Cone-Constrained Continuous-Time Markowitz Problems
Cone-Constrained Continuous-Time Markowitz Problems
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DOI：
10.2139/ssrn.2084753
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发表时间：
2012-06
期刊：
影响因子：
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通讯作者：
Christoph Czichowsky;M. Schweizer
中科院分区：
文献类型：
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作者：
Christoph Czichowsky;M. Schweizer
马科维茨问题包括在金融市场中寻找一种自融资交易策略，其最终财富具有最大均值和最小方差。我们在一般的半鞅模型和圆锥约束下以连续时间研究这一点：交易策略必须采用（可能是随机且与时间相关的）封闭圆锥中的值。我们首先通过证明约束终端增益空间（随机积分空间）在 L^2 中闭合来证明凸约束解的存在性。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。自然关联的约束线性二次最优控制问题的价值过程被分解为两个机会过程 L^{\pm} 作为系数出现的和。鞅最优性原理转化为 L^{\pm} 的半鞅特征的漂移条件，或者等效地转化为 L^{\pm} 的向后随机微分方程的耦合系统。我们展示了如何使用它来描述和构建最佳策略。我们的结果解释并概括了迄今为止文献中可用的所有结果。此外，我们甚至在无约束的情况下获得了新的清晰结果。
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.