In this paper,we aim to derive an averaging principle for stochastic differential equations driven by time-changed Lévy noise with variable delays.Under certain assumptions,we show that the solutions of stochasti...In this paper,we aim to derive an averaging principle for stochastic differential equations driven by time-changed Lévy noise with variable delays.Under certain assumptions,we show that the solutions of stochastic differential equations with time-changed Lévy noise can be approximated by solutions of the associated averaged stochastic differential equations in mean square convergence and in convergence in probability,respectively.The convergence order is also estimated in terms of noise intensity.Finally,an example with numerical simulation is given to illustrate the theoretical result.展开更多
In this paper,numerical methods for the time-changed stochastic differential equations of the form dY(t)=a(Y(t))dt+b(Y(t))dE(t)+s(Y(t))dB(E(t))are investigated,where all the coefficients a(·),b(·)and s(·...In this paper,numerical methods for the time-changed stochastic differential equations of the form dY(t)=a(Y(t))dt+b(Y(t))dE(t)+s(Y(t))dB(E(t))are investigated,where all the coefficients a(·),b(·)and s(·)are allowed to contain some super-linearly growing terms.An explicit method is proposed by using the idea of truncating terms that grow too fast.Strong convergence in the finite time of the proposed method is proved and the convergence rate is obtained.The proposed method is also proved to be able to reproduce the asymptotic stability of the underlying equation in the almost sure sense.Simulations are provided to demonstrate the theoretical results.展开更多
So far there have been few results presented on the exponential stability for time-changed stochastic differential equations.The main aim of this work is to fill this gap.By making use of general Lyapunov methods and ...So far there have been few results presented on the exponential stability for time-changed stochastic differential equations.The main aim of this work is to fill this gap.By making use of general Lyapunov methods and time-changed Itôformula,we establish the exponential stability and almost sure exponential stability of solution to time-changed SDEs.Finally,we construct some examples to illustrate the effectiveness of our established theory.展开更多
In this study,a multivariate ARMA–GARCH model with fractional generalized hyperbolic innovations exhibiting fat-tail,volatility clustering,and long-range dependence properties is introduced.To define the fractional g...In this study,a multivariate ARMA–GARCH model with fractional generalized hyperbolic innovations exhibiting fat-tail,volatility clustering,and long-range dependence properties is introduced.To define the fractional generalized hyperbolic process,the non-fractional variant is derived by subordinating time-changed Brownian motion to the generalized inverse Gaussian process,and thereafter,the fractional generalized hyperbolic process is obtained using the Volterra kernel.Based on the ARMA–GARCH model with standard normal innovations,the parameters are estimated for the high-frequency returns of six U.S.stocks.Subsequently,the residuals extracted from the estimated ARMA–GARCH parameters are fitted to the fractional and non-fractional generalized hyperbolic processes.The results show that the fractional generalized hyperbolic process performs better in describing the behavior of the residual process of high-frequency returns than the non-fractional processes considered in this study.展开更多
Background:In this paper,we study the right time for an investor to stop the investment over a given investment horizon so as to obtain as close to the highest possible wealth as possible,according to a Logarithmic ut...Background:In this paper,we study the right time for an investor to stop the investment over a given investment horizon so as to obtain as close to the highest possible wealth as possible,according to a Logarithmic utility-maximization objective involving the portfolio in the drift and volatility terms.The problem is formulated as an optimal stopping problem,although it is non-standard in the sense that the maximum wealth involved is not adapted to the information generated over time.Methods:By delicate stochastic analysis,the problem is converted to a standard optimal stopping one involving adapted processes.Results:Numerical examples shed light on the efficiency of the theoretical results.Conclusion:Our investment problem,which includes the portfolio in the drift and volatility terms of the dynamic systems,makes the problem including multi-dimensional financial assets more realistic and meaningful.展开更多
The purpose of this article is to study the rational evaluation of European options price when the underlying price process is described by a time-change Levy process. European option pricing formula is obtained under...The purpose of this article is to study the rational evaluation of European options price when the underlying price process is described by a time-change Levy process. European option pricing formula is obtained under the minimal entropy martingale measure (MEMM) and applied to several examples of particular time-change Levy processes. It can be seen that the framework in this paper encompasses the Black-Scholes model and almost all of the models proposed in the subordinated market.展开更多
Evaluating default correlation between securities in a portfolio is very important for credit derivatives pricing and risk management. Under the framework of the structural model proposed by Black and Cox, we assume t...Evaluating default correlation between securities in a portfolio is very important for credit derivatives pricing and risk management. Under the framework of the structural model proposed by Black and Cox, we assume that the asset values of companies are driven by Brownian motions in the worlds of the calendar time and the business time; they then could evolve continuously or by leap. We build the dynamic default correlations using the time-varying correlated Brownian motions in these processes. The sensitivity of default correlations to the key parameters is explored in the paper by numerical examples, and we apply the model to risk management as well. Because default times are unpredictable in the proposed model, the defaults might occur suddenly. Independent defaults and complete correlated defaults can be described in the model as well.展开更多
