Maximum process problems in optimal control theory

Given a standard Brownian motion (Bt)t≥0 and the equation of motion dXt=vtdt+2dBt, we set St=max0≤s≤tXs and consider the optimal control problem supvE(Sτ−Cτ), where c>0 and the supremum is taken over all admissible controls v satisfying vt∈[μ0,μ1] for all t up to τ=inf{t>0|Xt∉(ℓ0,ℓ1)} with μ0<0<μ1 and ℓ0<0<ℓ1 given and fixed. The following control v∗ is proved to be optimal: “pull as hard as possible,” that is, vt∗=μ0 if Xt<g∗(St), and “push as hard as possible,” that is, vt∗=μ1 if Xt>g∗(St), 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