The trap of complacency in predicting the maximum

Abstract

Given a standard Brownian motion Bμ=(Btμ)0≤t≤T with drift μ∈ℝ and letting Stμ=max0≤s≤t Bsμ 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 Bμ. 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.

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