ABSTRACT

We give explicit solutions for utility maximization of terminal wealth problem $u\left(X_T\right)$ in the presence of Knightian uncertainty $\underset{\pi \in {\mathrm{\Pi }}_{[0,T]}^{\mathrm{a}\mathrm{d}}}{sup}\underset{\theta \in {\mathrm{\Theta }}_{[0,T]}}{inf}\mathbb{E}\left[u\right(X_T^{\pi ,\theta }\left)\right]$ in continuous time $[0,T]$. We assume there is uncertainty on both drift and volatility of the underlying stocks, which induce nonequivalent measures on canonical space of continuous paths Ω. We take that the uncertainty set resides in compact sets that are time dependent. In this framework, we solve the robust optimization problem with logarithmic, power and exponential utility functions, explicitly. Numerical simulations revealing the effects of uncertainty on the dynamics are also presented.

摘要

我们给出了终端财富问题效用最大化的明确解决方案 $你\left(X_{吨}\right)$ 在存在奈特不确定性的情况下 $\underset{\pi \in {\mathrm{\Pi }}_{[0,吨]}^{\mathrm{一种}\mathrm{d}}}{超}\underset{\theta \in {\mathrm{\Theta }}_{[0,吨]}}{信息}\mathbb{乙}\left[你\right(X_{吨}^{\pi ,\theta }\left)\right]$ 在连续时间内 $[0,吨]$. 我们假设标的股票的漂移和波动性都存在不确定性，这会导致对连续路径 Ω 的规范空间的非等价测度。我们认为不确定性集存在于时间相关的紧致集合中。在这个框架中，我们明确地使用对数、幂和指数效用函数来解决鲁棒优化问题。还介绍了揭示不确定性对动力学影响的数值模拟。