Optimal reinsurance and investment strategies for insurer under interest rate and inflation risks

In this paper, we investigate an optimal reinsurance and investment problem for an insurer whose surplus process is approximated by a drifted Brownian motion. Proportional reinsurance is to hedge the risk of insurance. Interest rate risk and inflation risk are considered. We suppose that the instantaneous nominal interest rate follows an Ornstein–Uhlenbeck process, and the inflation index is given by a generalized Fisher equation. To make the market complete, zero-coupon bonds and Treasury Inflation Protected Securities (TIPS) are included in the market. The financial market consists of cash, zero-coupon bond, TIPS and stock. We employ the stochastic dynamic programming to derive the closed-forms of the optimal reinsurance and investment strategies as well as the optimal utility function under the constant relative risk aversion (CRRA) utility maximization. Sensitivity analysis is given to show the economic behavior of the optimal strategies and optimal utility.     

Optimal Consumption in a Growth Model with the CES Production Function

Abstract

This article studies the optimization problem of maximizing the expected discounted present value of lifetime utility of consumption in the framework of one-sector neoclassical growth model with the Constant Elasticity of Substitution (CES) production function. We establish the existence of a classical solution of the Hamilton–Jacobi–Bellman (HJB) equation associated with this problem by the technique of viscosity solutions under the strict concavity of the utility function, and hence derive an optimal consumption from the optimality conditions in the HJB equation.     

Optimal investment for the defined-contribution pension with stochastic salary under a CEV model

In this paper, we study the optimal investment strategy of defined-contribution pension with the stochastic salary. The investor is allowed to invest in a risk-free asset and a risky asset whose price process follows a constant elasticity of variance model. The stochastic salary follows a stochastic differential equation, whose instantaneous volatility changes with the risky asset price all the time. The HJB equation associated with the optimal investment problem is established, and the explicit solution of the corresponding optimization problem for the CARA utility function is obtained by applying power transform and variable change technique. Finally, we present a numerical analysis.     

通胀环境下考虑随机微分效用的最优消费和投资问题研究

本文研究了投资者在通胀环境下基于随机微分效用的最优消费和投资问题．首先对投资机会集进行描述．并用随机微分效用函数刻画了投资者的偏好．其次利用动态规划原理,考虑带通胀的最优消费和投资问题,并建立相应的HJB方程．接下来,根据假设的效用函数,推导出最优消费和投资策略,并分析参数对投资策略的影响．     

Portfolio selection problem with multiple risky assets under the constant elasticity of variance model

This paper focuses on the constant elasticity of variance (CEV) model for studying the utility maximization portfolio selection problem with multiple risky assets and a risk-free asset. The Hamilton-Jacobi-Bellman (HJB) equation associated with the portfolio optimization problem is established. By applying a power transform and a variable change technique, we derive the explicit solution for the constant absolute risk aversion (CARA) utility function when the elasticity coefficient is −1 or 0. In order to obtain a general optimal strategy for all values of the elasticity coefficient, we propose a model with two risky assets and one risk-free asset and solve it under a given assumption. Furthermore, we analyze the properties of the optimal strategies and discuss the effects of market parameters on the optimal strategies. Finally, a numerical simulation is presented to illustrate the similarities and differences between the results of the two models proposed in this paper.     

Portfolio Optimization with Stochastic Volatilities and Constraints: An Application in High Dimension

In this paper we are interested in an investment problem with stochastic volatilities and portfolio constraints on amounts. We model the risky assets by jump diffusion processes and we consider an exponential utility function. The objective is to maximize the expected utility from the investor terminal wealth. The value function is known to be a viscosity solution of an integro-differential Hamilton-Jacobi-Bellman (HJB in short) equation which could not be solved when the risky assets number exceeds three. Thanks to an exponential transformation, we reduce the nonlinearity of the HJB equation to a semilinear equation. We prove the existence of a smooth solution to the latter equation and we state a verification theorem which relates this solution to the value function. We present an example that shows the importance of this reduction for numerical study of the optimal portfolio. We then compute the optimal strategy of investment by solving the associated optimization problem.     

A control approach to robust utility maximization with logarithmic utility and time-consistent penalties

We propose a stochastic control approach to the dynamic maximization of robust utility functionals that are defined in terms of logarithmic utility and a dynamically consistent convex risk measure. The underlying market is modeled by a diffusion process whose coefficients are driven by an external stochastic factor process. In particular, the market model is incomplete. Our main results give conditions on the minimal penalty function of the robust utility functional under which the value function of our problem can be identified with the unique classical solution of a quasilinear PDE within a class of functions satisfying certain growth conditions. The fact that we obtain classical solutions rather than viscosity solutions facilitates the use of numerical algorithms, whose applicability is demonstrated in examples.     

具有生产性与福利性公共支出的随机内生增长模型

文章建立一个随机内生增长模型来阐明主要政策参数对经济增长与社会福利的影响.若对生产函数、效用函数、偏好及随机干扰作一些特殊的假设,我们证明了主要政策参数的均衡值能被模型参数唯一决定.进一步我们还得到了期望增长率与储蓄的清晰解.文章的最后,我们证明了政府支出直接影响个体决策者的决策:即提高经济增长率将减少福利;反之,增加福利将减少增长率.     

