MAXIMIZING THE PROBABILITY OF ACHIEVING A GOAL IN THE CASE OF A PARTIALLY OBSERVED DRIFT PROCESS

We consider an investor with initial wealth X₀<1, who wishes to maximize the probability of achieving a goal, X_T = 1, when the stock's drift — modeled as a linear mean-reverting diffusion — is not observed directly but only via the measurement process. Adopting a martingale approach, a generalized Cameron–Martin (1945) formula then enables explicit computation of the value of the problem as well as the wealth process. The dynamic optimal allocation can then be determined using Clark's formula.