Title: Optimal investment and hedging under partial information.
Abstract: After a brief introduction to linear filtering, we consider the Merton optimal investment problem when the agent does not know the drift parameter of the underlying stock. This is taken to be a random variable with a Gaussian prior distribution, which is updated via a Kalman filter. the resulting problem of optimal investment with a random drift can be treated as a full information problem, and an explicit solution is possible. We then apply the technique to an incomplete market hedging problem. A claim on a non-traded asset is hedged using a correlated traded asset, and the hedger is once again uncertain of the true values of the drifts of each asset. After filtering, the resulting problem with random drifts can be solved if each asset's prior distribution has the same variance. Analytic approximations for the optimal hedging strategy are used to examine the performance of the optimal hedge with learning. Finally, we examine an optimal investment problem with inside information, in which the insider does not know the true drift of the stock. Explicit solutions are possible, after first enlarging the filtration to accommodate the insider's additional knowledge, then filtering the asset price drift.
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