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Positivity

pp 1–35 | Cite as

Girsanov’s theorem in vector lattices

  • Jacobus J. GroblerEmail author
  • Coenraad C. A. Labuschagne
Article
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Abstract

In this paper we formulate and proof Girsanov’s theorem in vector lattices. To reach this goal, we develop the theory of cross-variation processes, derive the cross-variation formula and the Kunita–Watanabe inequality. Also needed and derived are properties of exponential processes, Itô’s rule for multi-dimensional processes and the integration by parts formula for martingales. After proving Girsanov’s theorem for the one-dimensional case, we also discuss the multi-dimensional case.

Keywords

Vector lattice Riesz space Stochastic process Brownian motion Itô integral Martingale Girsanov’s theorem 

Mathematics Subject Classification

46B40 46G10 47N30 60G20 

Notes

Acknowledgements

The first author added the last section to the original paper after comments made by the referee. I thank him for bringing the theory of G-expectations to my attention.

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Jacobus J. Grobler
    • 1
    Email author
  • Coenraad C. A. Labuschagne
    • 2
  1. 1.Unit for Business Mathematics and InformaticsNorth-West UniversityPotchefstroomSouth Africa
  2. 2.Department of Finance and Investment ManagementUniversity of JohannesburgAuckland Park, JohannesburgSouth Africa

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