锥约束连续时间Markowitz问题
Cone-Constrained Continuous-Time Markowitz Problems
摘要：
马科维茨问题是在金融市场中寻找一个自我融资的交易策略，其最终财富具有最大均值和最小方差。我们在连续时间的一般半鞅模型和锥约束下研究这一点：交易策略必须在一个（可能是随机的和时间相关的）闭锥中取值。我们首先证明了凸约束解的存在性，通过证明约束终端增益空间，这是一个随机积分空间，在L^2中是封闭的。然后，我们使用随机控制方法来描述最优策略的局部结构，如下所示。将自然关联约束线性二次型最优控制问题的值过程分解为两个机会过程L^{ \pm }作为系数的和.鞅最优性原理转化为L^{ \pm }的半鞅特征的漂移条件，或者等价地转化为L^{ \pm }的倒向随机微分方程的耦合系统。我们展示了如何使用这两个优化和构建最优策略。我们的研究结果解释和概括了文献中的所有结果。此外，我们甚至在无约束的情况下得到了新的尖锐的结果。
英文摘要：
The Markowitz problem consists of finding in a financial market a self-financing trading strategy whose final wealth has maximal mean and minimal variance. We study this in continuous time in a general semimartingale model and under cone constraints: Trading strategies must take values in a (possibly random and time-dependent) closed cone. We first prove existence of a solution for convex constraints by showing that the space of constrained terminal gains, which is a space of stochastic integrals, is closed in L^2. Then we use stochastic control methods to describe the local structure of the optimal strategy, as follows. The value process of a naturally associated constrained linear-quadratic optimal control problem is decomposed into a sum with two opportunity processes L^{\pm} appearing as coefficients. The martingale optimality principle translates into a drift condition for the semimartingale characteristics of L^{\pm} or equivalently into a coupled system of backward stochastic differential equations for L^{\pm}. We show how this can be used to both characterise and construct optimal strategies. Our results explain and generalise all the results available in the literature so far. Moreover, we even obtain new sharp results in the unconstrained case.
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