具有交易成本的不可分割市场的效用最大化
Utility Maximization of an Indivisible Market with Transaction Costs
摘要：
本文研究了在市场不可分割且包含交易成本的情况下效用最大化问题。首先，在问题公式中，最低保证金要求给出了一个所谓的偿付能力区域。然后，相关的效用最大化制定为一个最优切换问题。扩散是退化的，区域边界是无界集。由于简并性和随机终端时间对初始数据的依赖性，不再具有值函数的连续性，而不提出进一步的条件。本文给出了值函数连续的充分条件。我们的方法的本质是找到一个序列的连续函数局部一致收敛到所需的值函数。由于连续性，值函数可以用某些拟变分不等式的粘性解的概念来刻画。
英文摘要：
This work takes up the challenges of utility maximization problem when the market is indivisible and the transaction costs are included. First there is a so-called solvency region given by the minimum margin requirement in the problem formulation. Then the associated utility maximization is formulated as an optimal switching problem. The diffusion turns out to be degenerate and the boundary of domain is an unbounded set. One no longer has the continuity of the value function without posing further conditions due to the degeneracy and the dependence of the random terminal time on the initial data. This paper provides sufficient conditions under which the continuity of the value function is obtained. The essence of our approach is to find a sequence of continuous functions locally uniformly converging to the desired value function. Thanks to continuity, the value function can be characterized by using the notion of viscosity solution of certain quasi-variational inequality.
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