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In this paper we consider an optimal stopping problem for a time-homogeneous, onedimensional, regular diffusion. An essential tool in our approach is the MARTIN boundary theory. It is possible to determine explicitly the representing measure of a given p-excessive function. It is seen that this correspondence may be used to construct optimal stopping rules. I n some specific cases, .as demonstrated in the paper, the solution is reached directly and with ease. The SO called condition of "smooth pasting" is seen to be a simple consequence of our results.

Lett → h(t) be a smooth function on ℝ+, andB= {Bs;s≥ 0} a standard Brownian motion. In this paper we derive expressions for the distributions of the variablesTh: = inf {S;Bs=h(s)} and λth: = sup {s≦t; Bs= h(s)}, wheret>0 is given. Our formulas contain an expected value of a Brownian functional. It is seen that this can be computed, principally, using Feynman–Kac&s formula. Further, we discuss in our framework the familiar examples with linear and square root boundaries. Moreover our approach provides in some extent explicit solutions for the second-order boundaries.

In this paper we study Doob's transform of fractional Brownian motion (FBM). It is well known that Doob's transform of standard Brownian motion is identical in law with the Ornstein-Uhlenbeck diffusion defined as the solution of the (stochastic) Langevin equation where the driving process is a Brownian motion. It is also known that Doob's transform of FBM and the process obtained from the Langevin equation with FBM as the driving process are different. However, also the first one of these can be described as a solution of a Langevin equation but now with some other driving process than FBM. We are mainly interested in the properties of this new driving process denoted Y (1) . We also study the solution of the Langevin equation with Y (1) as the driving process. Moreover, we show that the covariance of Y (1) grows linearly; hence, in this respect Y (1) is more like a standard Brownian motion than a FBM. In fact, it is proved that a properly scaled version of Y (1) converges weakly to Brownian motion.