Hans Rudolf Lerche

First exit densities of Brownian motion through one-sided moving boundaries

SummaryLet {ψa; a ε ℝ} be a sequence of curved boundaries which tend to infinity as a increases. Let

Optimal Stopping of Regular Diffusions under Random Discounting

T_a = \inf \{ t > 0|W(t) \geqq \psi _a (t)\}

Sequential Bayes Detection of Trend Changes

where W(t) denotes the standard Brownian motion. Under regularity conditions on the boundaries uniform approximations for the first exit densities of Ta are derived. The consequences for upper and lower class functions are discussed. The approximations for the first exit densities of Brownian motion with drift, which are also derived, lead to uniform approximations for the power functions of sequential tests. The quality of the approximations is demonstrated by some figures.

On the structure of discounted optimal stopping problems for one-dimensional diffusions

Optimal stopping of diffusions and related processes is usually done by solving a free boundary problem. In this paper, we propagate an alternative way, which has already been described in two earlier papers of Beibel and Lerche; we call it the B–L approach. It can be viewed as optimal stopping via measure transformation. While we emphasized in Beibel and Lerche a rather algebraic view, we describe here more the analytic side of the approach. Finally, it is related to some recent Jamshidians results on a duality in optimal stopping.

Asymptotic densities of stopping times associated with tests of power one

Let X be a one-dimensional regular diffusion, A a positive continuous additive functional of X, and h a measurable real-valued function. A method is proposed to determine a stopping rule

A nonlinear parking problem

T^*

The Blackwell Prediction Algorithm for Infinite 0-1 Sequences, and a Generalization*

that maximizes {\bf E}

A refined large deviation principle for Brownian motion and its application to boundary crossing

\{e^{-A_T} h(X_T) \,1_{\{T < \infty\}}\}