Continuous-time mean–variance portfolio selection with liability and regime switching

A continuous-time mean–variance model for individual investors with stochastic liability in a Markovian regime switching financial market, is investigated as a generalization of the model of Zhou and Yin [Zhou, X.Y., Yin, G., 2003. Markowitz’s mean–variance portfolio selection with regime switching: A continuous-time model, SIAM J. Control Optim. 42 (4), 1466–1482]. We assume that the risky stock’s price is governed by a Markovian regime-switching geometric Brownian motion, and the liability follows a Markovian regime-switching Brownian motion with drift, respectively. The evolution of appreciation rates, volatility rates and the interest rates are modulated by the Markov chain, and the Markov switching diffusion is assumed to be independent of the underlying Brownian motion. The correlation between the risky asset and the liability is considered. The objective is to minimize the risk (measured by variance) of the terminal wealth subject to a given expected terminal wealth level. Using the Lagrange multiplier technique and the linear-quadratic control technique, we get the expressions of the optimal portfolio and the mean–variance efficient frontier in closed forms. Further, the results of our special case without liability is consistent with those results of Zhou and Yin [Zhou, X.Y., Yin, G., 2003. Markowitz’s mean–variance portfolio selection with regime switching: A continuous-time model, SIAM J. Control Optim. 42 (4), 1466–1482].     

Markowitz’s mean-variance asset-liability management with regime switching: A continuous-time model

This paper considers an asset-liability management (ALM) problem under a continuous-time Markov regime-switching model. By adopting the techniques of [Zhou, X.Y., Yin, G., 2003. Markowitz’s mean-variance portfolio selection with regime switching: A continuous-time model. SIAM J. Control Optim. 42, 1466–1482], we investigate the feasibility, obtain the optimal strategy, delineate the efficient frontier, and establish the associated mutual fund theorem.     

A uniform asymptotic estimate for discounted aggregate claims with subexponential tails

In this paper we study the tail probability of discounted aggregate claims in a continuous-time renewal model. For the case that the common claim-size distribution is subexponential, we obtain an asymptotic formula, which holds uniformly for all time horizons within a finite interval. Then, with some additional mild assumptions on the distributions of the claim sizes and inter-arrival times, we further prove that this formula holds uniformly for all time horizons. In this way, we significantly extend a recent result of Tang [Tang, Q., 2007. Heavy tails of discounted aggregate claims in the continuous-time renewal model. J. Appl. Probab. 44 (2), 285–294].     

Convergence analysis of waveform relaxation methods for neutral differential-functional systems

In this paper, the problems of convergence and superlinear convergence of continuous-time waveform relaxation method applied to Volterra type systems of neutral functional-differential equations are discussed. Under a Lipschitz condition with time- and delay-dependent right-hand side imposed on the so-called splitting function, more suitable conditions about convergence and superlinear convergence of continuous-time WR method are obtained. We also investigate the initial interval acceleration strategy for the practical implementation of the continuous-time waveform relaxation method, i.e., discrete-time waveform relaxation method. It is shown by numerical results that this strategy is efficacious and has the essential acceleration effect for the whole computation process.     

Power utility maximization in exponential Lévy models: convergence of discrete-time to continuous-time maximizers

We consider power utility maximization of terminal wealth in a 1-dimensional continuous-time exponential Lévy model with finite time horizon. We discretize the model by restricting portfolio adjustments to an equidistant discrete time grid. Under minimal assumptions we prove convergence of the optimal discrete-time strategies to the continuous-time counterpart. In addition, we provide and compare qualitative properties of the discrete-time and continuous-time optimizers.     

Indirect identification of continuous-time delay systems from step responses

In this paper, an indirect identification scheme is proposed for identifying the parameters of the continuous-time first-order plus time delay (FOPTD) model and the second-order plus time delay (SOPTD) model from step responses. Unlike the existing direct identification scheme, which identifies the parameters of the continuous-time FOPTD and SOPTD models directly from the continuous-time step response data, the proposed indirect scheme is to pre-identify discrete-time FOPTD and SOPTD models from the discretized continuous-time step response input–output data, then convert the obtained discrete-time models to the desirable continuous-time models. The proposed method is then extended to identify the afore-mentioned models from the step responses of the systems contaminated with input noise and constant output disturbance. The proposed simple alternative method exhibits good estimation performances in both the time domain and the frequency domain. Illustrative examples are presented to demonstrate the effectiveness of the proposed scheme.     

