Essays on optimal stopping in discrete and continuous time

This thesis deals with optimal stopping of Markov processes in discrete and
continuous time. In the first part we study discrete time random walks with
infinite time horizon. Here, a main problem of interest and guiding example
is the classical Chow-Robbins game, also known as the Sn/n-problem. We
derive tight upper and lower bounds for the value function of the Chow-
Robbins game and related problems and use these to approximate value
function and continuation set of the problems. For the Chow-Robbins game
we find all integer values in the continuation set for times n ≤ 489.241.
Starting from there we analyze analytic properties of value function and
stopping boundary. We show that the value function of the Chow-Robbins
game is non-smooth on a dense subset of the continuation set. We also show
that the continuation set is non-convex and find numeric evidence that the
stopping boundary is not-smooth either. Similar results hold for discrete
stopping problems in a fairly general setting.
The second part treats continuous time stopping problems with finite
time horizon, namely problems with a Brownian motion as a driving process.
For these we derive a new class of Fredholm-type integral equations for the
stopping set. For large problem classes of interest, we show by analytical
arguments that the equation uniquely characterizes the stopping boundary
of the problem. We then use the integral equations to rigorously find the
limit behavior of the stopping boundary close to the terminal time. We show
that the leading-order coefficient is universal for wide classes of problems.
We also use the representation for numerical purposes.


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