In this article, the joint distributions of several actuarial diagnostics which are important to insurers' running for the jump-diffusion risk process are examined. They include the ruin time, the time of the surplus...In this article, the joint distributions of several actuarial diagnostics which are important to insurers' running for the jump-diffusion risk process are examined. They include the ruin time, the time of the surplus process leaving zero ultimately (simply, the ultimately leaving-time), the surplus immediately prior to ruin, the supreme profits before ruin, the supreme profits and deficit until it leaves zero ultimately and so on. The explicit expressions for their distributions are obtained mainly by the various properties of Levy process, such as the homogeneous strong Markov property and the spatial homogeneity property etc, moveover, the many properties for Brownian motion.展开更多
In this paper, the optimal XL-reinsurance of an insurer with jump-diffusion risk process is studied. With the assumptions that the risk process is a compound Possion process perturbed by a standard Brownian motion and...In this paper, the optimal XL-reinsurance of an insurer with jump-diffusion risk process is studied. With the assumptions that the risk process is a compound Possion process perturbed by a standard Brownian motion and the reinsurance premium is calculated according to the variance principle, the implicit expression of the priority and corresponding value function when the utility function is exponential are obtained. At last, the value function is argued, the properties of the priority about parameters are discussed and numerical results of the priority for various claim-size distributions are shown.展开更多
A framework for the optimal sparse-control of the probability density function of a jump-diffusion process is presented. This framework is based on the partial integro-differential Fokker-Planck (FP) equation that gov...A framework for the optimal sparse-control of the probability density function of a jump-diffusion process is presented. This framework is based on the partial integro-differential Fokker-Planck (FP) equation that governs the time evolution of the probability density function of this process. In the stochastic process and, correspondingly, in the FP model the control function enters as a time-dependent coefficient. The objectives of the control are to minimize a discrete-in-time, resp. continuous-in-time, tracking functionals and its L2- and L1-costs, where the latter is considered to promote control sparsity. An efficient proximal scheme for solving these optimal control problems is considered. Results of numerical experiments are presented to validate the theoretical results and the computational effectiveness of the proposed control framework.展开更多
The classical Poisson risk model in ruin theory assumed that the interarrival times between two successive claims are mutually independent, and the claim sizes and claim intervals are also mutually independent. In thi...The classical Poisson risk model in ruin theory assumed that the interarrival times between two successive claims are mutually independent, and the claim sizes and claim intervals are also mutually independent. In this paper, we modify the classical Poisson risk model to describe the surplus process of an insurance portfolio. We consider a jump-diffusion risk process compounded by a geometric Brownian motion, and assume that the claim sizes and claim intervals are dependent. Using the properties of conditional expectation, we establish integro-differential equations for the Gerber-Shiu function and the ultimate ruin probability.展开更多
In this paper, we consider a hyper-exponential jump-diffusion model with a constant dividend barrier. Explicit solutions for the Laplace transform of the ruin time, and the Gerber- Shiu function are obtained via marti...In this paper, we consider a hyper-exponential jump-diffusion model with a constant dividend barrier. Explicit solutions for the Laplace transform of the ruin time, and the Gerber- Shiu function are obtained via martingale stopping.展开更多
In this paper, under the assumption that the exchange rate follows the extended Vasicek model, the pricing of the reset option in FBM model is investigated. Some interesting themes such as closed-form formulas for the...In this paper, under the assumption that the exchange rate follows the extended Vasicek model, the pricing of the reset option in FBM model is investigated. Some interesting themes such as closed-form formulas for the reset option with a single reset date and the phenomena of delta of the reset jumps existing in the reset option during the reset date are discussed. The closed-form formulae of pricing for two kinds of power options are derived in the end.展开更多
In this paper, it is assumed that an insurer with a jump-diffusion risk process would invest its surplus in a bond market, and the interest structure of the bond market is assumed to follow the Vasicek interest model....In this paper, it is assumed that an insurer with a jump-diffusion risk process would invest its surplus in a bond market, and the interest structure of the bond market is assumed to follow the Vasicek interest model. This paper focuses on the studying of the ruin problems in the above compounded process. In this compounded risk model, ruin may be caused by a claim or oscillation. We decompose the ruin probability for the compounded risk process into two probabilities: the probability that ruin caused by a claim and the probability that ruin caused by oscillation. Integro-differential equations for these ruin probabilities are derived. When the claim sizes are exponentially distributed, the above-mentioned integro-differential equations can be reduced into a three-order partial differential equation.展开更多
