Functional ANOVA,ultramodularity and monotonicity: Applications in multiattribute utility theory

Utility function properties as monotonicity and concavity play a fundamental role in reflecting a decision-maker’s preference structure. These properties are usually characterized via partial derivatives. However, elicitation methods do not necessarily lead to twice-differentiable utility functions. Furthermore, while in a single-attribute context concavity fully reflects risk aversion, in multiattribute problems such correspondence is not one-to-one. We show that Tsetlin and Winkler’s multivariate risk attitudes imply ultramodularity of the utility function. We demonstrate that geometric properties of a multivariate utility function can be successfully studied by utilizing an integral function expansion (functional ANOVA). The necessary and sufficient conditions under which monotonicity and/or ultramodularity of single-attribute functions imply the monotonicity and/or ultramodularity of the corresponding multiattribute function under additive, preferential and mutual utility independence are then established without reliance on the utility function differentiability. We also investigate the relationship between the presence of interactions among the attributes of a multiattribute utility function and the decision-maker’s multivariate risk attitudes.     

企业及市场经营效用函数形式研究

鉴于经营效用是企业制订战略决策和发展策略的重要影响因素,研究其基本效用类型,提出对应的函数形式,并在此基础上提出市场总体的综合经营效用测度方法。企业经营效用由自身经营效用和同业比较效用两部分线性合成,效用变量分别为企业单位盈利指标加权合成值及单位盈利水平与行业平均值的差,效用函数通过原点且单调,因此需将效用理论中对应于各种效用类型的对数函数、指数函数等进行坐标变换、旋转或对称。保守型和冒险型效用在定义域内分别为凹函数和凸函数,共组合为9种效用类型含81种基本函数形式,并给出各效用类型含义、经营特征和定价倾向。以市场份额作为各企业经营效用权重,构建幂平均效用合成模型作为市场总体综合经营效用测度。     

基于2可加模糊测度的多线性效用函数建模和求解

构建合适的多属性效用函数是多属性效用分析的关键。针对不同偏好假设,文献从可加独立、效用独立、效用依赖等分别进行了多属性效用函数构建的研究。然而,由于求解的复杂性,多属性效用理论的应用绝大部分限于可加效用函数和多乘效用函数。提出一种基于2可加模糊测度的多线性效用函数建模和求解方法。首先,证明多线性效用函数和基于模糊测度的多线性模型之间的等价性,提出利用基于模糊测度的多线性模型对多线性效用函数进行表示。其次,针对多线性模型的特点和模糊测度识别的复杂性,利用Banzhaf交互指数和2可加模糊测度对多线性模型进行表示,并利用最小方法差进行模糊测度和Banzhaf交互指数识别,进而实现多线性效用函数的求解。最后,将方法用于某可穿戴医疗设备基于顾客需求的多属性效用函数构建,确认了可行性。方法为多线性效用函数的求解提供了一种新思路。     

幂效用函数理论下的欧式期权定价

研究在不完全信息情况下,代理人根据自己的风险偏好建立效用函数理论,在幂效用函数下寻找资产的均衡价格和无风险折现因子,并在此基础上给出幂效用函数理论下的均衡的欧式期权定价公式.     

Pre-commitment vs. time-consistent strategies for the generalized multi-period portfolio optimization with stochastic cash flows

In this paper, we propose a multi-period portfolio optimization model with stochastic cash flows. Under the mean–variance preference, we derive the pre-commitment and time-consistent investment strategies by applying the embedding scheme and backward induction approach, respectively. We show that the time-consistent strategy is identical to the optimal open-loop strategy. Also, under the exponential utility preference, we develop the optimal strategy for multi-period investment, which is time-consistent. We show that the above two time-consistent strategies are equivalent in some cases. We compare the pre-commitment and time-consistent strategies under different situations with some numerical simulations. The results indicate that the time-consistent strategy is more stable and secure than pre-commitment strategy under the generalized mean–variance criterion.     

Risk-sensitive stopping problems for continuous-time Markov chains

In this paper we consider stopping problems for continuous-time Markov chains under a general risk-sensitive optimization criterion for problems with finite and infinite time horizon. More precisely our aim is to maximize the certainty equivalent of the stopping reward minus cost over the time horizon. We derive optimality equations for the value functions and prove the existence of optimal stopping times. The exponential utility is treated as a special case. In contrast to risk-neutral stopping problems it may be optimal to stop between jumps of the Markov chain. We briefly discuss the influence of the risk sensitivity on the optimal stopping time and consider a special house selling problem as an example.     

Some stability results of optimal investment in a simple Lévy market

We investigate some investment problems of maximizing the expected utility of the terminal wealth in a simple Lévy market, where the stock price is driven by a Brownian motion plus a Poisson process. The optimal investment portfolios are given explicitly under the hypotheses that the utility functions belong to the HARA, exponential and logarithmic classes. We show that the solutions for the HARA utility are stable in the sense of weak convergence when the parameters vary in a suitable way.     

