##
**The trap of complacency in predicting the maximum.**
*(English)*
Zbl 1120.60044

Let \(B^{\mu}=(B^{\mu}_{t})_{0\leq t \leq T}\) be a standard Brownian motion with drift \(\mu \in {\mathbb R}\). The authors consider the optimal prediction problem: \(V= \inf_{0\leq \tau \leq T} {\text E}(B^{\mu}_{\tau}- S^{\mu}_{T})^{2},\) where \(S^{\mu}_{t}=\max_{0\leq s \leq t} B^{\mu}_{s}\), for \(0\leq t \leq T\), and the infimum is taken over all stopping times \(\tau\) of \(B^{\mu}\).

First, the optimal prediction problem is reduced to a parabolic free-boundary problem. If \(\mu >0\), it is shown that the following stopping time is optimal: \({\tau}_{*}=\inf\{t_{*}\leq t\leq T \mid b_{1}(t)\leq S^{\mu}_{t} - B^{\mu}_{t}\leq b_{2}(t)\}\), where \(t_{*}\in [0,T)\) and the functions \(t\mapsto b_{1}(t)\) and \(t\mapsto b_{2}(t)\) are continuous on \([t_{*},T]\) with \(b_{1}(T)=0\) and \(b_{2}(T)=1/2{\mu}\), and \(b_{1}\) is decreasing and \(b_{2}\) is increasing on \([t_{*}, T]\) with \(b_{1}(t_{*})= b_{2}(t_{*})\) when \(t_{*}\neq 0\). If \(\mu\leq 0\), then the stopping time given by \({\tau}_{*}=\inf\{0\leq t\leq T \mid S^{\mu}_{t} - B^{\mu}_{t}\geq b_{1}(t)\}\) is optimal. In this case, the function \(t \mapsto b_{1}(t)\) is continuous on \([0,T]\), and decreasing on \([z_{*},T]\) and increasing on \([0,z_{*})\) for some \(z_{*}\in [0,T)\) with \(z_{*} =0\) if \(\mu = 0\).

The method of proof is based on local time-space calculus [G. Peskir, J. Theor. Probab. 18, No. 3, 499–535 (2005; Zbl 1085.60033), Math. Finance 15, No. 1, 169–181 (2005; Zbl 1109.91028) and Finance Stoch. 9, No. 2, 251–267 (2005; Zbl 1092.91029)]. In the case \(\mu >0\), the authors 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 \(\mu\leq 0\), the system of two Volterra equations reduces to one Volterra equation.

For \(\mu = 0\), the considered optimal prediction problem with a closed form expression for \(b_{1}\) was solved by S. E. Graversen, G. Peskir and A. N. Shiryaev [Theory Probab. Appl. 45, No. 1, 41–50 (2000) and Teor. Veroyatn. Primen. 45, No. 1, 125–136 (2000; Zbl 0982.60082)] using the method of time change [J. L. Pedersen, Stochastics Stochastics Rep. 75, No. 4, 205–219 (2003; Zbl 1032.60038)] which, however, cannot be extended to the case \(\mu \neq 0\).

First, the optimal prediction problem is reduced to a parabolic free-boundary problem. If \(\mu >0\), it is shown that the following stopping time is optimal: \({\tau}_{*}=\inf\{t_{*}\leq t\leq T \mid b_{1}(t)\leq S^{\mu}_{t} - B^{\mu}_{t}\leq b_{2}(t)\}\), where \(t_{*}\in [0,T)\) and the functions \(t\mapsto b_{1}(t)\) and \(t\mapsto b_{2}(t)\) are continuous on \([t_{*},T]\) with \(b_{1}(T)=0\) and \(b_{2}(T)=1/2{\mu}\), and \(b_{1}\) is decreasing and \(b_{2}\) is increasing on \([t_{*}, T]\) with \(b_{1}(t_{*})= b_{2}(t_{*})\) when \(t_{*}\neq 0\). If \(\mu\leq 0\), then the stopping time given by \({\tau}_{*}=\inf\{0\leq t\leq T \mid S^{\mu}_{t} - B^{\mu}_{t}\geq b_{1}(t)\}\) is optimal. In this case, the function \(t \mapsto b_{1}(t)\) is continuous on \([0,T]\), and decreasing on \([z_{*},T]\) and increasing on \([0,z_{*})\) for some \(z_{*}\in [0,T)\) with \(z_{*} =0\) if \(\mu = 0\).

The method of proof is based on local time-space calculus [G. Peskir, J. Theor. Probab. 18, No. 3, 499–535 (2005; Zbl 1085.60033), Math. Finance 15, No. 1, 169–181 (2005; Zbl 1109.91028) and Finance Stoch. 9, No. 2, 251–267 (2005; Zbl 1092.91029)]. In the case \(\mu >0\), the authors 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 \(\mu\leq 0\), the system of two Volterra equations reduces to one Volterra equation.

For \(\mu = 0\), the considered optimal prediction problem with a closed form expression for \(b_{1}\) was solved by S. E. Graversen, G. Peskir and A. N. Shiryaev [Theory Probab. Appl. 45, No. 1, 41–50 (2000) and Teor. Veroyatn. Primen. 45, No. 1, 125–136 (2000; Zbl 0982.60082)] using the method of time change [J. L. Pedersen, Stochastics Stochastics Rep. 75, No. 4, 205–219 (2003; Zbl 1032.60038)] which, however, cannot be extended to the case \(\mu \neq 0\).

Reviewer: Ryszard Doman (Poznan)

### MSC:

60G40 | Stopping times; optimal stopping problems; gambling theory |

60J65 | Brownian motion |

60J60 | Diffusion processes |

35R35 | Free boundary problems for PDEs |

45G15 | Systems of nonlinear integral equations |

### Keywords:

Brownian motion; optimal prediction; optimal stopping; local time-space calculus; nonlinear Volterra integral equation
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\textit{J. du Toit} and \textit{G. Peskir}, Ann. Probab. 35, No. 1, 340--365 (2007; Zbl 1120.60044)

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