On the solution of the optimal prediction problem for the maximum of a time-homogeneous diffusion (Q2925753)

Consider the optimal stopping problem \(\sup_{0\leq \tau\leq T}\operatorname{E}\,h(X_\tau, M_T)\), where \(h(x,s)\in C^{2,1}\), \(dX_t= b(X_t)+ \sigma(X_t) dB_t\), \(B\) Brownian motion, and \(M_t= \max_{0\leq u\leq t}X_u\). Leaving aside those cases in which the problem reduces to a problem to which the solution is known, one is left with the case where for \(M_t=s\) it is optimal stop as soon as \(X_t\) reaches some threshold \(g(s,t)< s\).NEWLINENEWLINE The author gives an integro-differential equation for \(g\) and a corresponding maximality principle for choosing the optimal stopping time. The proofs are not written out in all detail, but the line they follow is indicated, referring to [\textit{P. Milgram} and \textit{I. Segal}, Econometrica 70, No. 2, 583--601 (2002; Zbl 1103.90400)] and [\textit{G. Peskir}, Ann. Probab. 26, No. 4, 1614--1640 (1998; Zbl 0935.60025)].