研究了在模型不确定环境以及在一般的半鞅市场下的最优投资组合问题.首先,用鞅方法和对偶理论去寻求最优投资组合问题的解,证明了在适当的假设条件下,投资组合问题的对偶问题的HJB方程解的存在性和唯一性,并对这个唯一解进行相关的刻画.其次,推导出原问题和对偶问题的值函数是共轭函数.最后,考虑了一个跳扩散模型,其系数依赖于一个马尔柯夫链,且投资者对马尔柯夫链状态间的切换的速率是不确定的.当代理人具有对数效用函数时,用随机控制方法去推导相应的HJB方程,并得到对偶问题的解,从而推出最优投资组合问题的显式解.
The problem of optimal portfolio under model uncertainty and a general semimartingale market was studied. First, a solution to the investment problem was obtained using the martingale method and the dual theory. It was proven that under appropriate assumptions a unique solution to the investment problem exists and is characterized. Then, the value functions of the primal and dual problem are convex conjugate functions. Finally, a diffusion-jump-model was considered where the coefficients depend on the state of a Markov chain and the investor is uncertain about the intensity of the underlying Poisson process. For an agent with logarithmic utility function, the stochastic control method was adopted to derive the Hamilton- Jacobi-Bellmann-equation. Furthermore, the solution of the dual problem can be determined and it was shown how the optimal portfolio can be explicitly computed.