A lot of forms and applications have been found in Poincar6 inequality, but the optimum constants satisfying Poincar6 inequality have not been estimated. This paper estimates the optimum constants λ0 and λ1 satisfyi...A lot of forms and applications have been found in Poincar6 inequality, but the optimum constants satisfying Poincar6 inequality have not been estimated. This paper estimates the optimum constants λ0 and λ1 satisfying Poincaré inequality by using isoperimetric constants. It is λ0≥k0^2/（2R） and λ1 ≥k1^2/（2R）.展开更多
A variational formula for the lower bound of the principal eigenvalue of general Markov jump processes is presented.The result is complete in the sense that the condition is fulfilled and the resulting bound is sharp ...A variational formula for the lower bound of the principal eigenvalue of general Markov jump processes is presented.The result is complete in the sense that the condition is fulfilled and the resulting bound is sharp for Markov chains under some mild assumptions.展开更多
Let L be a L′evy process with characteristic measureν,which has an absolutely continuous lower bound w.r.t.the Lebesgue measure on Rn.By using Malliavin calculus for jump processes,we investigate Bismut formula,grad...Let L be a L′evy process with characteristic measureν,which has an absolutely continuous lower bound w.r.t.the Lebesgue measure on Rn.By using Malliavin calculus for jump processes,we investigate Bismut formula,gradient estimates and coupling property for the semigroups associated to semilinear SDEs forced by L′evy process L.展开更多
This is a sequel to our joint paper in which upper bound estimates for large deviations for Markov chains are studied.The purpose of this paper is to characterize the rate function of large devia- tions for jump proce...This is a sequel to our joint paper in which upper bound estimates for large deviations for Markov chains are studied.The purpose of this paper is to characterize the rate function of large devia- tions for jump processes.In particular,an explicit expression of the rate function is given in the case of the process being symmetrizable.展开更多
We survey the recent development of the DeGiorgi-Nash-Moser-Aronson type theory for a class of symmetric jump processes(or equivalently,a class of symmetric integro-differential operators).We focus on the sharp two-si...We survey the recent development of the DeGiorgi-Nash-Moser-Aronson type theory for a class of symmetric jump processes(or equivalently,a class of symmetric integro-differential operators).We focus on the sharp two-sided estimates for the transition density functions(or heat kernels) of the processes,a priori Hlder estimate and parabolic Harnack inequalities for their parabolic functions.In contrast to the second order elliptic differential operator case,the methods to establish these properties for symmetric integro-differential operators are mainly probabilistic.展开更多
Let X=(Ω,■,■,X_(t),θ_(t),P^(x))be a jump Markov process with q-pair q(x)-q(x,A).In this paper,the equilibrium principle is established and equilibrium functions,energy,capacity and related problems is investigated...Let X=(Ω,■,■,X_(t),θ_(t),P^(x))be a jump Markov process with q-pair q(x)-q(x,A).In this paper,the equilibrium principle is established and equilibrium functions,energy,capacity and related problems is investigated in terms of the q-pair q(x)-q(x,A).展开更多
In this paper, we discuss necessary and sufficient conditions on jumping kernels for a class of jump-type Markov processes on metric measure spaces to have scale-invariant finite range parabolic Harnack inequality.
Let （Xt）t≥0 be a symmetric strong Markov process generated by non-local regular Dirichlet form （D, （D）） as follows：where J（x, y） is a strictly positive and symmetric measurable function on Rd × Rd. We s...Let （Xt）t≥0 be a symmetric strong Markov process generated by non-local regular Dirichlet form （D, （D）） as follows：where J（x, y） is a strictly positive and symmetric measurable function on Rd × Rd. We study the intrinsic hypercontractivity, intrinsic supercontractivity, and intrinsic ultracontractivity for the Feynman-Kac semigroup展开更多
The representation of additive functionals and local times for jump Markov processes are obtained.The results of uniformly functional moderate deviation and their applications to birth-death processes are also presented.
