Path-Space Robust Bayesian Portfolio Selection

A Bayesian investor learns an unknown asset drift by Kalman--Bucy filtering and trades using the resulting estimate, but the observation model used for learning may itself be misspecified. We make the portfolio robust to an adversary who distorts the law of observed prices and pays a relative-entropy penalty. A single distortion affects both trading returns and learning, since wealth and the drift estimate respond to the same observed prices. The observation market is complete, so the fixed-target problem becomes an optimization over terminal tracking errors. Entropy duality removes the adversary's inner optimization, and the pricing budget reduces the investor's problem to one scalar equation for the budget multiplier. We show that the resulting payoff is replicable, and together with its worst-case measure forms an attained saddle. The robust position has the same sign as the non-robust one but a smaller size, and it tends to zero far from the target. A robust mean--variance investor also aims lower and starts with a smaller position. The non-robust strategy can fail badly --- with a known nonzero drift, its worst-case cost is infinite at every level of ambiguity. Learning can make this cost finite, but whether it does depends on the shape of the prior, not only on its mean and variance. The robust payoff remains worst-case optimal for any prior: it limits the error on price paths that a driftless model explains better than the reference model, which is where the non-robust error is largest.