12,507 research outputs found
Martingale representations in dynamic enlargement setting: the role of the accessible jump times
Let M and N be an F-martingale and an H-martingale respectively on the same
probability space, both enjoying the predictable representation property. We
discuss how, under the assumption of the existence of an equivalent decoupling
measure for F and H, the nature of the
jump times of M and N affects the representation of the FVH-martingales. More
precisely we show that the multiplicity of FVH depends on the behavior of the
common accessible jump times of the two martingales. Then we propose an
extension of Kusuoka's representation theorem to the case when the Brownian
Motion is replaced by a semi-martingale which may jump at the default time with
positive probability
On arbitrages arising from honest times
In the context of a general continuous financial market model, we study
whether the additional information associated with an honest time gives rise to
arbitrage profits. By relying on the theory of progressive enlargement of
filtrations, we explicitly show that no kind of arbitrage profit can ever be
realised strictly before an honest time, while classical arbitrage
opportunities can be realised exactly at an honest time as well as after an
honest time. Moreover, stronger arbitrages of the first kind can only be
obtained by trading as soon as an honest time occurs. We carefully study the
behavior of local martingale deflators and consider no-arbitrage-type
conditions weaker than NFLVR.Comment: 25 pages, revised versio
On strong solutions for positive definite jump-diffusions
We show the existence of unique global strong solutions of a class of
stochastic differential equations on the cone of symmetric positive definite
matrices. Our result includes affine diffusion processes and therefore extends
considerably the known statements concerning Wishart processes, which have
recently been extensively employed in financial mathematics. Moreover, we
consider stochastic differential equations where the diffusion coefficient is
given by the alpha-th positive semidefinite power of the process itself with
0.5<alpha<1 and obtain existence conditions for them. In the case of a
diffusion coefficient which is linear in the process we likewise get a positive
definite analogue of the univariate GARCH diffusions.Comment: version to appear in Stochastic Processes and Their Applications,
201
Filtration shrinkage, strict local martingales and the F\"{o}llmer measure
When a strict local martingale is projected onto a subfiltration to which it
is not adapted, the local martingale property may be lost, and the finite
variation part of the projection may have singular paths. This phenomenon has
consequences for arbitrage theory in mathematical finance. In this paper it is
shown that the loss of the local martingale property is related to a measure
extension problem for the associated F\"{o}llmer measure. When a solution
exists, the finite variation part of the projection can be interpreted as the
compensator, under the extended measure, of the explosion time of the original
local martingale. In a topological setting, this leads to intuitive conditions
under which its paths are singular. The measure extension problem is then
solved in a Brownian framework, allowing an explicit treatment of several
interesting examples.Comment: Published in at http://dx.doi.org/10.1214/13-AAP961 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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