We use an actuarial approach to estimate the valuation of the reload option for a non-tradable risk asset under the jump-diffusion processes and Hull-White interest rate. We verify the validity of the actuarial approa...We use an actuarial approach to estimate the valuation of the reload option for a non-tradable risk asset under the jump-diffusion processes and Hull-White interest rate. We verify the validity of the actuarial approach to the European vanilla option for non-tradable assets. The formulas of the actuarial approach to the reload option are derived from the fair premium principle and the obtained results are arbitrage. Numerical experiments are conducted to analyze the effects of different parameters on the results of valuation as well as their differences from those obtained by the no-arbitrage approach. Finally, we give the valuations of the reload options under different parameters.展开更多
This research explores upside and downside jumps in the dynamic processes of three rates:domestic interest rates,foreign interest rates,and exchange rates.To fill the gap between the asymmetric jump in the currency ma...This research explores upside and downside jumps in the dynamic processes of three rates:domestic interest rates,foreign interest rates,and exchange rates.To fill the gap between the asymmetric jump in the currency market and the current models,a correlated asymmetric jump model is proposed to capture the co-movement of the correlated jump risks for the three rates and identify the correlated jump risk premia.The likelihood ratio test results show that the new model performs best in 1-,3-,6-,and 12-month maturities.The in-and out-of-sample test results indicate that the new model can capture more risk factors with relatively small pricing errors.Finally,the risk factors captured by the new model can explain the exchange rate fluctuations for various economic events.展开更多
This paper studies the first passage time problem for a reflected two-sided jump-diffusion risk model with the jumps having a hyper-Erlang distribution.The authors give the explicit closed-form expression for the join...This paper studies the first passage time problem for a reflected two-sided jump-diffusion risk model with the jumps having a hyper-Erlang distribution.The authors give the explicit closed-form expression for the joint Laplace transform of the first passage time and the overshoot for the reflected process.Finally,the formula is applied to the ruin problem under the barrier dividend strategy and the pricing of the Russian option.展开更多
In this paper, we consider the finite time ruin probability for the jump-diffusion Poisson process. Under the assurnptions that the claimsizes are subexponentially distributed and that the interest force is constant, ...In this paper, we consider the finite time ruin probability for the jump-diffusion Poisson process. Under the assurnptions that the claimsizes are subexponentially distributed and that the interest force is constant, we obtain an asymptotic formula for the finite-time ruin probability. The results we obtain extends the corresponding results of Kliippelberg and Stadtmüller and Tang.展开更多
This article studies the optimal proportional reinsurance and investment problem under a constant elasticity of variance (CEV) model. Assume that the insurer's surplus process follows a jump-diffusion process, the ...This article studies the optimal proportional reinsurance and investment problem under a constant elasticity of variance (CEV) model. Assume that the insurer's surplus process follows a jump-diffusion process, the insurer can purchase proportional reinsurance from the reinsurer via the variance principle and invest in a risk-free asset and a risky asset whose price is modeled by a CEV model. The diffusion term can explain the uncertainty associated with the surplus of the insurer or the additional small claims. The objective of the insurer is to maximize the expected exponential utility of terminal wealth. This optimization problem is studied in two cases depending on the diffusion term's explanation. In all cases, by using techniques of stochastic control theory, closed-form expressions for the value functions and optimal strategies are obtained.展开更多
This paper investigates a dynamic asset allocation problem for loss-averse investors in a jumpdiffusion model where there are a riskless asset and N risky assets. Specifically, the prices of risky assets are governed ...This paper investigates a dynamic asset allocation problem for loss-averse investors in a jumpdiffusion model where there are a riskless asset and N risky assets. Specifically, the prices of risky assets are governed by jump-diffusion processes driven by an m-dimensional Brownian motion and a(N- m)-dimensional Poisson process. After converting the dynamic optimal portfolio problem to a static optimization problem in the terminal wealth, the optimal terminal wealth is first solved. Then the optimal wealth process and investment strategy are derived by using the martingale representation approach. The closed-form solutions for them are finally given in a special example.展开更多
