Results 1  10
of
12
Convergence of numerical methods for stochastic differential equations in mathematical finance
, 1204
"... ar ..."
(Show Context)
Multilevel Monte Carlo Quadrature of Discontinuous Payoffs in the Generalized Heston Model using Malliavin Integration by Parts
, 2013
"... The consecutive numbering of the publications is determined by their chronological order. The aim of this preprint series is to make new research rapidly available for scientific discussion. Therefore, the responsibility for the contents is solely due to the authors. The publications will be distrib ..."
Abstract

Cited by 14 (0 self)
 Add to MetaCart
(Show Context)
The consecutive numbering of the publications is determined by their chronological order. The aim of this preprint series is to make new research rapidly available for scientific discussion. Therefore, the responsibility for the contents is solely due to the authors. The publications will be distributed by the authors.
Loss of regularity for Kolmogorov equations
, 2012
"... The celebrated Hörmander condition is a sufficient (and nearly necessary) condition for a secondorder linear Kolmogorov partial differential equation (PDE) with smooth coefficients to be hypoelliptic. As a consequence, the solutions of Kolmogorov PDEs are smooth at all positive times if the coeffi ..."
Abstract

Cited by 4 (2 self)
 Add to MetaCart
(Show Context)
The celebrated Hörmander condition is a sufficient (and nearly necessary) condition for a secondorder linear Kolmogorov partial differential equation (PDE) with smooth coefficients to be hypoelliptic. As a consequence, the solutions of Kolmogorov PDEs are smooth at all positive times if the coefficients of the PDE are smooth and satisfy Hörmander’s condition even if the initial function is only continuous but not differentiable. Firstorder linear Kolmogorov PDEs with smooth coefficients do not have this smoothing effect but at least preserve regularity in the sense that solutions are smooth if their initial functions are smooth. In this article, we consider the intermediate regime of nonhypoelliptic secondorder Kolmogorov PDEs with smooth coefficients. The main observation of this article is that there exist counterexamples to regularity preservation in that case. More precisely, we give an example of a secondorder linear Kolmogorov PDE with globally bounded and smooth coefficients and a smooth initial function with compact support such that the unique globally bounded viscosity solution of the PDE is not even locally Hölder continuous and, thereby, we disprove the existence of globally bounded classical solutions of this PDE. From the perspective of probability theory, this observation has the consequence that there exists a stochastic differential equation (SDE) with globally bounded and smooth coefficients and a smooth function with compact support which is mapped by
Uniform approximation of the CIR process via exact simulation at random times∗
, 2015
"... In this paper we uniformly approximate the trajectories of the CoxIngersollRoss (CIR) process. At a sequence of random times the approximate trajectories will be even exact. In between, the approximation will be uniformly close to the exact trajectory. From a conceptual point of view the proposed ..."
Abstract
 Add to MetaCart
(Show Context)
In this paper we uniformly approximate the trajectories of the CoxIngersollRoss (CIR) process. At a sequence of random times the approximate trajectories will be even exact. In between, the approximation will be uniformly close to the exact trajectory. From a conceptual point of view the proposed method gives a better quality of approximation in a pathwise sense than standard, or even exact simulation of the CIR dynamics at some deterministic time grid. AMS 2010 subject classification. Primary 65C30; secondary 60H35.