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Markov projection of semimartingales — Application to comparison results

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  • Köpfer, Benedikt
  • Rüschendorf, Ludger

Abstract

In this paper we derive generalizations of comparison results for semimartingales. Our results are based on Markov projections and on known comparison results for Markov processes. The first part of the paper is concerned with an alternative method for the construction of Markov projections of semimartingales. In comparison to the construction in Bentata and Cont (2009) which is based on the solution of a well-posed martingale problem, we make essential use of pseudo-differential operators as investigated in Böttcher (2008) and of fundamental solutions of related evolution problems. This approach allows to dismiss with some boundedness assumptions on the differential characteristics in the martingale approach. As consequence of the construction of Markov projections, comparison results for path-independent functions (European options) of semimartingales can be reduced to the well investigated problem of comparison of Markovian semimartingales. The Markov projection approach to comparison results does not require one of the semimartingales to be Markovian, which is a common assumption in literature. An idea of Brunick and Shreve (2013) to mimick updated processes leads to a related reduction result to the Markovian case and thus to the comparison of related generators. As consequence, a general comparison result is also obtained for path-dependent functions of semimartingales.

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  • Köpfer, Benedikt & Rüschendorf, Ludger, 2023. "Markov projection of semimartingales — Application to comparison results," Stochastic Processes and their Applications, Elsevier, vol. 162(C), pages 361-386.
    • Handle: RePEc:eee:spapps:v:162:y:2023:i:c:p:361-386
    DOI: 10.1016/j.spa.2023.04.018
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    References listed on IDEAS

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    1. Amel Bentata & Rama Cont, 2015. "Forward equations for option prices in semimartingale models," Finance and Stochastics, Springer, vol. 19(3), pages 617-651, July.
    2. Jan Bergenthum & Ludger Rüschendorf, 2006. "Comparison of Option Prices in Semimartingale Models," Finance and Stochastics, Springer, vol. 10(2), pages 222-249, April.
    3. Neufeld, Ariel & Nutz, Marcel, 2014. "Measurability of semimartingale characteristics with respect to the probability law," Stochastic Processes and their Applications, Elsevier, vol. 124(11), pages 3819-3845.
    4. Ben Hambly & Matthieu Mariapragassam & Christoph Reisinger, 2014. "A Forward Equation for Barrier Options under the Brunick&Shreve Markovian Projection," Papers 1411.3618, arXiv.org, revised Sep 2016.
    5. Jan Bergenthum & Ludger Rüschendorf, 2006. "Comparison of Option Prices in Semimartingale Models," Finance and Stochastics, Springer, vol. 10(2), pages 222-249, April.
    6. Ben Hambly & Matthieu Mariapragassam & Christoph Reisinger, 2016. "A forward equation for barrier options under the Brunick & Shreve Markovian projection," Quantitative Finance, Taylor & Francis Journals, vol. 16(6), pages 827-838, June.
    7. Forde, Martin, 2014. "On the Markovian projection in the Brunick–Shreve mimicking result," Statistics & Probability Letters, Elsevier, vol. 85(C), pages 98-105.
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