The trap of complacency in predicting the maximum

Abstract

Given a standard Brownian motion B^μ=(B_t^μ)_0≤t≤T with drift μ∈ℝ and letting S_t^μ=max_0≤s≤t B_s^μ for 0≤t≤T, we consider the optimal prediction problem: $$V=\inf_{0\le \tau \le T}\mathsf{E}(B_{\tau}^{\mu}-S_{T}^{\mu})^{2}$$ where the infimum is taken over all stopping times τ of B^μ. Reducing the optimal prediction problem to a parabolic free-boundary problem we show that the following stopping time is optimal: $$τ_*=\inf \{t_*≤t≤T|b_1(t)≤S_t^μ−B_t-^μ≤b_2(t)\}$$ where t_*∈[0, T) and the functions t↦b₁(t) and t↦b₂(t) are continuous on [t_*, T] with b₁(T)=0 and b₂(T)=1/2μ. If μ>0, then b₁ is decreasing and b₂ is increasing on [t_*, T] with b₁(t_*)=b₂(t_*) when t_*≠0. Using local time-space calculus we derive a coupled system of nonlinear Volterra integral equations of the second kind and show that the pair of optimal boundaries b₁ and b₂ can be characterized as the unique solution to this system. This also leads to an explicit formula for V in terms of b₁ and b₂. If μ≤0, then t_*=0 and b₂≡+∞ so that τ_* is expressed in terms of b₁ only. In this case b₁ is decreasing on [z_*, T] and increasing on [0, z_*) for some z_*∈[0, T) with z_*=0 if μ=0, and the system of two Volterra equations reduces to one Volterra equation. If μ=0, then there is a closed form expression for b₁. This problem was solved in [Theory Probab. Appl. 45 (2001) 125–136] using the method of time change (i.e., change of variables). The method of time change cannot be extended to the case when μ≠0 and the present paper settles the remaining cases using a different approach.