Research reports

Years: 2025 2024 2023 2022 2021 2020 2019 2018 2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2004 2003 2002 2001 2000 1999 1998 1997 1996 1995 1994 1993 1992 1991

On the differentiability of solutions of stochastic evolution equations with respect to their initial values

by A. Andersson and A. Jentzen and R. Kurniawan and T. Welti

(Report number 2016-47)

Abstract
In this article we study the differentiability of solutions of parabolic semilinear stochastic evolution equations (SEEs) with respect to their initial values. We prove that if the nonlinear drift coefficients and the nonlinear diffusion coefficients of the considered SEEs are \(n\)-times continuously Fr\'{e}chet differentiable, then the solutions of the considered SEEs are also \(n\)-times continuously Fr\'{e}chet differentiable with respect to their initial values. Moreover, a key contribution of this work is to establish suitable enhanced regularity properties of the derivative processes of the considered SEE in the sense that the dominating linear operator appearing in the SEE smoothes the higher order derivative processes.

Keywords: stochastic evolution equations, differentiability, initial values

@Techreport{AJKW16_684,
  author = {A. Andersson and A. Jentzen and R. Kurniawan and T. Welti},
  title = {On the differentiability of solutions of stochastic evolution equations with respect to their initial values},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2016-47},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2016/2016-47.pdf },
  year = {2016}
}

Disclaimer
© Copyright for documents on this server remains with the authors. Copies of these documents made by electronic or mechanical means including information storage and retrieval systems, may only be employed for personal use. The administrators respectfully request that authors inform them when any paper is published to avoid copyright infringement. Note that unauthorised copying of copyright material is illegal and may lead to prosecution. Neither the administrators nor the Seminar for Applied Mathematics (SAM) accept any liability in this respect. The most recent version of a SAM report may differ in formatting and style from published journal version. Do reference the published version if possible (see SAM Publications).