## 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 $$\mu \neq 0$$.

### 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
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### References:

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