A Class of Itô Diffusions with Known Terminal Value and Specified Optimal Barrier

In this paper, we study the optimal stopping-time problems related to a class of Itô diffusions, modeling for example an investment gain, for which the terminal value is a priori known. This could be the case of an insider trading or of the pinning at expiration of stock options. We give the explicit solution to these optimization problems and in particular we provide a class of processes whose optimal barrier has the same form as the one of the Brownian bridge. These processes may be a possible alternative to the Brownian bridge in practice as they could better model real applications. Moreover, we discuss the existence of a process with a prescribed curve as optimal barrier, for any given (decreasing) curve. This gives a modeling approach for the optimal liquidation time, i.e., the optimal time at which the investor should liquidate a position to maximize the gain.

Which Rationals are Ratios of Pisot Sequences?

A Pisot sequence is a sequence of integers defined recursively by the formula - . If 0 < a0 < a1 then an+1/an converges to a limit θ. We ask whether any rational p/q other than an integer can ever occur as such a limit. For p/q > q/2, the answer is no. However, if p/q < q/2 then the question is shown to be equivalent to a stopping time problem related to the notorious 3x + 1 problem and to a question of Mahler concerning the powers of 3/2. Although some interesting statistical properties of these stopping time problems can be established, we are forced to conclude that the question raised in the title of this paper is perhaps more intractable than it might appear.