The primary goal of this paper is to price European options in the Merton's frame- work with underlying assets following jump-diffusion using fuzzy set theory. Owing to the vague fluctuation of the real financial mar...The primary goal of this paper is to price European options in the Merton's frame- work with underlying assets following jump-diffusion using fuzzy set theory. Owing to the vague fluctuation of the real financial market, the average jump rate and jump sizes cannot be recorded or collected accurately. So the main idea of this paper is to model the rate as a triangular fuzzy number and jump sizes as fuzzy random variables and use the property of fuzzy set to deduce two different jump-diffusion models underlying principle of rational expectations equilibrium price. Unlike many conventional models, the European option price will now turn into a fuzzy number. One of the major advantages of this model is that it allows investors to choose a reasonable European option price under an acceptable belief degree. The empirical results will serve as useful feedback information for improvements on the proposed model.展开更多
In this paper, the binomial tree method is introduced to price the European option under a class of jump-diffusion model. The purpose of the addressed problem is to find the parameters of the binomial tree and design ...In this paper, the binomial tree method is introduced to price the European option under a class of jump-diffusion model. The purpose of the addressed problem is to find the parameters of the binomial tree and design the pricing formula for European option. Compared with the continuous situation, the proposed value equation of option under the new binomial tree model converges to Merton’s accurate analytical solution, and the established binomial tree method can be proved to work better than the traditional binomial tree. Finally, a numerical example is presented to illustrate the effectiveness of the proposed pricing methods.展开更多
This paper studies incomplete stock market that includes discontinuous price processes. The discontinuity is modeled by very general point processes admitting only stochastic intensities. Prices are driven by jump-dif...This paper studies incomplete stock market that includes discontinuous price processes. The discontinuity is modeled by very general point processes admitting only stochastic intensities. Prices are driven by jump-diffusion uncertainty and have random but predictable jumps. The space of risk-neutral measures that are associated with the market is identified and related to fictitious completions. The construction of replicating portfolios is discussed, and convex duality methods are used to prove existence of optimal consumption and investment policies for a problem of utility maximization.展开更多
This paper considers a nonparametric diffusion process whose drift and diffusion coefficients are nonparametric functions of the state variable.A two-step approach to estimate the drift function of a jump-diffusion mo...This paper considers a nonparametric diffusion process whose drift and diffusion coefficients are nonparametric functions of the state variable.A two-step approach to estimate the drift function of a jump-diffusion model in noisy settings is proposed.The proposed estimator is shown to be consistent and asymptotically normal in the presence of finite activity jumps.Simulated experiments and a real data application are undertaken to assess the finite sample performance of the newly proposed method.展开更多
In this paper, we consider an insurer who wants to maximize its expected utility of terminal wealth by selecting optimal investment and risk control strategies. The insurer’s risk process is modeled by a jump-diffusi...In this paper, we consider an insurer who wants to maximize its expected utility of terminal wealth by selecting optimal investment and risk control strategies. The insurer’s risk process is modeled by a jump-diffusion process and is negatively correlated with the returns of securities and derivatives in the financial market. In the financial model, a part of insurers’ wealth is invested into the financial market. Using a martingale approach, we obtain an explicit solution of optimal strategy for the insurer under logarithmic utility function.展开更多
In this paper we consider the stochastic systems with jumps (random impulses) generated by Erlang flow of events that lead to discontinuities in paths. These systems may be used in various applications such as a contr...In this paper we consider the stochastic systems with jumps (random impulses) generated by Erlang flow of events that lead to discontinuities in paths. These systems may be used in various applications such as a control of complex technical systems, financial mathematics, mathematical biology and medicine. We propose to use a spectral method formalism to the probabilistic analysis problem for the stochastic systems with jumps. This method allows to get a solution of the analysis problem in an explicit form.展开更多
The problem of general exchange option pricing on jump-diffusion model is presented, we use the methods of the change of numeraire and martingale measure, and get the analytic solution of above option.
