989 research outputs found
The Uniform Integrability of Martingales. On a Question by Alexander Cherny
Let be a progressively measurable, almost surely right-continuous
stochastic process such that and for each
finite stopping time . In 2006, Cherny showed that is then a
uniformly integrable martingale provided that is additionally nonnegative.
Cherny then posed the question whether this implication also holds even if
is not necessarily nonnegative. We provide an example that illustrates that
this implication is wrong, in general. If, however, an additional integrability
assumption is made on the limit inferior of then the implication holds.
Finally, we argue that this integrability assumption holds if the stopping
times are allowed to be randomized in a suitable sense.Comment: Revised version. Accepted for publication in Stochastic Processes and
their Application
The Martingale Property in the Context of Stochastic Differential Equations
This note studies the martingale property of a nonnegative, continuous local
martingale Z, given as a nonanticipative functional of a solution to a
stochastic differential equation. The condition states that Z is a (uniformly
integrable) martingale if and only if an integral test of a related functional
holds.Comment: Revised version. Published in Electron. Commun. Proba
Conditioned Martingales
It is well known that upward conditioned Brownian motion is a
three-dimensional Bessel process, and that a downward conditioned Bessel
process is a Brownian motion. We give a simple proof for this result, which
generalizes to any continuous local martingale and clarifies the role of finite
versus infinite time in this setting. As a consequence, we can describe the law
of regular diffusions that are conditioned upward or downward.Comment: Corrected several typos, improved formulations. Accepted by
Electronic Communications in Probability; Electronic Communications in
Probability, 2012, Volume 17, Issue 4
A one-dimensional diffusion hits points fast
A one-dimensional, continuous, regular, and strong Markov process with
state space hits any point fast with positive probability. To
wit, if , then for all and
Convergence in Models with Bounded Expected Relative Hazard Rates
We provide a general framework to study stochastic sequences related to
individual learning in economics, learning automata in computer sciences,
social learning in marketing, and other applications. More precisely, we study
the asymptotic properties of a class of stochastic sequences that take values
in and satisfy a property called "bounded expected relative hazard
rates." Sequences that satisfy this property and feature "small step-size" or
"shrinking step-size" converge to 1 with high probability or almost surely,
respectively. These convergence results yield conditions for the learning
models in B\"orgers, Morales, and Sarin (2004), Erev and Roth (1998), and
Schlag (1998) to choose expected payoff maximizing actions with probability one
in the long run.Comment: After revision. Accepted for publication by Journal of Economic
Theor
Pathwise solvability of stochastic integral equations with generalized drift and non-smooth dispersion functions
We study one-dimensional stochastic integral equations with non-smooth
dispersion coefficients, and with drift components that are not restricted to
be absolutely continuous with respect to Lebesgue measure. In the spirit of
Lamperti, Doss and Sussmann, we relate solutions of such equations to solutions
of certain ordinary integral equations, indexed by a generic element of the
underlying probability space. This relation allows us to solve the stochastic
integral equations in a pathwise sense.Comment: Accepted for publication: Annales de l'Institut Henri Poincar\'
Supermartingales as Radon-Nikodym densities and related measure extensions
Certain countably and finitely additive measures can be associated to a given
nonnegative supermartingale. Under weak assumptions on the underlying
probability space, existence and (non)uniqueness results for such measures are
proven.Comment: Published at http://dx.doi.org/10.1214/14-AOP956 in the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
On the Hedging of Options On Exploding Exchange Rates
We study a novel pricing operator for complete, local martingale models. The
new pricing operator guarantees put-call parity to hold for model prices and
the value of a forward contract to match the buy-and-hold strategy, even if the
underlying follows strict local martingale dynamics. More precisely, we discuss
a change of num\'eraire (change of currency) technique when the underlying is
only a local martingale modelling for example an exchange rate. The new pricing
operator assigns prices to contingent claims according to the minimal cost for
superreplication strategies that succeed with probability one for both
currencies as num\'eraire. Within this context, we interpret the lack of the
martingale property of an exchange-rate as a reflection of the possibility that
the num\'eraire currency may devalue completely against the asset currency
(hyperinflation).Comment: Major revision. Accepted by Finance and Stochastics. The original
publication is available at http://link.springer.co
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