Hostname: page-component-78c5997874-s2hrs Total loading time: 0 Render date: 2024-11-10T13:54:11.498Z Has data issue: false hasContentIssue false
  • English
  • Français

On the double Laplace transform of the truncated variation of a Brownian motion with drift

Part of: Stochastic processes Distribution theory - Probability

Published online by Cambridge University Press:  01 October 2016

Rafał Marcin Łochowski
Rafał Marcin Łochowski*
Affiliation:
Department of Mathematics and Mathematical Economics, ul. Madalińskiego 6/8, 02-513 Warsaw, Poland email rlocho314@gmail.com African Institute for Mathematical Sciences, 5 Melrose Road, Muizenberg, Cape Town 7945, South Africa

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The aim of this paper is to find a formula for the double Laplace transform of the truncated variation of a Brownian motion with drift. In order to find the double Laplace transform, we also prove some identities for the Brownian motion with drift, which may be of independent interest.

MSC classification

Primary: 60E10: Characteristic functions; other transforms
Secondary: 60G15: Gaussian processes
Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 19 , Issue 1 , 2016 , pp. 281 - 292
Copyright
© The Author 2016 

References

Borodin, A. N. and Salminen, P., Handbook of Brownian motion – facts and formulae , 2nd edn (Birkhäuser, Basel–Berlin–Boston, 2002).CrossRefGoogle Scholar
Łochowski, R. M., ‘Truncated variation, upward truncated variation and downward truncated variation of Brownian motion with drift – their characteristics and applications’, Stochastic Process. Appl. 121 (2011) no. 2, 378–393.CrossRefGoogle Scholar
Łochowski, R. M., ‘On a generalisation of the Hahn–Jordan decomposition for real càdlàg functions’, Colloq. Math. 132 (2013) no. 1, 121–138.CrossRefGoogle Scholar
Łochowski, R. M. and Ghomrasni, R., ‘The play operator, the truncated variation and the generalisation of the Jordan decomposition’, Math. Methods Appl. Sci. 38 (2015) no. 3, 403–419.CrossRefGoogle Scholar
Łochowski, R. M. and Miłoś, P., ‘On truncated variation, upward truncated variation and downward truncated variation for diffusions’, Stochastic Process. Appl. 123 (2013) no. 2, 446–474.CrossRefGoogle Scholar
Łochowski, R. M. and Miłoś, P., ‘Limit theorems for the truncated variation and for numbers of interval crossings of Lévy and self-similar processes’, Preprint, 2014, http://web.sgh.waw.pl/∼rlocho/level_cross_Levy.pdf.Google Scholar
Miłoś, P., ‘Exact representation of truncated variation of Brownian motion’, Preprint, 2013, arXiv:1311.2415.Google Scholar
Taylor, H. M., ‘A stopped Brownian motion formula’, Ann. Probab. 3 (1975) no. 2, 234–246.CrossRefGoogle Scholar