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January 2007 The trap of complacency in predicting the maximum
J. du Toit, G. Peskir
Ann. Probab. 35(1): 340-365 (January 2007). DOI: 10.1214/009117906000000638

Abstract

Given a standard Brownian motion Bμ=(Btμ)0≤tT with drift μ∈ℝ and letting Stμ=max0≤stBsμ for 0≤tT, 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 tb1(t) and tb2(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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J. du Toit. G. Peskir. "The trap of complacency in predicting the maximum." Ann. Probab. 35 (1) 340 - 365, January 2007. https://doi.org/10.1214/009117906000000638

Information

Published: January 2007
First available in Project Euclid: 19 March 2007

zbMATH: 1120.60044
MathSciNet: MR2303953
Digital Object Identifier: 10.1214/009117906000000638

Subjects:
Primary: 35R35 , 60G40 , 62M20
Secondary: 45G15 , 60J60 , 60J65

Keywords: Brownian motion , curved boundary , diffusion , local time-space calculus , Markov process , nonlinear Volterra integral equation , normal reflection , optimal prediction , Optimal stopping , parabolic free-boundary problem , smooth fit , ultimate maximum

Rights: Copyright © 2007 Institute of Mathematical Statistics

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Vol.35 • No. 1 • January 2007
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