The Annals of Probability

The trap of complacency in predicting the maximum

J. du Toit and G. Peskir

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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*tT|b1(t)≤StμBtμb2(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.

Article information

Ann. Probab., Volume 35, Number 1 (2007), 340-365.

First available in Project Euclid: 19 March 2007

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60] 35R35: Free boundary problems 62M20: Prediction [See also 60G25]; filtering [See also 60G35, 93E10, 93E11]
Secondary: 60J65: Brownian motion [See also 58J65] 45G15: Systems of nonlinear integral equations 60J60: Diffusion processes [See also 58J65]

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


du Toit, J.; Peskir, G. The trap of complacency in predicting the maximum. Ann. Probab. 35 (2007), no. 1, 340--365. doi:10.1214/009117906000000638.

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