Optimal debt ratio and dividend payment strategies with reinsurance

This paper derives the optimal debt ratio and dividend payment strategies for an insurance company. Taking into account the impact of reinsurance policies and claims from the credit derivatives, the surplus process is stochastic that is jointly determined by the reinsurance strategies, debt levels, and unanticipated shocks. The objective is to maximize the total expected discounted utility of dividend payment until financial ruin. Using dynamic programming principle, the value function is the solution of a second-order nonlinear Hamilton–Jacobi–Bellman equation. The subsolution–supersolution method is used to verify the existence of classical solutions of the Hamilton–Jacobi–Bellman equation. The explicit solution of the value function is derived and the corresponding optimal debt ratio and dividend payment strategies are obtained in some special cases. An example is provided to illustrate the methodologies and some interesting economic insights.     

Portfolio optimization in a Lévy market with intertemporal substitution and transaction costs

We investigate an infinite horizon investment-consumption model in which a single agent consumes and distributes her wealth between a risk-free asset (bank account) and several risky assets (stocks) whose prices are governed by Lévy (jump-diffusion) processes. We suppose that transactions between the assets incur a transaction cost proportional to the size of the transaction. The problem is to maximize the total utility of consumption under Hindy-Huang-Kreps intertemporal preferences. This portfolio optimisation problem is formulated as a singular stochastic control problem and is solved using dynamic programming and the theory of viscosity solutions. The associated dynamic programming equation is a second order degenerate elliptic integro-differential variational inequality subject to a state constraint boundary condition. The main result is a characterization of the value function as the unique constrained viscosity solution of the dynamic programming equation. Emphasis is put on providing a framework that allows for a general class of Lévy processes. Owing to the complexity of our investment-consumption model, it is not possible to derive closed form solutions for the value function. Hence, the optimal policies cannot be obtained in closed form from the first order conditions for the dynamic programming equation. Therefore, we have to resort to numerical methods for computing the value function as well as the associated optimal policies. In view of the viscosity solution theory, the analysis found in this paper will ensure the convergence of a large class of numerical methods for the investment-consumption model in question.     

政府投资治理污染的随机增长模型

本文设定了两种不同的带有污染的生产函数.在此两种生产函数情形下,利用随机最优化的方法分别分析了由政府投资治理污染的随机增长模型,得到了以下结论:在宏观均衡条件下,增大污染的外部性指标促进经济增长却降低福利;提高政府的环保投资增加福利,但对增长的影响却与污染的外部性指标和污染的负福利效用权数的大小有关.     

Robust optimal reinsurance and investment strategies for an AAI with multiple risks

This paper studies the robust optimal reinsurance and investment problem for an ambiguity averse insurer (abbr. AAI). The AAI sells insurance contracts and has access to proportional reinsurance business. The AAI can invest in a financial market consisting of four assets: one risk-free asset, one bond, one inflation protected bond and one stock, and has different levels of ambiguity aversions towards the risks. The goal of the AAI is to seek the robust optimal reinsurance and investment strategies under the worst case scenario. Here, the nominal interest rate is characterized by the Vasicek model; the inflation index is introduced according to the Fisher’s equation; and the stock price is driven by the Heston’s stochastic volatility model. The explicit forms of the robust optimal strategies and value function are derived by introducing an auxiliary robust optimal control problem and stochastic dynamic programming method. In the end of this paper, a detailed sensitivity analysis is presented to show the effects of market parameters on the robust optimal reinsurance policy, the robust optimal investment strategy and the utility loss when ignoring ambiguity.     

Optimal proportional reinsurance and investment for stochastic factor models

In this work we investigate the optimal proportional reinsurance-investment strategy of an insurance company which wishes to maximize the expected exponential utility of its terminal wealth in a finite time horizon. Our goal is to extend the classical Cramér–Lundberg model introducing a stochastic factor which affects the intensity of the claims arrival process, described by a Cox process, as well as the insurance and reinsurance premia. The financial market is supposed not influenced by the stochastic factor, hence it is independent on the insurance market. Using the classical stochastic control approach based on the Hamilton–Jacobi–Bellman equation we characterize the optimal strategy and provide a verification result for the value function via classical solutions to two backward partial differential equations. Existence and uniqueness of these solutions are discussed. Results under various premium calculation principles are illustrated and a new premium calculation rule is proposed in order to get more realistic strategies and to better fit our stochastic factor model. Finally, numerical simulations are performed to obtain sensitivity analyses.     

A unified approach to portfolio optimization with linear transaction costs

In this paper we study the continuous time optimal portfolio selection problem for an investor with a finite horizon who maximizes expected utility of terminal wealth and faces transaction costs in the capital market. It is well known that, depending on a particular structure of transaction costs, such a problem is formulated and solved within either stochastic singular control or stochastic impulse control framework. In this paper we propose a unified framework, which generalizes the contemporary approaches and is capable to deal with any problem where transaction costs are a linear/piecewise-linear function of the volume of trade. We also discuss some methods for solving numerically the problem within our unified framework.     

