461 research outputs found
A note on lower bounds of martingale measure densities
For a given element and a convex cone , we give necessary and sufficient conditions for the existence
of an element lying in the polar of . This polar is taken in
and in . In the context of mathematical finance the main
result concerns the existence of martingale measures, whose densities are
bounded from below by prescribed random variable.Comment: 9 page
Strong supermartingales and limits of nonnegative martingales
Given a sequence of nonnegative martingales starting
at , we find a sequence of convex combinations
and a limiting process such that
converges in probability to
, for all finite stopping times . The limiting process then
is an optional strong supermartingale. A counterexample reveals that the
convergence in probability cannot be replaced by almost sure convergence in
this statement. We also give similar convergence results for sequences of
optional strong supermartingales , their left limits
and their stochastic integrals and explain the relation to the notion of the Fatou
limit.Comment: Published at http://dx.doi.org/10.1214/14-AOP970 in the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Duality Theory for Portfolio Optimisation under Transaction Costs
For portfolio optimisation under proportional transaction costs, we provide a
duality theory for general cadlag price processes. In this setting, we prove
the existence of a dual optimiser as well as a shadow price process in a
generalised sense. This shadow price is defined via a "sandwiched" process
consisting of a predictable and an optional strong supermartingale and pertains
to all strategies which remain solvent under transaction costs. We provide
examples showing that in the present general setting the shadow price process
has to be of this generalised form
Portfolio optimisation beyond semimartingales: shadow prices and fractional Brownian motion
While absence of arbitrage in frictionless financial markets requires price
processes to be semimartingales, non-semimartingales can be used to model
prices in an arbitrage-free way, if proportional transaction costs are taken
into account. In this paper, we show, for a class of price processes which are
not necessarily semimartingales, the existence of an optimal trading strategy
for utility maximisation under transaction costs by establishing the existence
of a so-called shadow price. This is a semimartingale price process, taking
values in the bid ask spread, such that frictionless trading for that price
process leads to the same optimal strategy and utility as the original problem
under transaction costs. Our results combine arguments from convex duality with
the stickiness condition introduced by P. Guasoni. They apply in particular to
exponential utility and geometric fractional Brownian motion. In this case, the
shadow price is an Ito process. As a consequence we obtain a rather surprising
result on the pathwise behaviour of fractional Brownian motion: the
trajectories may touch an Ito process in a one-sided manner without reflection.Comment: To appear in Annals of Applied Probability. We would like to thank
Junjian Yang for careful reading of the manuscript and pointing out a mistake
in an earlier versio
Shadow prices for continuous processes
In a financial market with a continuous price process and proportional
transaction costs we investigate the problem of utility maximization of
terminal wealth. We give sufficient conditions for the existence of a shadow
price process, i.e.~a least favorable frictionless market leading to the same
optimal strategy and utility as in the original market under transaction costs.
The crucial ingredients are the continuity of the price process and the
hypothesis of "no unbounded profit with bounded risk". A counter-example
reveals that these hypotheses cannot be relaxed
Hiding a drift
In this article we consider a Brownian motion with drift of the form
dS_t=\mu_t dt+dB_t\qquadfor t\ge0, with a specific nontrivial
, predictable with respect to , the natural
filtration of the Brownian motion . We construct a process
, also predictable with respect to , such that
is a Brownian motion in its own filtration.
Furthermore, for any , we refine this construction such that the
drift only takes values in , for
fixed .Comment: Published in at http://dx.doi.org/10.1214/09-AOP469 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
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