Skip to main content

Lund University Publications

LUND UNIVERSITY LIBRARIES

Two quadrature rules for stochastic Itô-integrals with fractional Sobolev regularity

Eisenmann, Monika LU orcid and Kruse, Raphael (2018) In Communications in Mathematical Sciences 16(8). p.2125-2146
Abstract
In this paper we study the numerical quadrature of a stochastic integral, where the temporal regularity of the integrand is measured in the fractional Sobolev-Slobodeckij norm in Wσ,p(0,T), σ ∈(0,2), p∈[2,∞). We introduce two quadrature rules: The first is best suited for the parameter range σ ∈(0,1) and consists of a Riemann-Maruyama approximation on a randomly shifted grid. The second quadrature rule considered in this paper applies to the case of a deterministic integrand of fractional Sobolev regularity with σ ∈(1,2). In both cases the order of convergence is equal to σ with respect to the Lp-norm. As an application, we consider the stochastic integration of a Poisson process, which has discontinuous sample paths. The theoretical... (More)
In this paper we study the numerical quadrature of a stochastic integral, where the temporal regularity of the integrand is measured in the fractional Sobolev-Slobodeckij norm in Wσ,p(0,T), σ ∈(0,2), p∈[2,∞). We introduce two quadrature rules: The first is best suited for the parameter range σ ∈(0,1) and consists of a Riemann-Maruyama approximation on a randomly shifted grid. The second quadrature rule considered in this paper applies to the case of a deterministic integrand of fractional Sobolev regularity with σ ∈(1,2). In both cases the order of convergence is equal to σ with respect to the Lp-norm. As an application, we consider the stochastic integration of a Poisson process, which has discontinuous sample paths. The theoretical results are accompanied by numerical experiments. (Less)
Please use this url to cite or link to this publication:
author
and
publishing date
type
Contribution to journal
publication status
published
subject
in
Communications in Mathematical Sciences
volume
16
issue
8
pages
2125 - 2146
publisher
International Press
external identifiers
  • scopus:85066479251
ISSN
1945-0796
DOI
10.4310/CMS.2018.v16.n8.a4
language
English
LU publication?
no
id
cd29c61a-9185-452f-9ea1-c7dda462600f
date added to LUP
2024-10-07 14:53:20
date last changed
2025-10-14 09:30:01
@article{cd29c61a-9185-452f-9ea1-c7dda462600f,
  abstract     = {{In this paper we study the numerical quadrature of a stochastic integral, where the temporal regularity of the integrand is measured in the fractional Sobolev-Slobodeckij norm in Wσ,p(0,T), σ ∈(0,2), p∈[2,∞). We introduce two quadrature rules: The first is best suited for the parameter range σ ∈(0,1) and consists of a Riemann-Maruyama approximation on a randomly shifted grid. The second quadrature rule considered in this paper applies to the case of a deterministic integrand of fractional Sobolev regularity with σ ∈(1,2). In both cases the order of convergence is equal to σ with respect to the Lp-norm. As an application, we consider the stochastic integration of a Poisson process, which has discontinuous sample paths. The theoretical results are accompanied by numerical experiments.}},
  author       = {{Eisenmann, Monika and Kruse, Raphael}},
  issn         = {{1945-0796}},
  language     = {{eng}},
  number       = {{8}},
  pages        = {{2125--2146}},
  publisher    = {{International Press}},
  series       = {{Communications in Mathematical Sciences}},
  title        = {{Two quadrature rules for stochastic Itô-integrals with fractional Sobolev regularity}},
  url          = {{http://dx.doi.org/10.4310/CMS.2018.v16.n8.a4}},
  doi          = {{10.4310/CMS.2018.v16.n8.a4}},
  volume       = {{16}},
  year         = {{2018}},
}