基金Supported by the National Natural Sci-ence Foundations of China (10271062 and 10471119)the Natural Science Foundation of Shandong Province(Y2004A06, Y2008A12, and ZR2009AL015)+1 种基金the Science Foundations of Shandong Provincial Education Department (J07yh05)the Science Foundations of Qufu Normal University (XJ0713, Bsqd200517)
文摘In this article, the joint distributions of several actuarial diagnostics which are important to insurers' running for the jump-diffusion risk process are examined. They include the ruin time, the time of the surplus process leaving zero ultimately (simply, the ultimately leaving-time), the surplus immediately prior to ruin, the supreme profits before ruin, the supreme profits and deficit until it leaves zero ultimately and so on. The explicit expressions for their distributions are obtained mainly by the various properties of Levy process, such as the homogeneous strong Markov property and the spatial homogeneity property etc, moveover, the many properties for Brownian motion.
基金Supported by the Humanity and Social Science Foundation of Ministry of Education of China(10YJC790296)Supported by the National Natural Science Foundation of China(71073020)
文摘In this paper, the optimal XL-reinsurance of an insurer with jump-diffusion risk process is studied. With the assumptions that the risk process is a compound Possion process perturbed by a standard Brownian motion and the reinsurance premium is calculated according to the variance principle, the implicit expression of the priority and corresponding value function when the utility function is exponential are obtained. At last, the value function is argued, the properties of the priority about parameters are discussed and numerical results of the priority for various claim-size distributions are shown.
文摘A framework for the optimal sparse-control of the probability density function of a jump-diffusion process is presented. This framework is based on the partial integro-differential Fokker-Planck (FP) equation that governs the time evolution of the probability density function of this process. In the stochastic process and, correspondingly, in the FP model the control function enters as a time-dependent coefficient. The objectives of the control are to minimize a discrete-in-time, resp. continuous-in-time, tracking functionals and its L2- and L1-costs, where the latter is considered to promote control sparsity. An efficient proximal scheme for solving these optimal control problems is considered. Results of numerical experiments are presented to validate the theoretical results and the computational effectiveness of the proposed control framework.
文摘The classical Poisson risk model in ruin theory assumed that the interarrival times between two successive claims are mutually independent, and the claim sizes and claim intervals are also mutually independent. In this paper, we modify the classical Poisson risk model to describe the surplus process of an insurance portfolio. We consider a jump-diffusion risk process compounded by a geometric Brownian motion, and assume that the claim sizes and claim intervals are dependent. Using the properties of conditional expectation, we establish integro-differential equations for the Gerber-Shiu function and the ultimate ruin probability.
基金Supported by the Natural Science Foundation of Jiangsu Province(BK20130260)the National Natural Science Foundation of China(11301369)the Postdoctoral Science Foundation of China(2013M540371)
文摘In this paper, we consider a hyper-exponential jump-diffusion model with a constant dividend barrier. Explicit solutions for the Laplace transform of the ruin time, and the Gerber- Shiu function are obtained via martingale stopping.
文摘In this paper, under the assumption that the exchange rate follows the extended Vasicek model, the pricing of the reset option in FBM model is investigated. Some interesting themes such as closed-form formulas for the reset option with a single reset date and the phenomena of delta of the reset jumps existing in the reset option during the reset date are discussed. The closed-form formulae of pricing for two kinds of power options are derived in the end.
基金The NNSF(10671072,10726075)of Chinathe Doctoral Program Foundation(20060269016)of the Ministry of Education of Chinathe National Basic Research Program(973 Program,2007CB814904)of China.