In this paper,we first establish the sharp two-sided heat kernel estimates and the gradient estimate for the truncated fractional Laplacian under gradient perturbation S^(b):=△^(α/2)+b·▽,where △^(α/2) is the...In this paper,we first establish the sharp two-sided heat kernel estimates and the gradient estimate for the truncated fractional Laplacian under gradient perturbation S^(b):=△^(α/2)+b·▽,where △^(α/2) is the truncated fractional Laplacian,α∈(1,2) and b ∈ K_(d)^(α-1).In the second part,for a more general finite range jump process,we present some sufficient conditions to allow that the two sided estimates of the heat kernel are comparable to the Poisson type function for large distance |x-y|in short time.展开更多
Identity by descent(IBD)sharing is a very important genomic feature in population genetics which can be used to reconstruct recent demographic history.In this paper we provide a framework to estimate IBD sharing for a...Identity by descent(IBD)sharing is a very important genomic feature in population genetics which can be used to reconstruct recent demographic history.In this paper we provide a framework to estimate IBD sharing for a demographic model called two-population model with migration.We adopt the structured coalescent theory and use a continuous-time Markov jump process{X(t),t≥0}to describe the genealogical process in such model.Then we apply Kolmogorov backward equation to calculate the distribution of coalescence time and develop a formula for estimating the IBD sharing.The simulation studies show that our method to estimate IBD sharing for this demographic model is robust and accurate.展开更多
The hedging problem for insiders is very important in the financial market.The locally risk minimizing hedging was adopted to solve this problem.Since the market was incomplete,the minimal martingale measure was chose...The hedging problem for insiders is very important in the financial market.The locally risk minimizing hedging was adopted to solve this problem.Since the market was incomplete,the minimal martingale measure was chosen as the equivalent martingale measure.By the F-S decomposition,the expression of the locally risk minimizing strategy was presented.Finally,the local risk minimization was applied to index tracking and its relationship with tracking error variance (TEV)-minimizing strategy was obtained.展开更多
Stochastic generalized porous media equation with jump is considered. The aim is to show the moment exponential stability and the almost certain exponential stability of the stochastic equation.
Structural models of credit risk are known to present vanishing spreads at very short maturities. This shortcoming, which is due to the diffusive behavior assumed for asset values, can be circumvented by considering d...Structural models of credit risk are known to present vanishing spreads at very short maturities. This shortcoming, which is due to the diffusive behavior assumed for asset values, can be circumvented by considering discontinuities of the jump type in their evolution over time. In this paper, we extend the pricing model for corporate bond and determine the default probability in jump-diffusion model to address this issue. To make the problem clearly, we first investigate the case that the firm value follows a geometric Brownian motion under similar assumptions to those in Black and Scholes（1973）, Briys and de Varenne（1997）, i.e, the default barrier is KD （t, T） and the recovery rate is （1 -w）, where D （t, T） is the price of zero coupon default free bond and w is a constant （0 〈 w 〈 1）. By changing the numeraire, we obtain the closed-form solution for both the price of bond and default probability. Further, we consider the case of jump-diffusion and suppose that a firm will go bankruptcy if its value Vt 〈 KD （t, T） and at the same time, the bondholder will receive （1 - w） vt/k By introducing the Green function of PDE with absorbing boundary and converting the problem to an II-type Volterra integral equation, we get the closed-form expressions in series form for bond price and corresponding default probability. Numerical results are presented to show the impact of different parameters to credit spread of bond.展开更多
Using approximation technique, we introduce the concepts of canonical extension and symmetrio integral for jump process and obtain some results in the chaotic form.
