时间不一致随机线性-二次控制：平衡点的刻画与唯一性
Time-Inconsistent Stochastic Linear--Quadratic Control: Characterization and Uniqueness of Equilibrium
摘要：
在本文中，我们继续我们的研究一般的时间不一致的随机线性-二次（LQ）控制问题最初在[6]中制定。我们通过一个前向-倒向随机微分方程流导出了平衡控制的一个充分必要条件。当状态为一维且问题中的系数均为确定性时，我们证明了在\cite{HJZ}中构造的显式平衡控制确实是唯一的。我们的证明是基于导出的平衡等价条件以及随机版本的Lebesgue微分定理。最后，我们证明了均衡策略是唯一的均值-方差投资组合选择模型在一个完整的金融市场中的无风险利率是一个确定的函数的时间，但所有其他的市场参数可能是随机过程。
英文摘要：
In this paper, we continue our study on a general time-inconsistent stochastic linear--quadratic (LQ) control problem originally formulated in [6]. We derive a necessary and sufficient condition for equilibrium controls via a flow of forward--backward stochastic differential equations. When the state is one dimensional and the coefficients in the problem are all deterministic, we prove that the explicit equilibrium control constructed in \cite{HJZ} is indeed unique. Our proof is based on the derived equivalent condition for equilibria as well as a stochastic version of the Lebesgue differentiation theorem. Finally, we show that the equilibrium strategy is unique for a mean--variance portfolio selection model in a complete financial market where the risk-free rate is a deterministic function of time but all the other market parameters are possibly stochastic processes.
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