The trap of complacency in predicting the maximum
成果类型：
Article
署名作者：
Du Toit, J.; Peskir, G.
署名单位：
University of Witwatersrand; University of Manchester
刊物名称：
ANNALS OF PROBABILITY
ISSN/ISSBN：
0091-1798
DOI：
10.1214/009117906000000638
发表日期：
2007
页码：
340-365
关键词：
摘要：
Given a standard Brownian motion B-mu = (B-t(mu))(0 <= t <= T) with drift mu is an element of R and letting S-t(mu) = max(0 <= s <= t) B-s(mu) for 0 <= t <= T, we consider the optimal prediction problem: V = inf (0 <=tau <= T) E(B-tau(mu) - S-T(mu))(2) where the infimum is taken over all stopping times tau of B-mu. Reducing the optimal prediction problem to a parabolic free-boundary problem we show that the following stopping time is optimal: tau(*) = inf{t(*) <= t <= T vertical bar b(1) (t) <= S-t(mu) - B-t(mu) <= b(2)(t)} where t(*) is an element of [0, T) and the functions t -> b(1) (t) and t -> b(2) (t) are continuous on [t(*), T] with b(1) (T) = 0 and b(2) (T) = 1/2 mu. If mu > 0, then b(1) is decreasing and b(2) is increasing on [t(*), T] with b(1) (t(*)) = b(2)(t(*)) when t(*) not equal 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 b(1) and b(2) can be characterized as the unique solution to this system. This also leads to an explicit formula for V in terms of b(1) and b(2). If mu <= 0, then t(*) = 0 and b(2) equivalent to +infinity so that tau(*) is expressed in terms of b(1) only. In this case b(1) is decreasing on [z(*), T] and increasing on [0, z(*)) for some z(*) is an element of [0, T) with z(*) = 0 if mu = 0, and the system of two Volterra equations reduces to one Volterra equation. If mu = 0, then there is a closed form expression for b(1). 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 mu not equal 0 and the present paper settles the remaining cases using a different approach.