Although Geometric Brownian Motion and Jump Diffusion Models have largely dominated the literature on asset price modeling, studies of the empirical stock price data on the Ghana Stock Exchange have led to the conclus...Although Geometric Brownian Motion and Jump Diffusion Models have largely dominated the literature on asset price modeling, studies of the empirical stock price data on the Ghana Stock Exchange have led to the conclusion that there are some stocks in which the return processes consistently depart from these models in theory as well as in its statistical properties. This paper gives a fundamental review of the development of a stock price model based on pure jump processes to capture the unique behavior exhibited by some stocks on the Exchange. Although pure jump processes have been examined thoroughly by other authors, there is a lack of mathematical clarity in terms of deriving the underlying stock price process. This paper provides a link between stock prices existing on a measure space to its development as a pure jump Levy process. We test the suitability of the model to the empirical evidence using numerical procedures. The simulation results show that the trajectories of the model are a better fit for the empirical data than those produced by the diffusion and jump diffusion models.展开更多
In this paper the insurer＇s solvency ratio model with or without jump diffusion process in the presence of financial distress cost is constructed, where an insurer＇s solvency ratio is characterized by a Markov-modul...In this paper the insurer＇s solvency ratio model with or without jump diffusion process in the presence of financial distress cost is constructed, where an insurer＇s solvency ratio is characterized by a Markov-modulated dynamics. By Girsanov＇s theorem and the option pricing formula, the expected present value of shareholders＇ terminal payoff is provided.展开更多
This paper presents limit theorems for realized power variation of processes of the form Xt=t0φsdGs+ξt as the sampling frequency within a fixed interval increases to infinity.Here G is a Gaussian process with statio...This paper presents limit theorems for realized power variation of processes of the form Xt=t0φsdGs+ξt as the sampling frequency within a fixed interval increases to infinity.Here G is a Gaussian process with stationary increments,ξis a purely non-Gaussian L′evy process independent from G,andφis a stochastic process ensuring that the integral is well defined as a pathwise Riemann-Stieltjes integral.We obtain the central limit theorems for the case that both the continuous term and the jump term are presented simultaneously in the law of large numbers.展开更多
A Markovian risk process is considered in this paper, which is the generalization of the classical risk model. It is proper that a risk process with large claims is modelled as the Markovian risk model. In such a mode...A Markovian risk process is considered in this paper, which is the generalization of the classical risk model. It is proper that a risk process with large claims is modelled as the Markovian risk model. In such a model, the occurrence of claims is described by a point process {N（t）}t≥0 with N（t） being the number of jumps during the interval （0, t] for a Markov jump process. The ruin probability ψ（u） of a company facing such a risk model is mainly studied. An integral equation satisfied by the ruin probability function ψ（u） is obtained and the bounds for the convergence rate of the ruin probability ψ（u） are given by using a generalized renewal technique developed in the paper.展开更多
基金The Science and Technology Foundation of Chongqing Municipal Education Commission (No.KJ071106)
文摘A lot of forms and applications have been found in Poincar6 inequality, but the optimum constants satisfying Poincar6 inequality have not been estimated. This paper estimates the optimum constants λ0 and λ1 satisfying Poincaré inequality by using isoperimetric constants. It is λ0≥k0^2/（2R） and λ1 ≥k1^2/（2R）.
基金Research supported in part by NSFC (No.19631060)Math.Tian Yuan Found.,Qiu Shi Sci.& Tech.Found.,RFDP and MCME
文摘A variational formula for the lower bound of the principal eigenvalue of general Markov jump processes is presented.The result is complete in the sense that the condition is fulfilled and the resulting bound is sharp for Markov chains under some mild assumptions.
文摘Let L be a L′evy process with characteristic measureν,which has an absolutely continuous lower bound w.r.t.the Lebesgue measure on Rn.By using Malliavin calculus for jump processes,we investigate Bismut formula,gradient estimates and coupling property for the semigroups associated to semilinear SDEs forced by L′evy process L.
文摘This is a sequel to our joint paper in which upper bound estimates for large deviations for Markov chains are studied.The purpose of this paper is to characterize the rate function of large devia- tions for jump processes.In particular,an explicit expression of the rate function is given in the case of the process being symmetrizable.
基金supported by National Science Foundation of USA(Grant No.DMS-0600206)
文摘We survey the recent development of the DeGiorgi-Nash-Moser-Aronson type theory for a class of symmetric jump processes(or equivalently,a class of symmetric integro-differential operators).We focus on the sharp two-sided estimates for the transition density functions(or heat kernels) of the processes,a priori Hlder estimate and parabolic Harnack inequalities for their parabolic functions.In contrast to the second order elliptic differential operator case,the methods to establish these properties for symmetric integro-differential operators are mainly probabilistic.
