256 research outputs found
On the value of optimal stopping games
We show, under weaker assumptions than in the previous literature, that a
perpetual optimal stopping game always has a value. We also show that there
exists an optimal stopping time for the seller, but not necessarily for the
buyer. Moreover, conditions are provided under which the existence of an
optimal stopping time for the buyer is guaranteed. The results are illustrated
explicitly in two examples.Comment: Published at http://dx.doi.org/10.1214/105051606000000204 in the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org
Boundary conditions for the single-factor term structure equation
We study the term structure equation for single-factor models that predict
nonnegative short rates. In particular, we show that the price of a bond or a
bond option is the unique classical solution to a parabolic differential
equation with a certain boundary behavior for vanishing values of the short
rate. If the boundary is attainable then this boundary behavior serves as a
boundary condition and guarantees uniqueness of solutions. On the other hand,
if the boundary is nonattainable then the boundary behavior is not needed to
guarantee uniqueness but it is nevertheless very useful, for instance, from a
numerical perspective.Comment: Published in at http://dx.doi.org/10.1214/10-AAP698 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Bayesian sequential testing of the drift of a Brownian motion
We study a classical Bayesian statistics problem of sequentially testing the
sign of the drift of an arithmetic Brownian motion with the - loss
function and a constant cost of observation per unit of time for general prior
distributions. The statistical problem is reformulated as an optimal stopping
problem with the current conditional probability that the drift is non-negative
as the underlying process. The volatility of this conditional probability
process is shown to be non-increasing in time, which enables us to prove
monotonicity and continuity of the optimal stopping boundaries as well as to
characterize them completely in the finite-horizon case as the unique
continuous solution to a pair of integral equations. In the infinite-horizon
case, the boundaries are shown to solve another pair of integral equations and
a convergent approximation scheme for the boundaries is provided. Also, we
describe the dependence between the prior distribution and the long-term
asymptotic behaviour of the boundaries.Comment: 28 page
The inverse first-passage problem and optimal stopping
Given a survival distribution on the positive half-axis and a Brownian
motion, a solution of the inverse first-passage problem consists of a boundary
so that the first passage time over the boundary has the given distribution. We
show that the solution of the inverse first- passage problem coincides with the
solution of a related optimal stopping problem. Consequently, methods from
optimal stopping theory may be applied in the study of the inverse
first-passage problem. We illustrate this with a study of the associated
integral equation for the boundary
Convexity preserving jump-diffusion models for option pricing
We investigate which jump-diffusion models are convexity preserving. The
study of convexity preserving models is motivated by monotonicity results for
such models in the volatility and in the jump parameters. We give a necessary
condition for convexity to be preserved in several-dimensional jump-diffusion
models. This necessary condition is then used to show that, within a large
class of possible models, the only convexity preserving models are the ones
with linear coefficients.Comment: 14 page
Dynkin games with incomplete and asymmetric information
We study the value and the optimal strategies for a two-player zero-sum
optimal stopping game with incomplete and asymmetric information. In our
Bayesian set-up, the drift of the underlying diffusion process is unknown to
one player (incomplete information feature), but known to the other one
(asymmetric information feature). We formulate the problem and reduce it to a
fully Markovian setup where the uninformed player optimises over stopping times
and the informed one uses randomised stopping times in order to hide their
informational advantage. Then we provide a general verification result which
allows us to find the value of the game and players' optimal strategies by
solving suitable quasi-variational inequalities with some non-standard
constraints. Finally, we study an example with linear payoffs, in which an
explicit solution of the corresponding quasi-variational inequalities can be
obtained.Comment: 31 pages, 5 figures, small changes in the terminology from game
theor
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