Abstract: Due to the complexity of a financial market and information incompleteness, it is necessary to consider a financial market with the incomplete information. It is well-known that the inflation is an important factor which affects an investor's making-decision. This paper discusses the problem of the optimal portfolio with inflation under partial information. Firstly, by using the theory of stochastic differential equations, we establish the dynamics of stock prices. Moreover, we obtain the dynamics of consumer-basket-price by It\^o formula, and give the inflation discounted stock price dynamics equation. Secondly, the related derivation with the case of partial information is transformed to that with the case of full information by applying the nonlinear filtering method and the martingale technique. Finally, with value function of maximizing the expected utility, we derive an explicit representation of the optimal trading strategy based on the hidden Markov filtering results and Malliavin calculus, and present the numerical simulation and its economic analysis under a special case. The obtained results show that the inflation indeed has a significant effect on an investor making-decision.

Key words: portfolio optimization, partial information, inflation, Malliavin calculus, HMM filtering