文摘Let X=(Ω,■,■,X_(t),θ_(t),P^(x))be a jump Markov process with q-pair q(x)-q(x,A).In this paper,the equilibrium principle is established and equilibrium functions,energy,capacity and related problems is investigated in terms of the q-pair q(x)-q(x,A).
基金supported by NSF (Grant No. DMS-0600206)supported by the Korea Science Engineering Foundation (KOSEF) Grant funded by the Korea government (MEST) (No. R01-2008-000-20010-0)supported by the Grant-in-Aid for Scientific Research (B) 18340027
文摘In this paper, we discuss necessary and sufficient conditions on jumping kernels for a class of jump-type Markov processes on metric measure spaces to have scale-invariant finite range parabolic Harnack inequality.
基金The authors would like to thank Professor Mu-Fa Chen and Professor Feng-Yu Wang for introducing them the field of functional inequalities when they studied in Beijing Normal University, and for their continuous encouragement and great help in the past few years. The authors are also indebted to the referees for valuable comments on the draft. This work was supported by the National Natural Science Foundation of China （Grant No. 11201073）, Japan Society for the Promotion of Science （No. 26.04021）, the Natural Science Foundation of Fujian Province （No. 2015J01003）, and the Program for Nonlinear Analysis and Its Applications （No. IRTL1206） （for Jian Wang）.
文摘Let （Xt）t≥0 be a symmetric strong Markov process generated by non-local regular Dirichlet form （D, （D）） as follows：where J（x, y） is a strictly positive and symmetric measurable function on Rd × Rd. We study the intrinsic hypercontractivity, intrinsic supercontractivity, and intrinsic ultracontractivity for the Feynman-Kac semigroup
基金supported by the National Nature Science Foundation of China(10271091)
文摘The representation of additive functionals and local times for jump Markov processes are obtained.The results of uniformly functional moderate deviation and their applications to birth-death processes are also presented.
基金Partially supported by NSFC(Grant Nos.11731009 and 11401025)。
文摘In this paper,we first establish the sharp two-sided heat kernel estimates and the gradient estimate for the truncated fractional Laplacian under gradient perturbation S^(b):=△^(α/2)+b·▽,where △^(α/2) is the truncated fractional Laplacian,α∈(1,2) and b ∈ K_(d)^(α-1).In the second part,for a more general finite range jump process,we present some sufficient conditions to allow that the two sided estimates of the heat kernel are comparable to the Poisson type function for large distance |x-y|in short time.
基金supported by the Fundamental Research Funds for the Central Universities(2020RC001)。
文摘Identity by descent(IBD)sharing is a very important genomic feature in population genetics which can be used to reconstruct recent demographic history.In this paper we provide a framework to estimate IBD sharing for a demographic model called two-population model with migration.We adopt the structured coalescent theory and use a continuous-time Markov jump process{X(t),t≥0}to describe the genealogical process in such model.Then we apply Kolmogorov backward equation to calculate the distribution of coalescence time and develop a formula for estimating the IBD sharing.The simulation studies show that our method to estimate IBD sharing for this demographic model is robust and accurate.
基金National Natural Science Foundations of China (No. 11071076,No. 11126124)
文摘The hedging problem for insiders is very important in the financial market.The locally risk minimizing hedging was adopted to solve this problem.Since the market was incomplete,the minimal martingale measure was chosen as the equivalent martingale measure.By the F-S decomposition,the expression of the locally risk minimizing strategy was presented.Finally,the local risk minimization was applied to index tracking and its relationship with tracking error variance (TEV)-minimizing strategy was obtained.
基金Project supported by the Tianyuan Foundation of National Natural Science of China(No.11126079)the China Postdoctoral Science Foundation(No.2013M530559)the Fundamental Research Funds for the Central Universities(No.CDJRC10100011)
文摘Stochastic generalized porous media equation with jump is considered. The aim is to show the moment exponential stability and the almost certain exponential stability of the stochastic equation.
