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Predicting the time of the ultimate maximum for Brownian motion with drift. (English) Zbl 1149.60311
Sarychev, Andrey (ed.) et al., Mathematical control theory and finance. Proceedings of the workshop, Lisbon, April 10–14, 2007. Berlin: Springer (ISBN 978-3-540-69531-8/hbk). 95-112 (2008).
Summary: Given a standard Brownian motion \(B^\mu = (B^\mu_t)_{0\leq t\leq 1}\) with drift \(\mu\in \mathbb R\), letting \(S^\mu_t = \max_{0\leq s\leq t} B^\mu_s\) for \(t\in [0,1]\), and denoting by \(\theta\) the time at which \(S^\mu_1\) is attained, we consider the optimal prediction problem
\[ V_*= \inf_{0\leq\tau\leq 1} {\mathbf E}|\theta-\tau| \]
where the infimum is taken over all stopping times \(\tau\) of \(B^\mu\). Reducing the optimal prediction problem to a parabolic free-boundary problem and making use of local time-space calculus techniques, we show that the following stopping time is optimal:
\[ \tau_*=\inf\{0\leq t\leq 1\mid S^\mu_t-B^\mu_t\geq b(t)\} \]
where \(b : [0,1]\to \mathbb R\) is a continuous decreasing function with \(b(1) = 0\) that is characterized as the unique solution to a nonlinear Volterra integral equation. This also yields an explicit formula for \(V_*\), in terms of \(b\). lf \(\mu\neq 0\) then there is a closed form expression for \(b\). This problem was solved in [M. A. Urusov, Theory Probab. Appl. 49, No. 1, 169–176 (2005); translation from Teor. Veroyatn. Primen. 49, No. 1, 184–190 (2004; Zbl 1090.60072)] and [S. E. Graversen, G. Peskir and A. N. Shiryaev, Theory Probab. Appl. 45, 41–50 (2000) and Teor. Veroyatn. Primen. 45, No. 1, 125–136 (2000; Zbl 0982.60082)] using the method of time change. The latter method cannot be extended to the case when \(\mu\neq 0\) and the present paper settles the remaining cases using a different approach. It is also shown that the shape of the optimal stopping set remains preserved for all Lévy processes.
For the entire collection see [Zbl 1143.91005].

MSC:
60G40 Stopping times; optimal stopping problems; gambling theory
60J65 Brownian motion
35R35 Free boundary problems for PDEs
62M20 Inference from stochastic processes and prediction
45G10 Other nonlinear integral equations
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