连续时间复合二项模型多维精算量的联合分布

连续时间复合二项模型是由文献首先提出的.作为离散时间复合二项模型的连续化版本,连续时间复合二项模型的极限形式即为经典风险模型.为了得到该模型多维精算量的联合分布,该文引入了一列上穿零点,推导出该列上穿零点所构成的缺陷(defective)更新序列的更新质量函数.利用此更新质量函数及余额过程的强马氏性可以得到破产概率和包含破产时间,破产前余额,破产严重程度,破产前最大盈余,破产到恢复的最大赤字,整个过程的最大赤字等多维精算量的联合分布.由此联合分布得到其1-骨架链—离散时间复合二项模型的对应的联合分布,最后给出在1-骨架链中索赔额服从指数分布时这一特殊情况下相应多维精算量的联合分布的明确表达式.     

Fitting continuous-time and discrete-time models using discrete-time data and their applications

An exponential function scheme, which is an extension of the time-domain prony method, and a mixed-matching method are developed for fitting the coefficients of both continuous-time and discrete-time transfer functions, using the discrete-time data of either continuous-time or discrete-time systems. When the discrete-time data are obtained from a continuous-time (discrete-time) system and the discrete-time (continuous-time) models are desirable, the proposed method can be applied to perform the model conversions. If the discrete-time data are obtained from a high-degree system, the proposed method can be applied to determine the reduced-degree models.     

Comments on ‘Fitting continuous-time and discrete-time models using discrete-time data and their application’

This short paper extends the results of the paper on ‘Fitting continuous-time and discrete-time models using discrete-time data and their application’ (Zhu J., Shieh, L. S. and Yates, R. E.: Appl. Math. Modelling 1985, 9, 93) to systems with multiple poles.     

Model conversions of uncertain linear system using the improved block-pulse function approach

This paper presents an improved block-pulse function approach to convert a continuous-time (respectively, discrete-time) structured uncertain linear system into an equivalent discrete-time (respectively, continuous-time) structured uncertain linear model. The concept of the principle of equivalent areas is utilized for the uncertain model conversions. This allows the use of well-established theorems and algorithms in the discrete-time (respectively, continuous-time) domain to indirectly solve the continuous-time (respectively, discrete-time) domain problems. A numerical example is given to demonstrate the effectiveness of the proposed method.     

EXISTENCE OF POSITIVE PERIODIC SOLUTION OF A DISCRETE TIME MUTUALISM SYSTEM WITH DELAYS

Discrete-time analogue of mutualism model is introduced. The discrete-time analogues is considered to be numerical discretization of the continuous-time models and we study their dynamical characteristics. It is shown that the discrete-time analogues preserve the periodicity of the continuous-time models.     

Digital modeling and hybrid control of sampled-data uncertain system with input time delay using the law of mean

This paper utilizes an interval Pade approximate method together with interval arithmetic operation to convert a continuous-time uncertain system with input time-delay to an equivalent discrete-time interval model and transforms the robust control law of a continuous-time uncertain system with input time delay into an equivalent one of a sampled-data uncertain system with input time delay. The developed discrete-time interval model tightly encloses the exact discrete-time uncertain system with input time delay. Based on the law of mean and inclusion theory, a perturbed digital control law of input time-delay sampled-data uncertain system is newly presented, so that the states of the digitally controlled sample-data uncertain system closely match those of the originally well-designed continuous-time uncertain system.     

Digital Stabilizer Design for a Switched Linear Control Delay System

The problem of designing a digital controller stabilizing a continuous-time switched linear control delay system is studied. The approach to stabilization successively includes the construction of a continuous-time–discrete-time closed-loop system with a digital controller, the transition to its discrete-time model, and the construction of a discrete-time controller by simultaneous stabilization methods.     