基金Supported by the National Basic Research Program of China(973 Program)(2007CB814903)
文摘Structural models of credit risk are known to present vanishing spreads at very short maturities. This shortcoming, which is due to the diffusive behavior assumed for asset values, can be circumvented by considering discontinuities of the jump type in their evolution over time. In this paper, we extend the pricing model for corporate bond and determine the default probability in jump-diffusion model to address this issue. To make the problem clearly, we first investigate the case that the firm value follows a geometric Brownian motion under similar assumptions to those in Black and Scholes（1973）, Briys and de Varenne（1997）, i.e, the default barrier is KD （t, T） and the recovery rate is （1 -w）, where D （t, T） is the price of zero coupon default free bond and w is a constant （0 〈 w 〈 1）. By changing the numeraire, we obtain the closed-form solution for both the price of bond and default probability. Further, we consider the case of jump-diffusion and suppose that a firm will go bankruptcy if its value Vt 〈 KD （t, T） and at the same time, the bondholder will receive （1 - w） vt/k By introducing the Green function of PDE with absorbing boundary and converting the problem to an II-type Volterra integral equation, we get the closed-form expressions in series form for bond price and corresponding default probability. Numerical results are presented to show the impact of different parameters to credit spread of bond.
基金Supported in part by National Natural Science Foundation of China.
文摘Using approximation technique, we introduce the concepts of canonical extension and symmetrio integral for jump process and obtain some results in the chaotic form.
文摘Although Geometric Brownian Motion and Jump Diffusion Models have largely dominated the literature on asset price modeling, studies of the empirical stock price data on the Ghana Stock Exchange have led to the conclusion that there are some stocks in which the return processes consistently depart from these models in theory as well as in its statistical properties. This paper gives a fundamental review of the development of a stock price model based on pure jump processes to capture the unique behavior exhibited by some stocks on the Exchange. Although pure jump processes have been examined thoroughly by other authors, there is a lack of mathematical clarity in terms of deriving the underlying stock price process. This paper provides a link between stock prices existing on a measure space to its development as a pure jump Levy process. We test the suitability of the model to the empirical evidence using numerical procedures. The simulation results show that the trajectories of the model are a better fit for the empirical data than those produced by the diffusion and jump diffusion models.
基金Supported by National Natural Science Foundation of China (10671182)Anhui Natural Science Foundation (090416225)+1 种基金Anhui Natural Science Foundation of Universities (KJ2010A037, KJ2010B026)Anhui Natural Science Foundation (10040606Q03)
文摘In this paper the insurer＇s solvency ratio model with or without jump diffusion process in the presence of financial distress cost is constructed, where an insurer＇s solvency ratio is characterized by a Markov-modulated dynamics. By Girsanov＇s theorem and the option pricing formula, the expected present value of shareholders＇ terminal payoff is provided.
基金supported by National Natural Science Foundation of China(Grant Nos.11071045 and 11226201)Natural Science Foundation of Jiangsu Province of China(Grant No.BK20131340)+1 种基金Social Science Foundation of Chinese Ministry of Education(Grant No.12YJCZH128)QingLan Project ofthe Priority Academic Program Development of Jiangsu Higher Education Institutions(Auditing Science and Technology)
文摘This paper presents limit theorems for realized power variation of processes of the form Xt=t0φsdGs+ξt as the sampling frequency within a fixed interval increases to infinity.Here G is a Gaussian process with stationary increments,ξis a purely non-Gaussian L′evy process independent from G,andφis a stochastic process ensuring that the integral is well defined as a pathwise Riemann-Stieltjes integral.We obtain the central limit theorems for the case that both the continuous term and the jump term are presented simultaneously in the law of large numbers.
文摘A Markovian risk process is considered in this paper, which is the generalization of the classical risk model. It is proper that a risk process with large claims is modelled as the Markovian risk model. In such a model, the occurrence of claims is described by a point process {N（t）}t≥0 with N（t） being the number of jumps during the interval （0, t] for a Markov jump process. The ruin probability ψ（u） of a company facing such a risk model is mainly studied. An integral equation satisfied by the ruin probability function ψ（u） is obtained and the bounds for the convergence rate of the ruin probability ψ（u） are given by using a generalized renewal technique developed in the paper.