文摘In this paper, it is assumed that an insurer with a jump-diffusion risk process would invest its surplus in a bond market, and the interest structure of the bond market is assumed to follow the Vasicek interest model. This paper focuses on the studying of the ruin problems in the above compounded process. In this compounded risk model, ruin may be caused by a claim or oscillation. We decompose the ruin probability for the compounded risk process into two probabilities: the probability that ruin caused by a claim and the probability that ruin caused by oscillation. Integro-differential equations for these ruin probabilities are derived. When the claim sizes are exponentially distributed, the above-mentioned integro-differential equations can be reduced into a three-order partial differential equation.
基金Supported by the National Natural Science Foundation of China(No.11571365,11171349)
文摘We use an actuarial approach to estimate the valuation of the reload option for a non-tradable risk asset under the jump-diffusion processes and Hull-White interest rate. We verify the validity of the actuarial approach to the European vanilla option for non-tradable assets. The formulas of the actuarial approach to the reload option are derived from the fair premium principle and the obtained results are arbitrage. Numerical experiments are conducted to analyze the effects of different parameters on the results of valuation as well as their differences from those obtained by the no-arbitrage approach. Finally, we give the valuations of the reload options under different parameters.
文摘This research explores upside and downside jumps in the dynamic processes of three rates:domestic interest rates,foreign interest rates,and exchange rates.To fill the gap between the asymmetric jump in the currency market and the current models,a correlated asymmetric jump model is proposed to capture the co-movement of the correlated jump risks for the three rates and identify the correlated jump risk premia.The likelihood ratio test results show that the new model performs best in 1-,3-,6-,and 12-month maturities.The in-and out-of-sample test results indicate that the new model can capture more risk factors with relatively small pricing errors.Finally,the risk factors captured by the new model can explain the exchange rate fluctuations for various economic events.
基金supported by the Natural Science Foundation of China under Grant Nos.11301369,11401419the Natural Science Foundation of Jiangsu Province under Grant Nos.BK20130260,BK20140279
文摘This paper studies the first passage time problem for a reflected two-sided jump-diffusion risk model with the jumps having a hyper-Erlang distribution.The authors give the explicit closed-form expression for the joint Laplace transform of the first passage time and the overshoot for the reflected process.Finally,the formula is applied to the ruin problem under the barrier dividend strategy and the pricing of the Russian option.
基金Supported by the National Natural Science Foundation of China(No.70471071)Philosophy and Social Science Foundation of the Education Anthority of Jiangsu Province(No.04SJB630005)
文摘In this paper, we consider the finite time ruin probability for the jump-diffusion Poisson process. Under the assurnptions that the claimsizes are subexponentially distributed and that the interest force is constant, we obtain an asymptotic formula for the finite-time ruin probability. The results we obtain extends the corresponding results of Kliippelberg and Stadtmüller and Tang.
文摘This article studies the optimal proportional reinsurance and investment problem under a constant elasticity of variance (CEV) model. Assume that the insurer's surplus process follows a jump-diffusion process, the insurer can purchase proportional reinsurance from the reinsurer via the variance principle and invest in a risk-free asset and a risky asset whose price is modeled by a CEV model. The diffusion term can explain the uncertainty associated with the surplus of the insurer or the additional small claims. The objective of the insurer is to maximize the expected exponential utility of terminal wealth. This optimization problem is studied in two cases depending on the diffusion term's explanation. In all cases, by using techniques of stochastic control theory, closed-form expressions for the value functions and optimal strategies are obtained.
基金Supported by the National Natural Science Foundation of China(No.61304065,11471304,11401556)the Natural Science Foundation of the Jiangsu Higher Education Institutions of China(No.12KJB110011)
文摘This paper investigates a dynamic asset allocation problem for loss-averse investors in a jumpdiffusion model where there are a riskless asset and N risky assets. Specifically, the prices of risky assets are governed by jump-diffusion processes driven by an m-dimensional Brownian motion and a(N- m)-dimensional Poisson process. After converting the dynamic optimal portfolio problem to a static optimization problem in the terminal wealth, the optimal terminal wealth is first solved. Then the optimal wealth process and investment strategy are derived by using the martingale representation approach. The closed-form solutions for them are finally given in a special example.