Exponential utility maximization for an insurer with time-inconsistent preferences

This paper studies the optimal consumption–investment–reinsurance problem for an insurer with a general discount function and exponential utility function in a non-Markovian model. The appreciation rate and volatility of the stock, the premium rate and volatility of the risk process of the insurer are assumed to be adapted stochastic processes, while the interest rate is assumed to be deterministic. The object is to maximize the utility of intertemporal consumption and terminal wealth. By the method of multi-person differential game, we show that the time-consistent equilibrium strategy and the corresponding equilibrium value function can be characterized by the unique solutions of a BSDE and an integral equation. Under appropriate conditions, we show that this integral equation admits a unique solution. Furthermore, we compare the time-consistent equilibrium strategies with the optimal strategy for exponential discount function, and with the strategies for naive insurers in two special cases.     

Dynamic programming in stochastic control of systems with delay

We consider optimal control problems for systems described by stochastic differential equations with delay (SDDE). We prove a version of Bellman's principle of optimality (the dynamic programming principle) for a general class of such problems. That the class in general means that both the dynamics and the cost depends on the past in a general way. As an application, we study systems where the value function depends on the past only through some weighted average. For such systems we obtain a Hamilton-Jacobi-Bellman partial differential equation that the value function must solve if it is smooth enough. The weak uniqueness of the SDDEs we consider is our main tool in proving the result. Notions of strong and weak uniqueness for SDDEs are introduced, and we prove that strong uniqueness implies weak uniqueness, just as for ordinary stochastic differential equations.     

Homogenization of a stochastically forced Hamilton-Jacobi equation

We study the homogenization of a Hamilton-Jacobi equation forced by rapidly oscillating noise that is colored in space and white in time. It is shown that the homogenized equation is deterministic, and, in general, the noise has an enhancement effect, for which we provide a quantitative estimate. As an application, we perform a noise sensitivity analysis for Hamilton-Jacobi equations forced by a noise term with small amplitude, and identify the scaling at which the macroscopic enhancement effect is felt. The results depend on new, probabilistic estimates for the large scale Hölder regularity of the solutions, which are of independent interest.     

具有共同冲击相依性的跳扩散金融市场中有着延迟和违约风险的鲁棒最优再保险和投资策略

在本文中,我们考虑跳扩散模型下具有延迟和违约风险的鲁棒最优再保险和投资问题,保险人可以投资无风险资产,可违约的债券和两个风险资产,其中两个风险资产遵循跳跃扩散模型且受到同种因素带来共同影响而相互关联.假设允许保险人购买比例再保险,特别地再保险保费利用均值方差保费原则来计算.在考虑与绩效相关的资本流入/流出下,保险公司的财富过程通过随机微分延迟方程建模.保险公司的目标是最大程度地发挥终端财富和平均绩效财富组合的预期指数效用,以分别研究违约前和违约后的情况.此外,推导了最优策略的闭式表达式和相应的价值函数.最后通过数值算例和敏感性分析,表明了各种参数对最优策略的影响.另外对于模糊厌恶投资者,忽视模型模糊性风险会带来显著的效用损失.     

Time-consistent reinsurance–investment strategy for a mean–variance insurer under stochastic interest rate model and inflation risk

In this paper, we consider the time-consistent reinsurance–investment strategy under the mean–variance criterion for an insurer whose surplus process is described by a Brownian motion with drift. The insurer can transfer part of the risk to a reinsurer via proportional reinsurance or acquire new business. Moreover, stochastic interest rate and inflation risks are taken into account. To reduce the two kinds of risks, not only a risk-free asset and a risky asset, but also a zero-coupon bond and Treasury Inflation Protected Securities (TIPS) are available to invest in for the insurer. Applying stochastic control theory, we provide and prove a verification theorem and establish the corresponding extended Hamilton–Jacobi–Bellman (HJB) equation. By solving the extended HJB equation, we derive the time-consistent reinsurance–investment strategy as well as the corresponding value function for the mean–variance problem, explicitly. Furthermore, we formulate a precommitment mean–variance problem and obtain the corresponding time-inconsistent strategy to compare with the time-consistent strategy. Finally, numerical simulations are presented to illustrate the effects of model parameters on the time-consistent strategy.     

Optimal investment and reinsurance for an insurer under Markov-modulated financial market

This study examines optimal investment and reinsurance policies for an insurer with the classical surplus process. It assumes that the financial market is driven by a drifted Brownian motion with coefficients modulated by an external Markov process specified by the solution to a stochastic differential equation. The goal of the insurer is to maximize the expected terminal utility. This paper derives the Hamilton–Jacobi–Bellman (HJB) equation associated with the control problem using a dynamic programming method. When the insurer admits an exponential utility function, we prove that there exists a unique and smooth solution to the HJB equation. We derive the explicit optimal investment policy by solving the HJB equation. We can also find that the optimal reinsurance policy optimizes a deterministic function. We also obtain the upper bound for ruin probability in finite time for the insurer when the insurer adopts optimal policies.