Optimal prediction of resistance and support levels

Assuming that the asset price X follows a geometric Brownian motion, we study the optimal prediction problem (Formula presented.) where the infimum is taken over stopping times τ of X and ℓ is a hidden aspiration level (having a potential of creating a resistance or support level for X). Adopting the ‘aspiration-level hypothesis’ and assuming that ℓ is independent from X, we show that a wide class of admissible (non-oscillatory) laws of ℓ lead to unique optimal trading boundaries that can be viewed as the ‘conditional median curves’ for the resistance and support levels (with respect to X and T). We prove the existence of these boundaries and derive the (nonlinear) integral equations which characterize them uniquely. The results are illustrated through some specific examples of admissible laws and their conditional median curves.