基金Supported by the Key Grant Project of Chinese Ministry of Education(309018)National Natural Science Foundation of China(70973104 and 11171304)the Zhejiang Natural Science Foundation of China(Y6110023)
文摘The primary goal of this paper is to price European options in the Merton's frame- work with underlying assets following jump-diffusion using fuzzy set theory. Owing to the vague fluctuation of the real financial market, the average jump rate and jump sizes cannot be recorded or collected accurately. So the main idea of this paper is to model the rate as a triangular fuzzy number and jump sizes as fuzzy random variables and use the property of fuzzy set to deduce two different jump-diffusion models underlying principle of rational expectations equilibrium price. Unlike many conventional models, the European option price will now turn into a fuzzy number. One of the major advantages of this model is that it allows investors to choose a reasonable European option price under an acceptable belief degree. The empirical results will serve as useful feedback information for improvements on the proposed model.
文摘In this paper, the binomial tree method is introduced to price the European option under a class of jump-diffusion model. The purpose of the addressed problem is to find the parameters of the binomial tree and design the pricing formula for European option. Compared with the continuous situation, the proposed value equation of option under the new binomial tree model converges to Merton’s accurate analytical solution, and the established binomial tree method can be proved to work better than the traditional binomial tree. Finally, a numerical example is presented to illustrate the effectiveness of the proposed pricing methods.
基金This research is partially supported by NSF under DMI-9908294 and DMI-0196084.
文摘This paper studies incomplete stock market that includes discontinuous price processes. The discontinuity is modeled by very general point processes admitting only stochastic intensities. Prices are driven by jump-diffusion uncertainty and have random but predictable jumps. The space of risk-neutral measures that are associated with the market is identified and related to fictitious completions. The construction of replicating portfolios is discussed, and convex duality methods are used to prove existence of optimal consumption and investment policies for a problem of utility maximization.
基金the National Natural Science Foundation of China under Grant No.11961038Cultivating Project of National Natural Science Foundation(QianKeHe talent-development platform[2017]No.5723,QianKeHe talent-development platform[2017]No.5723-02)+7 种基金supported by the National Natural Science Foundation of China under Grant Nos.12071220,11701286supported by the National Natural Science Foundation of China under Grant Nos.11831008,11971235Young Talents Project of Science and Technology Research Program of Education Department in Guizhou Province(Qianjiao KYword[2018]364)Science and Technology Foundation of Guizhou Province(QianKeHejichu[2019]1286)Social Science Foundation of Jiangsu Province under Grant No.20EYC008the National Statistical Research Project of China under Grant No.2020LZ35the National Statistical Research Project of China under Grant No.2020LZ19Open Project of Jiangsu Key Laboratory of Financial Engineering under Grant No.NSK2021-12。
文摘This paper considers a nonparametric diffusion process whose drift and diffusion coefficients are nonparametric functions of the state variable.A two-step approach to estimate the drift function of a jump-diffusion model in noisy settings is proposed.The proposed estimator is shown to be consistent and asymptotically normal in the presence of finite activity jumps.Simulated experiments and a real data application are undertaken to assess the finite sample performance of the newly proposed method.
文摘In this paper, we consider an insurer who wants to maximize its expected utility of terminal wealth by selecting optimal investment and risk control strategies. The insurer’s risk process is modeled by a jump-diffusion process and is negatively correlated with the returns of securities and derivatives in the financial market. In the financial model, a part of insurers’ wealth is invested into the financial market. Using a martingale approach, we obtain an explicit solution of optimal strategy for the insurer under logarithmic utility function.
文摘In this paper we consider the stochastic systems with jumps (random impulses) generated by Erlang flow of events that lead to discontinuities in paths. These systems may be used in various applications such as a control of complex technical systems, financial mathematics, mathematical biology and medicine. We propose to use a spectral method formalism to the probabilistic analysis problem for the stochastic systems with jumps. This method allows to get a solution of the analysis problem in an explicit form.
文摘The problem of general exchange option pricing on jump-diffusion model is presented, we use the methods of the change of numeraire and martingale measure, and get the analytic solution of above option.