Portfolio selection in stochastic markets with exponential utility functions

We consider the optimal portfolio selection problem in a multiple period setting where the investor maximizes the expected utility of the terminal wealth in a stochastic market. The utility function has an exponential structure and the market states change according to a Markov chain. The states of the market describe the prevailing economic, financial, social and other conditions that affect the deterministic and probabilistic parameters of the model. This includes the distributions of the random asset returns as well as the utility function. The problem is solved using the dynamic programming approach to obtain the optimal solution and an explicit characterization of the optimal policy. We also discuss the stochastic structure of the wealth process under the optimal policy and determine various quantities of interest including its Fourier transform. The exponential return-risk frontier of the terminal wealth is shown to have a linear form. Special cases of multivariate normal and exponential returns are disussed together with a numerical illustration.     

HARA frontiers of optimal portfolios in stochastic markets

In this paper, we consider the optimal portfolio selection problem in continuous-time settings where the investor maximizes the expected utility of the terminal wealth in a stochastic market. The utility function has the structure of the HARA family and the market states change according to a Markov process. The states of the market describe the prevailing economic, financial, social and other conditions that affect the deterministic and probabilistic parameters of the model. This includes the distributions of the random asset returns as well as the utility function. We analyzed Black–Scholes type continuous-time models where the market parameters are driven by Markov processes. The Markov process that affects the state of the market is independent of the underlying Brownian motion that drives the stock prices. The problem of maximizing the expected utility of the terminal wealth is investigated and solved by stochastic optimal control methods for exponential, logarithmic and power utility functions. We found explicit solutions for optimal policy and the associated value functions. We also constructed the optimal wealth process explicitly and discussed some of its properties. In particular, it is shown that the optimal policy provides linear frontiers.     

基于指数效用无差异价值过程的不完备市场下的可违约期权的定价模型

本文研究不完备市场情况下的可违约期权的动态指数效用无差异定价。不同于大多数的可违约期权定价文献,本文没有假定鞅的不变性,即通常的H 假设,而是通过信息流的扩张和测度的变换,将信用风险敏感的资产转换为一个G 局部鞅,其后引入一个具体的倒向随机微分方程（BSDE）,并证明该方程解的存在性与唯一性;然后利用无差异价值过程C_t（B,α）在最小熵鞅测度下对一般的投资策略为上鞅,而在最优投资策略下为鞅的事实,证明无差异价值过程C_t（B,α）就是BSDE 的解,从而给出可违约期权的定价。     

Optimal dynamic excess-of-loss reinsurance and multidimensional portfolio selection

In this paper, the surplus process of the insurance company is described by a Brownian motion with drift. In addition, the insurer is allowed to invest in a risk-free asset and n risky assets and purchase excess-of-loss reinsurance. Under short-selling prohibition, we consider two optimization problems: the problem of maximizing the expected exponential utility of terminal wealth and the problem of minimizing the probability of ruin. We first show that the excess-of-loss reinsurance strategy is always better than the proportional reinsurance under two objective functions. Then, by solving the corresponding Hamilton-Jacobi-Bellman equations, the closed-form solutions of their optimal value functions and the corresponding optimal strategies are obtained. In particular, when there is no risky-free interest rate, the results indicate that the optimal strategies, under maximizing the expected exponential utility and minimizing the probability of ruin, are equivalent for some special parameter. This validates Ferguson’s longstanding conjecture about the relation between the two problems.     

基于加权效用和VaR-PI约束下DC型养老金计划的最优资产配置

本文从养老金计划参与人和基金经理的双重视角出发,以最大化双方加权的期望效用为目标,研究了在最低保障和VaR约束下,DC养老金计划的最优资产配置问题。假设养老金计划参与人和基金经理均是损失厌恶的,分别用两个S型的效用函数来刻画双方的损失厌恶行为。VaR约束和加权的效用函数使得本文所研究的优化问题成为一个复杂的非凹效用最大化问题。利用拉格朗日对偶理论和凹化方法求得了最优财富和最优投资组合的封闭解。数值结论表明当更为看重养老金计划参与人的利益时,基金经理会采取更为激进的投资策略,VaR约束可以改进对DC养老金计划的风险管理。     

基于再保险和投资的随机微分博弈

本文研究了具有再保险和投资的随机微分博弈.应用线性-二次控制的理论,在指数效用和幂效用下,求得了最优再保险策略、最优投资策略、最优市场策略和值函数的显示解,推广了文[8]的结果.通过本文的研究,当市场出现最坏的情况时,可以指导保险公司选择恰当的再保险和投资策略使自身所获得的财富最大化.     

