Markov切换具有Knight不确定下最优消费和投资组合研究

Optimal Consumption and Portfolio with Ambiguity to Markovian Switching

本文在模型不确定环境和一般的半鞅市场条件下,考虑来自于消费和终端财富预期效用最大化问题.代理人以一初始资本和一随机禀赋(endowment)进行投资.我们用鞅方法和对偶理论去寻求最优消费和投资组合问题的解,首先,利用对偶原理,给出在适当的假设条件下,该投资组合问题唯一解存在性的证明,同时对该解进行刻画,并推导出原问题和对偶问题的值函数是互为共轭的.此外,我们还考虑了一个跳扩散模型,该跳扩散模型的系数依赖于一个Markov链,且投资者对Markov链状态间的切换的速率是Knight不确定的.在该模型中我们考虑代理人具有对数效用函数时,可用随机控制方法推导其HJB方程,并能给出HJB方程的数值解,进而能推出最优消费和投资策略.

This paper considers the problem of maximizing expected utility from consumption and terminal wealth under model uncertainty for a general semimartingale market, where the agent with an initial capital and a random endowment can invest. To find a solution to the investment problem we use the martingale method. We first prove that under appropriate assumptions a unique solution to the investment problem exists. Then we deduce that the value functions of primal problem and dual problem are convex conjugate functions. Furthermore we consider a diffusion-jump-model where the coefficients depend on the state of a Markov chain and the investor is ambiguity to the intensity of the underlying Poisson process. Finally, for an agent with the logarithmic utility function, we use the stochastic control method to derive the Hamilton-Jacobi-Bellmann （HJB） equation. And the solution to this HJB equation can be determined numerically. We also show how thereby the optimal investment strategy can be computed.