31 research outputs found
BS\Delta Es and BSDEs with non-Lipschitz drivers: Comparison, convergence and robustness
We provide existence results and comparison principles for solutions of
backward stochastic difference equations (BSEs) and then prove
convergence of these to solutions of backward stochastic differential equations
(BSDEs) when the mesh size of the time-discretizaton goes to zero. The
BSEs and BSDEs are governed by drivers and
respectively. The new feature of this paper is that they may
be non-Lipschitz in z. For the convergence results it is assumed that the
BSEs are based on d-dimensional random walks approximating the
d-dimensional Brownian motion W underlying the BSDE and that converges to
f. Conditions are given under which for any bounded terminal condition
for the BSDE, there exist bounded terminal conditions for the sequence
of BSEs converging to , such that the corresponding solutions
converge to the solution of the limiting BSDE. An important special case is
when and f are convex in z. We show that in this situation, the solutions
of the BSEs converge to the solution of the BSDE for every uniformly
bounded sequence converging to . As a consequence, one obtains
that the BSDE is robust in the sense that if is close to
in distribution, then the solution of the Nth BSE is close to
the solution of the BSDE in distribution too.Comment: Published in at http://dx.doi.org/10.3150/12-BEJ445 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
On dynamic spectral risk measures, a limit theorem and optimal portfolio allocation
In this paper we propose the notion of continuous-time dynamic spectral
risk-measure (DSR). Adopting a Poisson random measure setting, we define this
class of dynamic coherent risk-measures in terms of certain backward stochastic
differential equations. By establishing a functional limit theorem, we show
that DSRs may be considered to be (strongly) time-consistent continuous-time
extensions of iterated spectral risk-measures, which are obtained by iterating
a given spectral risk-measure (such as Expected Shortfall) along a given
time-grid. Specifically, we demonstrate that any DSR arises in the limit of a
sequence of such iterated spectral risk-measures driven by lattice-random
walks, under suitable scaling and vanishing time- and spatial-mesh sizes. To
illustrate its use in financial optimisation problems, we analyse a dynamic
portfolio optimisation problem under a DSR.Comment: To appear in Finance and Stochastic