离散型最优投资停止问题到连续型最优投资停止问题的收敛速度
The convergence rate from discrete to continuous optimal investment stopping problem
摘要：
研究了连续和离散两种情形下的最优投资停止问题，其中投资者需要同时选择最优交易策略和最优停时，以最大化终端财富的期望效用.在文献[9]的基础上，引入了一个附加的随机支付函数，利用二次反射倒向随机微分方程（简称Bracket）的无界终端条件理论，刻画了连续问题的值函数.对于离散问题，在马尔可夫框架和有界障碍假设下，得到了由分段二次倒向随机微分方程递归组成的离散形式，并利用辅助的前向-后向系统和Malliavin演算给出了解的先验估计.最后，通过上述刻画，得到了由离散型到连续型的二次反射型的一致收敛性和收敛速度，这是由相应的最优投资停止问题引起的.
英文摘要：
We study the optimal investment stopping problem in both continuous and discrete case, where the investor needs to choose the optimal trading strategy and optimal stopping time concurrently to maximize the expected utility of terminal wealth. Based on the work [9] with an additional stochastic payoff function, we characterize the value function for the continuous problem via the theory of quadratic reflected backward stochastic differential equation (BSDE for short) with unbounded terminal condition. In regard to discrete problem, we get the discretization form composed of piecewise quadratic BSDEs recursively under Markovian framework and the assumption of bounded obstacle, and provide some useful prior estimates about the solutions with the help of auxiliary forward-backward SDE system and Malliavin calculus. Finally, we obtain the uniform convergence and relevant rate from discretely to continuously quadratic reflected BSDE, which arise from corresponding optimal investment stopping problem through above characterization.
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