基于二次效用函数的投资组合问题研究

设无风险利率、股票收益率和波动率都是一致有界随机过程,在股票价格服从跳跃一扩散过程时,同时考虑具有随机资金流的介入,研究了二次效用的动态投资组合选择优化问题,通过随机线性二次控制和倒向随机微分方程得到了最优投资组合策略的解析表达式.     

均值-方差准则下具有负债的随机微分博弈

研究了均值-方差准则下,具有负债的随机微分博弈.研究目标是:在终值财富的均值等于k的限制下,在市场出现最坏的情况下找到最优的投资策略使终值财富的方差最小.即:基于均值-方差随机微分博弈的投资组合选择问题.使用线性-二次控制的理论解决了该问题,获得了最优的投资策略、最优市场策略和有效边界的显示解.并通过对所得结果进行进一步分析,在经济上给出了进一步的解释.通过本文的研究,可以指导金融公司在面临负债和金融市场情况恶劣时,选择恰当的投资策略使自身获得一定的财富而面临的风险最小.     

考虑组合风险的指数化投资与随机线性二次最优控制

指数化投资使投资者享有市场平均收益水平,具有投资风险分散化、投资组合透明化、投资成本低廉等优势,日益受到投资者的亲睐。由于通常指数化投资者不愿意承担较大风险,本文考虑极小化跟踪误差与投资组合的风险之和(其中风险用风险资产的累积方差来衡量)。本文证明了无论是连续时间或离散时间、有限时区或无限时区的情形,在一定的条件下,最优控制都唯一存在,即利用随机线性二次最优控制进行指数化投资,最优投资策略都唯一存在。     

具有随机资金流的一类效用优化问题研究

现实的金融市场上,当有重大信息出现时,会对股价产生冲击,使得股价产生跳跃,同时投资过程会有随机资金流的介入,考虑股价出现跳跃与随机资金流介入的投资组合优化问题,通过构造倒向-前向随机微分方程并结合随机最优控制理论研究了一般效用函数下的投资组合选择问题,获得最优投资组合策略,然后针对二次效用函数,给出显式表示的最优投资组合策略.     

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.     

二次损失下随机回归系数和参数的线性Minimax估计

对带有随机效应的一般线性模型,本文提出了随机回归系数和参数线性组合的Minimax估计问题. 在二次损失下,研究了线性估计的极小极大性.关于适当的假设,得到了可估函数的唯一线性Mjnimax 估计.     

二次损失下随机回归系数和参数的线性Minimax估计

对带有随机效应的一般线性模型,本文提出了随机回归系数和参数线性组合的Minimax估计问题.在二次损失下,研究了线性估计的极小极大性.关于适当的假设,得到了可估函数的唯一线性Minimax估计.     

随机利率环境下基于二次效用函数的动态投资组合(英文)

研究随机利率环境下基于效用最大化的动态投资组合,并假设利率是服从Ho-Lee利率模型和Vasicek利率模型的随机过程.应用动态规划原理得到值函数满足的HJB方程,并应用Legendre变换得到其对偶方程.最后,应用变量替换对二次效用函数下的最优投资策略进行研究,得到了最优投资策略的显示解.     

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.     

条件平均场随机微分方程的最优控制问题

作者研究了一个条件平均场随机微分方程的最优控制问题.这种方程和某些部分信息下的随机最优控制问题有关,并且可以看做是平均场随机微分方程的推广.作者以庞特里雅金最大值原理的形式给出最优控制满足的必要和充分条件.此外,文中给出一个线性二次最优控制问题来说明理论结果的应用.     

A mixed integer programming model for multistage mean–variance post-tax optimization

In this paper, we introduce a mixed integer stochastic programming approach to mean–variance post-tax portfolio management. This approach takes into account of risk in a multistage setting and allows general withdrawals from original capital. The uncertainty on asset returns is specified as a scenario tree. The risk across scenarios is addressed using the probabilistic approach of classical stochastic programming. The tax rules are used with stochastic linear and mixed integer quadratic programming models to compute an overall tax and return-risk efficient multistage portfolio. The incorporation of the risk term in the model provides robustness and leads to diversification over wrappers and assets within each wrapper. General withdrawals and risk aversion have an impact on the distribution of assets among wrappers. Computational results are presented using a study with different scenario trees in order to show the performance of these models.     

Continuity of parametric mixed-integer quadratic programs and its application to stability analysis of two-stage quadratic stochastic programs with mixed-integer recourse

For mixed-integer quadratic program where all coefficients in the objective function and the right-hand sides of constraints vary simultaneously, we show locally Lipschitz continuity of its optimal value function, and derive the corresponding global estimation; furthermore, we also obtain quantitative estimation about the change of its optimal solutions. Applying these results to two-stage quadratic stochastic program with mixed-integer recourse, we establish quantitative stability of the optimal value function and the optimal solution set with respect to the Fortet-Mourier probability metric, when the underlying probability distribution is perturbed. The obtained results generalize available results on continuity properties of mixed-integer quadratic programs and extend current results on quantitative stability of two-stage quadratic stochastic programs with mixed-integer recourse.     

Doubly stochastic quadratic operators and Birkhoff’s problem

In the present we introduce a concept of doubly stochastic quadratic operator. We study necessary and sufficient conditions for doubly stochasticity of operator. Moreover, we study analogue of Birkhoff’s theorem for the class of doubly stochastic operators.     

Stochastic moment problem and hedging of generalized Black-Scholes options

In mathematical finance one is interested in the quadratic error which occurs while replacing a continuously adjusted portfolio by a discretely adjusted one. We first study higher order approximations of stochastic integrals. Then we apply the results to quantify quadratic error which occurs in estimating the discretely adjusted hedging risk in pricing European options in a generalized Black-Scholes market.     

Almost-periodic solutions for Riccati equations of stochastic control

We consider an average quadratic cost criteria for affine stochastic differential equations with almost-periodic coefficients. Under stabilizability and detectability conditions we show that the Riccati equation associated with the quadratic control problem has a unique almost-periodic solution. In the periodic case the corresponding result is proved in [4].     

Backward Stochastic Riccati Equations and Infinite Horizon L-Q Optimal Control with Infinite Dimensional State Space and Random Coefficients

We study the Riccati equation arising in a class of quadratic optimal control problems with infinite dimensional stochastic differential state equation and infinite horizon cost functional. We allow the coefficients, both in the state equation and in the cost, to be random. In such a context backward stochastic Riccati equations are backward stochastic differential equations in the whole positive real axis that involve quadratic non-linearities and take values in a non-Hilbertian space. We prove existence of a minimal non-negative solution and, under additional assumptions, its uniqueness. We show that such a solution allows to perform the synthesis of the optimal control and investigate its attractivity properties. Finally the case where the coefficients are stationary is addressed and an example concerning a controlled wave equation in random media is proposed.     

A Stochastic Linear Quadratic Optimal Control Problem with Generalized Expectation

Abstract

In this article, we initiate a study on optimal control problem for linear stochastic differential equations with quadratic cost functionals under generalized expectation via backward stochastic differential equations.