跳扩散系统的零和线性二次随机微分对策问题

发布时间：2018-03-06 22:12

本文选题：线性二次　切入点：微分对策　出处：《山东大学》2017年硕士论文　论文类型：学位论文

【摘要】：随机线性二次问题是一类经典的最优控制问题,一些非线性控制问题可以用线性二次问题做逼近。讨论加上跳扩散的系统更加符合现实应用的发展需要,特别是在日益繁荣的金融市场中,一些均值方差投资组合问题和风险控制问题可以转化为随机线性二次控制问题;真实市场中存在竞争对手,竞争对手共同参与市场并影响市场,这启发我们将部分此类问题转化为零和微分对策问题,而考虑跳扩散的系统则更加符合金融市场随机性的规律。本文研究了一类带泊松跳的零和线性二次随机微分对策问题,且其扩散项系数不为零。基于对相关黎卡堤方程的研究,给出了一类闭环形式的最优反馈控制策略对,在一定程度上拓展了带跳线性二次问题的结果。我们运用Hamadene线性变换将对策问题与另一个控制问题联系起来,讨论了对策问题的哈密顿系统及黎卡堤方程并给出了解的存在唯一性证明。在跳扩散系统的零和线性二次随机微分对策问题中,控制变量由两部分组成,可以看成是两个玩家同时参与系统且共同影响状态变量;状态方程是线性的正向随机微分方程,并且它是由一个d-维标准布朗运动和一个泊松随机鞅测度所驱动的;目标函数是关于状态变量和控制变量的二次形式,玩家一希望目标函数一达到最大(或最小),玩家二则希望目标函数二达到最小(或最大)。玩家一和玩家二相互制约、相互影响。由于对策问题的特殊性,我们需要考虑容许控制、容许策略、容许反馈控制以及容许反馈策略定义的合理性。此外,我们分别定义了玩家一、玩家二以及对策问题的值,最终能够在这种动态的博弈中取得一个均衡点,即最优反馈控制策略对。
[Abstract]:Stochastic linear quadratic problem is a classical optimal control problem, and some nonlinear control problems can be approximated by linear quadratic problem. Especially in the increasingly prosperous financial markets, some mean-variance portfolio problems and risk control problems can be transformed into stochastic linear quadratic control problems. Competitors participate in the market and influence the market, which inspires us to transform some of these problems into zero-sum differential game problems. In this paper, a class of zero sum linear quadratic stochastic differential games with Poisson jump is studied, and the diffusion coefficient is not zero. In this paper, a class of closed-loop optimal feedback control strategy pairs is given, which extends the results of quadratic problem with jumper to a certain extent. We use Hamadene linear transformation to associate the game problem with another control problem. In this paper, the Hamiltonian system and Riccati equation of the game problem are discussed and the existence and uniqueness of the solution are proved. In the zero sum linear quadratic stochastic differential game problem of the jump diffusion system, the control variable is composed of two parts. The equation of state is a linear forward stochastic differential equation driven by a d- dimensional standard Brownian motion and a Poisson random martingale measure. The objective function is the quadratic form of the state variable and the control variable. Player 1 wants the objective function to reach the maximum (or minimum), while player 2 wants the objective function 2 to reach the minimum (or maximum). Player 1 and player 2 restrict each other. Because of the particularity of the game problem, we need to consider the rationality of admissible control, admissible strategy, admissible feedback control and allowable feedback strategy definition. Player two and the value of game problem can finally obtain an equilibrium point in this dynamic game, that is, the optimal feedback control strategy pair.
【学位授予单位】：山东大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O232;F224

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