含不动产项目的保险公司再保险-投资策略

站在保险公司管理者的角度, 考虑存在不动产项目投资机会时保险公司的再保险--投资策略问题. 假定保险公司可以投资于不动产项目、风险证券和无风险证券, 并通过比例再保险控制风险, 目标是最小化保险公司破产概率并求得相应最佳策略, 包括: 不动产项目投资时机、 再保险比例以及投资于风险证券的金额. 运用混合随机控制-最优停时方法, 得到最优值函数及最佳策略的显式解. 结果表明, 当且仅当其盈余资金多于某一水平(称为投资阈值)时保险公司投资于不动产项目. 进一步的数值算例分析表明: (a)~不动产项目投资的阈值主要受项目收益率影响而与投资金额无明显关系, 收益率越高则投资阈值越低; (b)~市场环境较好(牛市)时项目的投资阈值降低; 反之, 当市场环境较差(熊市)时投资阈值提高.     

含有随机波动的非线性的最优投资和消费模型

在连续时间模型假设下,研究风险资产价格服从一个带有随机波动的几何布朗运动的最优消费和投资问题．首先建立了最优消费和投资同题随机最优控制数学模型;然后运用随机最优控制理论,得到了最优投资和消费随机最优控制问题的值函数所满足的线性抛物线偏微分方程和非线性抛物线偏微分方程．     

The Ruin Probability in the Presence of Extended Regular Variation and Optimal Investment

Considering the classical model with risky investment, we are interested in the ruin probability that is minimized by a suitably chosen investment strategy for a capital market index. For claim sizes with common distribution of extended regular variation, starting from an integro-differential equation for the maximal survival probability, we find that the corresponding ruin probability as a function of the initial surplus is also extended regular variation.     

最小化破产概率的保险人鲁棒投资再保险策略研究

在模型不确定条件下,研究以破产概率最小化为目标的模糊厌恶型保险公司的最优投资再保险问题. 假设保险公司可投资于一种风险资产,也可购买比例再保险. 分别考虑风险资产的价格过程服从随机波动率模型和非随机波动率模型的两种情况,根据动态规划原理建立相应的HJB方程,得到保险公司的最优鲁棒投资再保险策略和价值函数的解析解. 最后,通过数值模拟分析了各模型参数对最优策略和价值函数的影响.     

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.     

Optimal control of a big financial company with debt liability under bankrupt probability constraints

This paper considers an optimal control of a big financial company with debt liability under bankrupt probability constraints. The company, which faces constant liability payments and has choices to choose various production/business policies from an available set of control policies with different expected profits and risks, controls the business policy and dividend payout process to maximize the expected present value of the dividends until the time of bankruptcy. However, if the dividend payout barrier is too low to be acceptable, it may result in the company’s bankruptcy soon. In order to protect the shareholders’ profits, the managements of the company impose a reasonable and normal constraint on their dividend strategy, that is, the bankrupt probability associated with the optimal dividend payout barrier should be smaller than a given risk level within a fixed time horizon. This paper aims at working out the optimal control policy as well as optimal return function for the company under bankrupt probability constraint by stochastic analysis, partial differential equation and variational inequality approach. Moreover, we establish a riskbased capital standard to ensure the capital requirement can cover the total given risk by numerical analysis, and give reasonable economic interpretation for the results.     

不确定情况下两部门经济模型最优投资率的确定

本依据索罗的经济增长模型，推导出两部门经济模型的最优投资率，并在此基础上讨论了生产函数为柯布—道格拉斯函数时，最优投资率满足的条件。     

Optimal investment with transaction costs based on exponential utility function: a parabolic double obstacle problem

This paper concerns optimal investment problem with proportional transaction costs and finite time horizon based on exponential utility function. Using a partial differential equation approach, we reveal that the problem is equivalent to a parabolic double obstacle problem involving two free boundaries that correspond to the optimal buying and selling policies. Numerical examples are obtained by the binomial method.     

Minimizing the probability of lifetime ruin under stochastic volatility

We assume that an individual invests in a financial market with one riskless and one risky asset, with the latter’s price following a diffusion with stochastic volatility. Given the rate of consumption, we find the optimal investment strategy for the individual who wishes to minimize the probability of going bankrupt. To solve this minimization problem, we use techniques from stochastic optimal control.     

一类最优投资随机控制模型的马氏链算法

现代金融经济中的很多问题可以构建成随机控制模型,而随机控制的求解却存在一定的困难.马氏链算法应该是一种有效的求解随机控制问题的数值方法.本文以Claus Munk的工作为基础,针对一类最优投资模型,具体确定了马氏链的转移矩阵并证明其满足算法收敛条件,并用MATLAB语言编成一个程序实现.     