Let (Zt)t≥o be a one-dimensional symmetric α-stable process with α ∈ (0, 2), and let a be a bounded (from above and from below) and 1/(α V 1)- Holder continuous function on R. Consider the stochastic diff...Let (Zt)t≥o be a one-dimensional symmetric α-stable process with α ∈ (0, 2), and let a be a bounded (from above and from below) and 1/(α V 1)- Holder continuous function on R. Consider the stochastic differential equation dXt = σ(Xt-)dZt, which admits a unique strong solution. By using the splitting technique and the coupling method, we derive the HSlder continuity of the associated semigroup.展开更多
The Gamma-Dirichlet algebra corresponds to the decomposition of the gamma process into the independent product of a gamma random variable and a Dirichlet process. This structure allows us to study the properties of th...The Gamma-Dirichlet algebra corresponds to the decomposition of the gamma process into the independent product of a gamma random variable and a Dirichlet process. This structure allows us to study the properties of the Dirichlet process through the gamma process and vice versa. In this article, we begin with a brief survey of several existing results concerning this structure. New results are then obtained for the large deviations of the jump sizes of the gamma process and the quasi-invariance of the two-parameter Poisson-Dirichlet distribution. We finish the paper with the derivation of the transition function of the Fleming-Viot process with parent independent mutation from the transition function of the measure-valued branching diffusion with immigration by exploring the Gamma-Dirichlet algebra embedded in these processes. This last result is motivated by an open R. C. Gritfiths. problem proposed by S. N. Ethier and展开更多
Conditional expectations(like,e.g.,discounted prices in financial applications)are martingales under an appropriate filtration and probability measure.When the information flow arrives in a punctual way,a reasonable a...Conditional expectations(like,e.g.,discounted prices in financial applications)are martingales under an appropriate filtration and probability measure.When the information flow arrives in a punctual way,a reasonable assumption is to suppose the latter to have piecewise constant sample paths between the random times of information updates.Providing a way to find and construct piecewise constant martingales evolving in a connected subset of R is the purpose of this paper.After a brief review of possible standard techniques,we propose a construction scheme based on the sampling of latent martingalesZ with lazy clocksθ.Theseθare time-change processes staying in arrears of the true time but that can synchronize at random times to the real(calendar)clock.This specific choice makes the resulting time-changed process Zt=Zθt a martingale(called a lazy martingale)without any assumption onZ,and in most cases,the lazy clockθis adapted to the filtration of the lazy martingale Z,so that sample paths of Z on[0,T]only requires sample paths ofθ,Zup to T.This would not be the case if the stochastic clockθcould be ahead of the real clock,as is typically the case using standard time-change processes.The proposed approach yields an easy way to construct analytically tractable lazy martingales evolving on(interval of)R.展开更多
基金supported by the National NaturalScience Foundation of China(12071003,11901005)the Natural Science Foundation of Anhui Province(2008085QA20)。
文摘In this paper,we aim to derive an averaging principle for stochastic differential equations driven by time-changed Lévy noise with variable delays.Under certain assumptions,we show that the solutions of stochastic differential equations with time-changed Lévy noise can be approximated by solutions of the associated averaged stochastic differential equations in mean square convergence and in convergence in probability,respectively.The convergence order is also estimated in terms of noise intensity.Finally,an example with numerical simulation is given to illustrate the theoretical result.
基金Wei Liu would like to thank Shanghai Rising-Star Program(Grant No.22QA1406900)Science and Technology Innovation Plan of Shanghai(Grant No.20JC1414200)the National Natural Science Foundation of China(Grant Nos.11871343,11971316 and 12271368)for their financial support.
文摘In this paper,numerical methods for the time-changed stochastic differential equations of the form dY(t)=a(Y(t))dt+b(Y(t))dE(t)+s(Y(t))dB(E(t))are investigated,where all the coefficients a(·),b(·)and s(·)are allowed to contain some super-linearly growing terms.An explicit method is proposed by using the idea of truncating terms that grow too fast.Strong convergence in the finite time of the proposed method is proved and the convergence rate is obtained.The proposed method is also proved to be able to reproduce the asymptotic stability of the underlying equation in the almost sure sense.Simulations are provided to demonstrate the theoretical results.
基金by the National Natural Science Foundation of China(Nos.11901188,61703001)by the Scientific Research Funds of Hunan Provincial Education Department of China(No.19B156).
文摘So far there have been few results presented on the exponential stability for time-changed stochastic differential equations.The main aim of this work is to fill this gap.By making use of general Lyapunov methods and time-changed Itôformula,we establish the exponential stability and almost sure exponential stability of solution to time-changed SDEs.Finally,we construct some examples to illustrate the effectiveness of our established theory.
文摘In this study,a multivariate ARMA–GARCH model with fractional generalized hyperbolic innovations exhibiting fat-tail,volatility clustering,and long-range dependence properties is introduced.To define the fractional generalized hyperbolic process,the non-fractional variant is derived by subordinating time-changed Brownian motion to the generalized inverse Gaussian process,and thereafter,the fractional generalized hyperbolic process is obtained using the Volterra kernel.Based on the ARMA–GARCH model with standard normal innovations,the parameters are estimated for the high-frequency returns of six U.S.stocks.Subsequently,the residuals extracted from the estimated ARMA–GARCH parameters are fitted to the fractional and non-fractional generalized hyperbolic processes.The results show that the fractional generalized hyperbolic process performs better in describing the behavior of the residual process of high-frequency returns than the non-fractional processes considered in this study.
