Optimal Stopping Rules and Maximal Inequalities for Bessel Processes

We consider, for Bessel processes X ∈ Bes^α (x) with arbitrary order (dimension) α ∈ R, the problem of the optimal stopping (1.4) for which the gain is determined by the value of the maximum of the process X and the cost which is proportional to the duration of the observation time. We give a description of the optimal stopping rule structure (Theorem 1) and the price (Theorem 2). These results are used for the proof of maximal inequalities of the type \[ { E} {max }r ≤ τ X_r ≤ γ (α )√ {{ E}τ }, \] where X ∈ Bes^α (0), τ is arbitrary stopping time, γ (α ) is a constant depending on the dimension (order) α. It is shown that γ (α ) ~ √ α at α → ∞.