G-布朗运动驱动过程的随机最大值原理及其在金融中的应用
A Stochastic Maximum Principle for Processes Driven by G-Brownian Motion and Applications to Finance
摘要：
本文研究了由不确定波动引起的模型风险下的随机最优控制问题。为了有一个数学上一致的框架，我们使用了G-期望的概念和Peng（2007）引入的相应的G-Brwonian运动。基于次线性期望空间$(Ω,\mathcal{H},\hat{\mathbb{E}})$上的随机微分方程理论，证明了受G-布朗运动驱动的受控过程的随机最大值原理。然后我们得到了由$\mathcal{H}$ -函数表示的极大条件，加上一些凸性条件构成了最优性的充分条件。最后，我们解决了具有模糊波动率的投资组合最优化问题，作为主要结果的显式说明示例。
英文摘要：
In this paper, we consider the stochastic optimal control problems under model risk caused by uncertain volatilities. To have a mathematical consistent framework we use the notion of G-expectation and its corresponding G-Brwonian motion introduced by Peng(2007). Based on the theory of stochastic differential equations on a sublinear expectation space $(Ω,\mathcal{H},\hat{\mathbb{E}})$, we prove a stochastic maximum principle for controlled processes driven by G-Brownian motion. Then we obtain the maximum condition in terms of the $\mathcal{H}$-function plus some convexity conditions constitute sufficient conditions for optimality. Finally, we solve a portfolio optimization problem with ambiguous volatility as an explicitly illustrated example of the main result.
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