基金This work is supported by Research Grants Council of Hong Kong under grant no.519913 and 15224215National Natural Science Foundation of China(No.11571124).
文摘Background:In this paper,we study the right time for an investor to stop the investment over a given investment horizon so as to obtain as close to the highest possible wealth as possible,according to a Logarithmic utility-maximization objective involving the portfolio in the drift and volatility terms.The problem is formulated as an optimal stopping problem,although it is non-standard in the sense that the maximum wealth involved is not adapted to the information generated over time.Methods:By delicate stochastic analysis,the problem is converted to a standard optimal stopping one involving adapted processes.Results:Numerical examples shed light on the efficiency of the theoretical results.Conclusion:Our investment problem,which includes the portfolio in the drift and volatility terms of the dynamic systems,makes the problem including multi-dimensional financial assets more realistic and meaningful.
文摘The purpose of this article is to study the rational evaluation of European options price when the underlying price process is described by a time-change Levy process. European option pricing formula is obtained under the minimal entropy martingale measure (MEMM) and applied to several examples of particular time-change Levy processes. It can be seen that the framework in this paper encompasses the Black-Scholes model and almost all of the models proposed in the subordinated market.
基金Supported by the Fundamental Research Funds for the Central Universities
文摘Evaluating default correlation between securities in a portfolio is very important for credit derivatives pricing and risk management. Under the framework of the structural model proposed by Black and Cox, we assume that the asset values of companies are driven by Brownian motions in the worlds of the calendar time and the business time; they then could evolve continuously or by leap. We build the dynamic default correlations using the time-varying correlated Brownian motions in these processes. The sensitivity of default correlations to the key parameters is explored in the paper by numerical examples, and we apply the model to risk management as well. Because default times are unpredictable in the proposed model, the defaults might occur suddenly. Independent defaults and complete correlated defaults can be described in the model as well.
基金The authors were indebted to the referees for their helpful comments and careful corrections. The first author's work was supported by the Key Laboratory of Random Complex Structures and Data Sciences, Chinese Academy of Sciences (2008DP173182), the National Natural Science Foundation of China (Grant No. 11571347), and Academy of Mathematics and Systems Science (Y129161ZZ1). The second author's work was supported by the National Natural Science Foundation of China (Grant Nos. 11201073 and 11522106), the National Science Foundation of Fujian Province (2015J01003), and the Program for Nonlinear Analysis and Its Applications (IRTL1206).
文摘Let (Zt)t≥o be a one-dimensional symmetric α-stable process with α ∈ (0, 2), and let a be a bounded (from above and from below) and 1/(α V 1)- Holder continuous function on R. Consider the stochastic differential equation dXt = σ(Xt-)dZt, which admits a unique strong solution. By using the splitting technique and the coupling method, we derive the HSlder continuity of the associated semigroup.
文摘The Gamma-Dirichlet algebra corresponds to the decomposition of the gamma process into the independent product of a gamma random variable and a Dirichlet process. This structure allows us to study the properties of the Dirichlet process through the gamma process and vice versa. In this article, we begin with a brief survey of several existing results concerning this structure. New results are then obtained for the large deviations of the jump sizes of the gamma process and the quasi-invariance of the two-parameter Poisson-Dirichlet distribution. We finish the paper with the derivation of the transition function of the Fleming-Viot process with parent independent mutation from the transition function of the measure-valued branching diffusion with immigration by exploring the Gamma-Dirichlet algebra embedded in these processes. This last result is motivated by an open R. C. Gritfiths. problem proposed by S. N. Ethier and
基金the Fonds de la Recherche Scientifique-FNRS under Grant J.0037.18.
文摘Conditional expectations(like,e.g.,discounted prices in financial applications)are martingales under an appropriate filtration and probability measure.When the information flow arrives in a punctual way,a reasonable assumption is to suppose the latter to have piecewise constant sample paths between the random times of information updates.Providing a way to find and construct piecewise constant martingales evolving in a connected subset of R is the purpose of this paper.After a brief review of possible standard techniques,we propose a construction scheme based on the sampling of latent martingalesZ with lazy clocksθ.Theseθare time-change processes staying in arrears of the true time but that can synchronize at random times to the real(calendar)clock.This specific choice makes the resulting time-changed process Zt=Zθt a martingale(called a lazy martingale)without any assumption onZ,and in most cases,the lazy clockθis adapted to the filtration of the lazy martingale Z,so that sample paths of Z on[0,T]only requires sample paths ofθ,Zup to T.This would not be the case if the stochastic clockθcould be ahead of the real clock,as is typically the case using standard time-change processes.The proposed approach yields an easy way to construct analytically tractable lazy martingales evolving on(interval of)R.