Optimal investment policy: An application of Stokes' theorem

An application of the Stokes' theorem is illustrated by solving the two-state problem, with inequality constraints, of Dobell and Ho concerning the optimal investment of resources. Whenever applicable, the Stokes' theorem approach seems to be elegant and parsimonious.     

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.     

Optimal consumption and investment with insurer default risk

We solve the optimal consumption and investment problem in an incomplete market, where borrowing constraints and insurer default risk are considered jointly. We derive in closed-form the optimal consumption and investment strategies. We find two main results by quantitative analysis. As insurer default risk increases, the proportion of wealth invested in stocks could increase when wealth is small, and decrease when wealth is large. As risk aversion increases, the voluntary annuity demand could increase when insurer default risk is low, and decrease when this risk is high.     

Optimal threshold probability and expectation in semi-Markov decision processes

We consider undiscounted semi-Markov decision process with a target set and our main concern is a problem minimizing threshold probability. We formulate the problem as an infinite horizon case with a recurrent class. We show that an optimal value function is a unique solution to an optimality equation and there exists a stationary optimal policy. Also several value iteration methods and a policy improvement method are given in our model. Furthermore, we investigate a relationship between threshold probabilities and expectations for total rewards.     

Optimal investment strategy to minimize the ruin probability of an insurance company under borrowing constraints

We consider that the surplus of an insurance company follows a Cramér-Lundberg process. The management has the possibility of investing part of the surplus in a risky asset. We consider that the risky asset is a stock whose price process is a geometric Brownian motion. Our aim is to find a dynamic choice of the investment policy which minimizes the ruin probability of the company. We impose that the ratio between the amount invested in the risky asset and the surplus should be smaller than a given positive bound a. For instance the case a=1 means that the management cannot borrow money to buy stocks.[Hipp, C., Plum, M., 2000. Optimal investment for insurers. Insurance: Mathematics and Economics 27, 215-228] and [Schmidli, H., 2002. On minimizing the ruin probability by investment and reinsurance. Ann. Appl. Probab. 12, 890-907] solved this problem without borrowing constraints. They found that the ratio between the amount invested in the risky asset and the surplus goes to infinity as the surplus approaches zero, so the optimal strategies of the constrained and unconstrained problems never coincide.We characterize the optimal value function as the classical solution of the associated Hamilton-Jacobi-Bellman equation. This equation is a second-order non-linear integro-differential equation. We obtain numerical solutions for some claim-size distributions and compare our results with those of the unconstrained case.     

具有随机工资的养老金最优投资问题

在三种目标函数下, 研究了具有随机工资的养老金最优投资问题. 第一种是均值-方差准则, 第二种基于效用的随机微分博弈, 第三种基于均值-方差准则的随机微分博弈. 随机微分博弈问题中博弈的双方为养老金计划投资者和金融市场, 金融市场是博弈的虚拟手. 应用线性二次控制理论求得了三种目标函数下的最优策略和值函数的显式解.     

道德风险影响下的保险最优投资研究

在考虑道德风险的情况下,以均值方差准则为目标研究保险人最优投资问题.假设保险盈余过程服从C-L模型,金融市场上存在一种无风险资产和一种风险资产可供投资,其中风险资产的价格过程服从几何布朗运动.在纯道德风险保险契约设计中,借鉴相关研究对努力水平和效用化努力成本的假设,量化道德风险对盈余过程的影响.在均值方差目标下,建立保险人最优投资问题的广义Hamilton-Jacobi-Bellman(HJB)方程,给出保险人时间一致的均衡投资策略和价值函数.结果显示累计索赔比例参数越大,公司对最优努力水平越敏感,采取措施降低道德风险有利于公司收益提升;努力成本参数越大,公司会降低努力水平减少支出,避免损失.     

Optimal proportional reinsurance and investment with minimum probability of ruin

The paper concerns a problem of optimal reinsurance and investment in order to minimizing the probability of ruin. In the whole paper, the cedent’s surplus is allowed to invest in a risk-free asset and a risky asset and the company’s risk is reduced through proportional reinsurance, while in addition the claim process is assumed to follow a Brownian motion with drift. By solving the corresponding Hamilton-Jacobi-Bellman equations, the optimal reinsurance-investment strategy is derived. The presented results generalize those by Taksar [1].     

Optimal Market Making in the Foreign Exchange Market

Abstract

This paper is concerned with optimal market making in the foreign exchange market. The market maker's holdings in the different currencies are modelled as stochastic processes that are influenced by both the stochastic exchange rates and the stochastic customer buy and sell orders. The market maker can control their own bid and ask price quotes and, additionally, can buy and sell at other market participants' quotes. The resulting stochastic control problem consists of a controlled diffusion problem for the optimal quotes and a singular control problem for optimal trades at other market participants' quotes. A Markov chain approximation is used to derive optimal strategies.