Optimal prediction of the last r-excursion time of Brownian motion models

This paper investigates the optimal prediction of the last $r$-excursion time for a Brownian motion model. The last $r$-excursion time, denoted by $l_{r}$, refers to the right endpoint of the last negative excursion lasting longer than a constant $r>0$. It reduces to the standard last passage time when $r\downarrow 0$. For a Brownian motion with drift $\mu >0$ and volatility $\sigma >0$, our goal is to identify an optimal stopping time that minimizes the ($L_{1}$) distance from the last $r$-excursion time $l_{r}$. We find that the optimal stopping barrier exhibits two distinct structures: a constant barrier (characterized as a solution of a non-linear equation) or a moving barrier (characterized by the unique solution to an integral equation) depending on the ratio $R=\frac{\mu \sqrt{r}}{\sigma }$ which integrates a firm's financial profitability, volatility, and risk tolerance to financial distress. To obtain the optimal stopping time, we examine the smooth fit condition, Lipschitz continuity of the barrier, and probability regularity of the boundary points. As an application in risk management, we develop a decision rule that informs the timing of business expansion and contraction.