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Abstract
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We study an optimal investment problem under incomplete information for an investor with constant relative risk aversion. We assume that the investor can only observe the asset prices, but not the instantaneous returns. We further assume that the instantaneous returns follow an Ornstein--Uhlenbeck process, and that their initial distribution is Gaussian. We analytically solve the Bellman equation for this problem, and identify the optimal investment strategy under incomplete information. We study the relationship between the value function under partial observation and the value function under full observation, and derive a formula for the economic value of information. Furthermore, we outline how the optimal strategy under partial observation can be computed from the optimal strategy for an investor with full observation. In market model with only one risky asset, we derive explicit formulas for the value functions under both partial and full observation. We also provide an explicit formula for the economic value of information. Finally, we point out how our results in the Gaussian case can be extended to general non-Gaussian prior distributions. |
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