On the optimality of the rule "Buy-and-Hold"

Let $(B, S)$ be a financial structure with a bank account $B=(B_t)_{t\geq0}, dB_t=rB_tdt, B_0=1$ and a stock $S=(S_t)_{t\geq0}, dS_t=S_t(\mu dt + \sigma dW_t), S_0=1,$ where $W=(W_t)_{t\geq0}$ is a standard Wiener process("Black-Scholes model"). Denote $P_t=S_t/B_t, t\in[0,T], M_T=\max\limits_{t\in[0,T]}P_t$ and let $v=v(x), x\geq0,$ be an utility function (e.g., $v(x)=\log x, v(x)=x$).
In the talk we present results of finding optimal stopping time (a time of selling the stock) in the problem

$$\sup_{\tau\in \mathcal{M}_T} E v(\frac{P_{\tau}}{M_T}),$$

where $\mathcal{M}_T$ is the class of stopping times that take values in $[0,T].$
Also we consider a problem where $\mu$ changes its values by jumps.