can effectively reduce the final spread of rumors.Finally,the above results are verified by numerical simulation.
基金Supported by the National Natural Science Foundation of China(11201221)Supported by the Natural Science Foundation of Jiangsu Province(BK2012468)
文摘In this paper, we consider a Markov switching Lévy process model in which the underlying risky assets are driven by the stochastic exponential of Markov switching Lévy process and then apply the model to option pricing and hedging. In this model, the market interest rate, the volatility of the underlying risky assets and the N-state compensator,depend on unobservable states of the economy which are modeled by a continuous-time Hidden Markov process. We use the MEMM(minimal entropy martingale measure) as the equivalent martingale measure. The option price using this model is obtained by the Fourier transform method. We obtain a closed-form solution for the hedge ratio by applying the local risk minimizing hedging.
基金supported by the National Key R&D Program of China(Grant No.2018YFA0703900)the National Natural Science Foundation of China(Grant No.11671231)+2 种基金the Qilu Young Scholars Program of Shandong Universitysupported by the Tian Yuan Projection of the National Natural Science Foundation of China(Grant Nos.11526205,11626247)the National Basic Research Program of China(973 Program)(Grant No.2007CB814900(Financial Risk)).
文摘This article establishes a universal robust limit theorem under a sublinear expectation framework.Under moment and consistency conditions,we show that,forα∈(1,2),the i.i.d.sequence{(1/√∑_(i=1)^(n)X_(i),1/n∑_(i=1)^(n)X_(i)Y_(i),1/α√n∑_(i=1)^(n)X_(i))}_(n=1)^(∞)converges in distribution to L_(1),where L_(t=(ε_(t),η_(t),ζ_(t))),t∈[0,1],is a multidimensional nonlinear Lévy process with an uncertainty■set as a set of Lévy triplets.This nonlinear Lévy process is characterized by a fully nonlinear and possibly degenerate partial integro-differential equation(PIDE){δ_(t)u(t,x,y,z)-sup_(F_(μ),q,Q)∈■{∫_(R^(d)δλu(t,x,y,z)(dλ)with.To construct the limit process,we develop a novel weak convergence approach based on the notions of tightness and weak compactness on a sublinear expectation space.We further prove a new type of Lévy-Khintchine representation formula to characterize.As a byproduct,we also provide a probabilistic approach to prove the existence of the above fully nonlinear degenerate PIDE.
基金Supported by NSFC(Grant Nos.11522104,11871132 and 11925102)Xinghai Jieqing and DUT19TD14 funds from Dalian University of Technology。
文摘In this paper,we use a unified framework to study Poisson stable(including stationary,periodic,quasi-periodic,almost periodic,almost automorphic,Birkhoff recurrent,almost recurrent in the sense of Bebutov,Levitan almost periodic,pseudo-periodic,pseudo-recurrent and Poisson stable)solutions for semilinear stochastic differential equations driven by infinite dimensional L′evy noise with large jumps.Under suitable conditions on drift,diffusion and jump coefficients,we prove that there exist solutions which inherit the Poisson stability of coefficients.Further we show that these solutions are globally asymptotically stable in square-mean sense.Finally,we illustrate our theoretical results by several examples.
基金This work was supported by National Key R&D Program of China(Grant No.2018YFA0703900)National Natural Science Foundation of China(Grant No.11671231)+1 种基金Tian Yuan Fund of the National Natural Science Foundation of China(Grant Nos.11526205 and 11626247)National Basic Research Program of China(973 Program)(Grant No.2007CB814900).
文摘We introduce G-Lévy processes which develop the theory of processes with independent and stationary increments under the framework of sublinear expectations.We then obtain the Lévy-Khintchine formula and the existence for G-Lévy processes.We also introduce G-Poisson processes.
基金Acknowledgements The authors thank two anonymous referees for their constructive suggestions which have led to much improvement on the paper. The first author is grateful to Professor Xiaowen Zhou for useful discussion. The research of Yuen was supported by a university research grant of the University of Hong Kong. The research of Yin was supported by the National Natural Science Foundation of China (No. 11171179), the Research Fund for the Doctoral Program of Higher Education of China (No. 20133705110002), and the Program for Scientific Research Innovation Team in Colleges and Universities of Shandong Province.
文摘We consider the spectrally negative L@vy processes and determine the joint laws for the quantities such as the first and last passage times over a fixed level, the overshoots and undershoots at first passage, the minimum, the maximum, and the duration of negative values. We apply our results to insurance risk theory to find an explicit expression for the generalized expected discounted penalty function in terms of scale functions. Furthermore, a new expression for the generalized Dickson's formula is provided.
基金supported by the Major Basic Research Program of Natural Science Foundation of Shandong Province under Grant No.2019A01the Natural Science Foundation of Shandong Province of China under Grant No.ZR2020MF062。
文摘This paper is concerned with a class of mean-field type stochastic optimal control systems,which are governed by fully coupled mean-field forward-backward stochastic differential equations with Teugels martingales associated to Lévy processes.In these systems,the coefficients contain not only the state processes but also their marginal distribution,and the cost function is of mean-field type as well.The necessary and sufficient conditions for such optimal problems are obtained.Furthermore,the applications to the linear quadratic stochastic optimization control problem are investigated.
基金supported by TWAS Research Grants to individuals (No. 09-100 RG/MATHS/AF/AC-IUNESCO FR: 3240230311)
文摘A new class of generalized backward doubly stochastic differential equations (GBDSDEs in short) driven by Teugels martingales associated with Levy process are investigated. We establish a comparison theorem which allows us to derive an existence result of solutions under continuous and linear growth conditions.
基金supported by the National Natural Science Foundation(Grant No.10671188)Special Foundation of USTC
文摘Let {X(t), t ≥ 0} be a Lévy process with EX(1) = 0 and EX^2(1) 〈 ∞. In this paper, we shall give two precise asymptotic theorems for {X(t), t 〉 0}. By the way, we prove the corresponding conclusions for strictly stable processes and a general precise asymptotic proposition for sums of i.i.d. random variables.
基金Supported by the National Natural Science Foundation of China (No.10771119)the Research Fund forthe Doctoral Program of Higher Education of China (No.20093705110002)
基金Man Chen was supported by the China Scholarship Council(No.201908110314)Xianyuan Wu was supported by the National Natural Science Foundation of China(Grant No.11471222)Man Chen and Xianyuan Wu were supported by the Academy for Multidisciplinary Studies,Capital Normal University,and Man Chen and Xiaowen Zhou were supported by RGPIN-2016-06704.
文摘For spectrally negative Lévy process (SNLP), we find an expression, in terms of scale functions, for a potential measure involving the maximum and the last time of reaching the maximum up to a draw-down time. As applications, we obtain a potential measure for the reflected SNLP and recover a joint Laplace transform for the Wiener-Hopf factorization for SNLP.
基金supported by Science Engineering Research Board(SERB),DST,GovtYSS Project F.No:YSS/2014/000447 dated 20.11.2015UGC,New Delhi,for providing BSR fellowship for the year 2015.
文摘This paper focuses on optimal control of nonlinear stochastic delay system constructed through Teugels martingales associated with Lévy processes and standard Brownian motion,in which finite horizon is extended to infinite horizon.In order to describe the interacting many-body system,the expectation values of state processes are added to the concerned system.Further,sufficient and necessary conditions are established under convexity assumptions of the control domain.Finally,an example is given to demonstrate the application of the theory.