Portfolio selection in stochastic markets with HARA utility functions

In this paper, we consider the optimal portfolio selection problem where the investor maximizes the expected utility of the terminal wealth. The utility function belongs to the HARA family which includes exponential, logarithmic, and power utility functions. The main feature of the model is that returns of the risky assets and the utility function all depend on an external process that represents the stochastic market. The states of the market describe the prevailing economic, financial, social, political and other conditions that affect the deterministic and probabilistic parameters of the model. We suppose that the random changes in the market states are depicted by a Markov chain. Dynamic programming is used to obtain an explicit characterization of the optimal policy. In particular, it is shown that optimal portfolios satisfy the separation property and the composition of the risky portfolio does not depend on the wealth of the investor. We also provide an explicit construction of the optimal wealth process and use it to determine various quantities of interest. The return-risk frontiers of the terminal wealth are shown to have linear forms. Special cases are discussed together with numerical illustrations.     

保险公司和再保险公司之间的停止损失再保险策略选择博弈

本文在扩散逼近风险模型下考虑保险公司和再保险公司之间的停止损失再保险策略选择博弈问题.假设保险公司和再保险公司都以期望终端盈余效用增加作为购买停止损失再保险和接受承保的条件.在保险公司和再保险公司都具有指数效用函数条件下,运用动态规划原理,通过求解其对应的Hamilton-Jacobi-Bellman方程,得到了三种博弈情形下保险公司和再保险公司之间的停止损失再保险策略和值函数的显示解,以及再保险合约能够成交时再保费满足的条件.结果显示,在适当的条件下,保险公司和再保险公司之间的停止再保险合约是可以成交的.最后,通过灵敏性分析给出了最优停止损失再保险策略和再保费,以及效用损益与模型主要参数之间的关系,并给出相应的经济分析.     

Optimal proportional reinsurance and investment with multiple risky assets and no-shorting constraint

In this paper, the basic claim process is assumed to follow a Brownian motion with drift. In addition, the insurer is allowed to invest in a risk-free asset and n risky assets and to purchase proportional reinsurance. Under the constraint of no-shorting, we consider two optimization problems: the problem of maximizing the expected exponential utility of terminal wealth and the problem of minimizing the probability of ruin. By solving the corresponding Hamilton–Jacobi–Bellman equations, explicit expressions for their optimal value functions and the corresponding optimal strategies are obtained. In particular, when there is no risk-free interest rate, the results indicate that the optimal strategies, under maximizing the expected exponential utility and minimizing the probability of ruin, are equivalent for some special parameter. This validates Ferguson’s longstanding conjecture about the relation between the two problems.     

Would a risk-averse newsvendor order less at a higher selling price?

We model a risk-averse newsvendor’s decision-making behavior with some commonly used classes of utility functions within the expected utility theory (EUT) framework. Under fairly general conditions of EUT, we show that a risk-averse newsvendor will order less than an arbitrarily small quantity as selling price gets larger if price is higher than a threshold value, i.e., the optimal order quantity decreases as the selling price increases.     

Decomposition of multiattribute expected-utility functions

This paper integrates and extends the theory of the decomposition of multiattribute expected-utility functions based on utility independence. In a preliminary section, the standard decision model of expected utility is briefly discussed, including the fact that the decision maker's preference forlotteries with two outcomes determines the utility function uniquely. The decomposition possibilities of a utility function are captured by the concept ofautonomous sets of attributes, an affine separability of some kind known as generalized utility independence.Overlapping autonomous sets lead to biaffine-associative, i.e.multiplicative oradditive decompositions. The multiplicative representation shows that autonomy has strongerclosure properties than utility independence, for instance with respect to set-theoretic difference. Autonomy is also a concept with a wider scope since it applies to the decomposition of Boolean functions, games and a number of other topics in combinatorial optimization. This relationship to the well-known theory ofsubstitution decomposition in discrete mathematics also reveals a kind of discrete core behind the decomposition of utility functions. The entirety of autonomous sets can be represented by a compact data structure, the so-calledcomposition tree, which frequently corresponds to a natural hierarchy of attributes. Multiplicative/additive ormulti-affine functions correspond to the hierarchy steps. The known representation of multi-affine functions is shown to be given by aMoebius inversion formula. The entire approach has the advantage that it allows the application of more sophisticated representation methods on a detailed level, whereas it employs onlyfinite set theory andarithmetic on the main levels in the hierarchy.     

基于新型负指数期望效用的投资组合选择模型

结合现有文献对最优投资决策问题的讨论, 提出了一类满足单调性和凹性的新型负指数效用函数, 并给出数学和经济学上的合理解释. 通过多种类型的加权函数以及对尾部的恰当描述, 损失分布的厚尾现象得到更加有效地控制. 利用$-统计量估计新型期望效用, 并说明其合理性. 进一步地, 构建了兼顾多种市场摩擦因素的实际投资组合选择模型. 选用中国和美国股票市场的数据进行实证研究. 结果表明了新期望效用的优越性和鲁棒性.     

变额年金的最优控制

在幂效用函数和指数效用函数的条件下,讨论保险人在年金积累期和年金给付期的投资策略,建立保险人变额年金投资的最优控制模型,得出变额年金的最优控制策略.