Continuous-time limit of repeated interactions for a system in a confining potential

We study the continuous-time limit of a class of Markov chains coming from the evolution of classical open systems undergoing repeated interactions. This repeated interaction model has been initially developed for dissipative quantum systems in Attal and Pautrat (2006) and was recently set up for the first time in Deschamps (2012) for classical dynamics. It was particularly shown in the latter that this scheme furnishes a new kind of Markovian evolutions based on Hamilton’s equations of motion. The system is also proved to evolve in the continuous-time limit with a stochastic differential equation. We here extend the convergence of the evolution of the system to more general dynamics, that is, to more general Hamiltonians and probability measures in the definition of the model. We also present a natural way to directly renormalize the initial Hamiltonian in order to obtain the relevant process in a study of the continuous-time limit. Then, even if Hamilton’s equations have no explicit solution in general, we obtain some bounds on the dynamics allowing us to prove the convergence in law of the Markov chain on the system to the solution of a stochastic differential equation, via the infinitesimal generators.     

Stochastic modelling of combat with reinforcement

The homogeneous combat when reinforcements are made at prespecified force levels has been modelled as a continuous-time discrete-state space Markov process. The effect of reinforcement on various combat characteristics, viz., the mean combat duration, the winning probabilities, the expected survivors, etc., is studied using the uniformization method. Two new concepts, i.e., Reinforcement Effectiveness Index and Reinforcement Parity Curve, have also been defined and are illustrated numerically.     

Spectral gap for open Jackson networks

We generalize the decomposition method of the finite Markov chains for Poincaré inequality in Jerrum et al. (Ann. Appl. Probab., 14, 1741-1765 (2004)) to the reversible continuous-time Markov chains. And inductively, we give the lower bound of spectral gap for the ergodic open Jackson network by the decomposition method and the symmetrization procedure. The upper bound of the spectral gap is also presented.     

Discrete-continuous model conversion

This paper presents methods for model conversions of continuous-time state-space equations and discrete-time state-space equations. An improved geometric-series method is presented for converting continuous-time models to equivalent discrete-time models. Also, a direct truncation method, a matrix continued fraction method and a geometric-series method are presented for converting discrete models to equivalent continuous models. As a result, many well-developed theorems and methods in either continuous or discrete domains can be effectively applied to a suitable model in either domain.     

Pricing catastrophe options in discrete operational time

We employ a doubly-binomial process as in Gerber [Gerber, H.U., 1988. Mathematical fun with the compound binomial process. ASTIN Bull. 18, 161-168] to discretize and generalize the continuous “randomized operational time” model of Chang et al. ([Chang, C.W., Chang, J.S.K., Yu, M.T., 1996. Pricing catastrophe insurance futures call spreads: A randomized operational time approach. J. Risk Insurance 63, 599-616] and CCY hereafter) from a complete-market continuous-time setting to an incomplete-market discrete-time setting, so as to price a richer set of catastrophe (CAT) options. For futures options, we derive the equivalent martingale probability measures by benchmarking to the shadow price of a bond to span arrival uncertainty, and the underlying futures price to span price uncertainty. With a time change from calendar time to the operational transaction-time dimension, we derive CCY as a limiting case under risk-neutrality when both calendar-time and transaction-time intervals shrink to zero. For a cash option with non-traded underlying loss index, we benchmark to the market reinsurance premiums to span claim uncertainty, and with a time change to claim time, we derive the cash option price as a binomial sum of claim-time binomial Asian option prices under the martingale measures.     

On pinning control of some typical discrete-time dynamical networks

Recently, the pinning control of complex dynamical networks to their homogeneous states has been studied by many researchers, most of the dynamical networks are continuous-time ones, i.e., their dynamical behavior can be described by ODEs. An interesting result is that, for a continuous-time network, its desired (homogeneous) state can be achieved by pinning some nodes with relatively large degrees (also called the specifically pinning scheme [Wang XF, Chen GR. Pinning control of scale-free dynamical networks. Physica A 2002;310:521–31]). Is this specifically pinning scheme also effective for the discrete-time dynamical networks? In this paper, we demonstrate that the pinning control for a discrete-time dynamical network is difficult, and sometimes it is impossible to achieve the desired state just by controlling the nodes with larger degrees. In order to control the discrete-time dynamical networks successfully, we may need to control all the nodes. Finally, we also consider how to extend the interval for the feedback gain d for successful control.