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
In this article, we consider the long time behavior of the solutions to stochastic wave equations driven by a non-Gaussian Lévy process. We shall prove that under some appropriate conditions, the exponential stab...In this article, we consider the long time behavior of the solutions to stochastic wave equations driven by a non-Gaussian Lévy process. We shall prove that under some appropriate conditions, the exponential stability of the solutions holds. Finally, we give two examples to illustrate our results.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
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
In this study,we are interested in stochastic differential equations driven by GLévy processes.We illustrate that a certain class of additive functionals of the equations of interest exhibits the path-independent...In this study,we are interested in stochastic differential equations driven by GLévy processes.We illustrate that a certain class of additive functionals of the equations of interest exhibits the path-independent property,generalizing a few known findings in the literature.The study is ended with many examples.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
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 study the option price theory of stochastic differential equations under G-Lévy process. By using G-It<span style="font-size:12px;white-space:nowrap;">ô</span> for...In this paper, we study the option price theory of stochastic differential equations under G-Lévy process. By using G-It<span style="font-size:12px;white-space:nowrap;">ô</span> formula and G-expectation property, we give the proof of Black-Scholes equations (Integro-PDE) under G-Lévy process. Finally, we give the simulation of G-Lévy process and the explicit solution of Black-Scholes under G-Lévy process.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
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
In this paper we investigate an integration by parts formula for Lévy processes by using lower bound conditions of the corresponding Lévy measure. As applications, derivative formula and coupling property ar...In this paper we investigate an integration by parts formula for Lévy processes by using lower bound conditions of the corresponding Lévy measure. As applications, derivative formula and coupling property are derived for transition semigroups of linear SDEs driven by Lévy processes.展开更多
In this paper, we propose numerical schemes for stochastic differential equations driven by G-Lévy process under the G-expectation framework. By using G-Itôformula and G-expectation property, we propose ...In this paper, we propose numerical schemes for stochastic differential equations driven by G-Lévy process under the G-expectation framework. By using G-Itôformula and G-expectation property, we propose Euler scheme and Milstein scheme which have order-1.0 convergence rate. And two numerical experiments including Ornstein-Uhlenbeck and Black-Scholes cases are given.展开更多
In this paper, we first present an option pricing model of stochastic differential equations driven by the G-Lévy process under the G-expectation framework, and prove the generalized Black-Scholes equations. Then...In this paper, we first present an option pricing model of stochastic differential equations driven by the G-Lévy process under the G-expectation framework, and prove the generalized Black-Scholes equations. Then, we present the algorithm for the time-homogeneous Poisson process versus the non-time-homogeneous Poisson process. Finally, we provide an explicit solution of generalized Black-Scholes equations and simulate it numerically with Matlab software.展开更多
In this paper, according to G-Brownian motion and other related concepts and properties, we define multiple Itôintegrals driven by G-Brownian motion and G-Lévy process. By using the G-Itôformula...In this paper, according to G-Brownian motion and other related concepts and properties, we define multiple Itôintegrals driven by G-Brownian motion and G-Lévy process. By using the G-Itôformula and the properties of G-expectation, two main theorems about Itôintegral are obtained and proved. These two theorems provide powerful help for the subsequent research on jump process.展开更多
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.
基金supported by National Natural Science Foundation of China(11571190)the Fundamental Research Funds for the Central Universities+3 种基金supported by the China Scholarship Council(201807315008)National Natural Science Foundation of China(11501565)the Youth Project of Humanities and Social Sciences of Ministry of Education(19YJCZH251)supported by National Natural Science Foundation of China(11701084 and 11671084)
文摘In this article, we consider the long time behavior of the solutions to stochastic wave equations driven by a non-Gaussian Lévy process. We shall prove that under some appropriate conditions, the exponential stability of the solutions holds. Finally, we give two examples to illustrate our results.
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.
文摘In this study,we are interested in stochastic differential equations driven by GLévy processes.We illustrate that a certain class of additive functionals of the equations of interest exhibits the path-independent property,generalizing a few known findings in the literature.The study is ended with many examples.
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 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.
文摘In this paper, we study the option price theory of stochastic differential equations under G-Lévy process. By using G-It<span style="font-size:12px;white-space:nowrap;">ô</span> formula and G-expectation property, we give the proof of Black-Scholes equations (Integro-PDE) under G-Lévy process. Finally, we give the simulation of G-Lévy process and the explicit solution of Black-Scholes under G-Lévy process.
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.
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(10971180),(11271169)A Project Funded by the Priority Academic Program Development(PAPD) of Jiangsu Higher Education Institutions
文摘In this paper we investigate an integration by parts formula for Lévy processes by using lower bound conditions of the corresponding Lévy measure. As applications, derivative formula and coupling property are derived for transition semigroups of linear SDEs driven by Lévy processes.
文摘In this paper, we propose numerical schemes for stochastic differential equations driven by G-Lévy process under the G-expectation framework. By using G-Itôformula and G-expectation property, we propose Euler scheme and Milstein scheme which have order-1.0 convergence rate. And two numerical experiments including Ornstein-Uhlenbeck and Black-Scholes cases are given.
文摘In this paper, we first present an option pricing model of stochastic differential equations driven by the G-Lévy process under the G-expectation framework, and prove the generalized Black-Scholes equations. Then, we present the algorithm for the time-homogeneous Poisson process versus the non-time-homogeneous Poisson process. Finally, we provide an explicit solution of generalized Black-Scholes equations and simulate it numerically with Matlab software.
文摘In this paper, according to G-Brownian motion and other related concepts and properties, we define multiple Itôintegrals driven by G-Brownian motion and G-Lévy process. By using the G-Itôformula and the properties of G-expectation, two main theorems about Itôintegral are obtained and proved. These two theorems provide powerful help for the subsequent research on jump process.
