Maximum process problems in optimal control theory

Given a standard Brownian motion ( B t ) t ≥ 0 and the equation of motion d X t = v t d t + 2 d B t , we set S t = max 0 ≤ s ≤ t X s and consider the optimal control problem sup v E ( S τ − C τ ) , where c > 0 and the supremum is taken over all admissible controls v satisfying v t ∈ [ μ 0 , μ 1 ] for all t up to τ = inf { t > 0 | X t ∉ ( ℓ 0 , ℓ 1 ) } with μ 0 < 0 < μ 1 and ℓ 0 < 0 < ℓ 1 given and fixed. The following control v ∗ is proved to be optimal: “pull as hard as possible,” that is, v t ∗ = μ 0 if X t < g ∗ ( S t ) , and “push as hard as possible,” that is, v t ∗ = μ 1 if X t > g ∗ ( S t ) , where s ↦ g ∗ ( s ) is a switching curve that is determined explicitly (as the unique solution to a nonlinear differential equation). The solution found demonstrates that the problem formulations based on a maximum functional can be successfully included in optimal control theory (calculus of variations) in addition to the classic problem formulations due to Lagrange, Mayer, and Bolza.