pp 1–35 | Cite as

Girsanov’s theorem in vector lattices

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


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.


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

Mathematics Subject Classification

46B40 46G10 47N30 60G20 



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.


  1. 1.
    Aliprantis, C.D., Burkinshaw, O.: Locally Solid Riesz Spaces. Academic Press, New York (1978)zbMATHGoogle Scholar
  2. 2.
    Aliprantis, C.D., Burkinshaw, O.: Positive Operators. Academic Press Inc., Orlando (1985)zbMATHGoogle Scholar
  3. 3.
    Ash, R.B.: Basic Probability Theory. Wiley, New York (1970)zbMATHGoogle Scholar
  4. 4.
    Ash, R.B., Doléans-Dade, C.A.: Probability & Measure Theory, 2nd edn. Harcourt/Academic Press, Massachusetts (2000)zbMATHGoogle Scholar
  5. 5.
    Azouzi, Y., Ramdane, K.: On the distribution function with respect to conditional expectation on Riesz spaces. Quaest. Math. 41, 257–264 (2017). MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Cameron, R.H., Martin, W.P.: Transformation of Wiener integrals under translations. Ann. Math. 45, 386–396 (1944)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Denis, L., Hu, M., Peng, S.: Function spaces and capacity related to a sublinear expectation: application to \(G\)-Brownian motion paths. Potential Anal. 34(2), 139–161 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Donner, K.: Extension of Positive Operators and Korovkin Theorems. Lecture Notes in Mathematics, vol. 904. Springer, Berlin (1982)CrossRefzbMATHGoogle Scholar
  9. 9.
    Fremlin, D.H.: Topological Riesz Spaces and Measure Theory. Cambridge University Press, Cambridge (1974)CrossRefzbMATHGoogle Scholar
  10. 10.
    Girsanov, I.V.: On transforming a certain class of stochastic processes by absolutely continuous substitution of measures. Theory Probab. Appl. 5, 285–301 (1960)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Grobler, J.J.: Continuous stochastic processes in Riesz spaces: the Doob–Meyer decomposition. Positivity 14, 731–751 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Grobler, J.J.: Doob’s optional sampling theorem in Riesz spaces. Positivity 15, 617–637 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Grobler, J.J.: Jensen’s and Martingale Inequalities in Riesz spaces. Indag. Math. 25, 275–295 (2014). MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Grobler, J.J.: The Kolmogorov–C̆entsov theorem and Brownian motion in vector lattices. J. Math. Anal. Appl. 410, 891–901 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Grobler, J.J.: Corrigendum to “The Kolmogorov-C̆entsov theorem and Brownian motion in vector lattices” [J. Math. Anal. Appl. 410 (2014) 891–901]. J. Math. Anal. Appl. 420, 878 (2014). MathSciNetCrossRefGoogle Scholar
  16. 16.
    Grobler, J.J., Labuschagne, C.C.A., Maraffa, V.: Quadratic variation of martingales in Riesz spaces. J. Math. Anal. Appl. 410, 418–426 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Grobler, J.J., Labuschagne, C.C.A.: The Itô integral for Brownian motion in vector lattices: Part 1. J. Math. Anal. Appl. 423, 797–819 (2015). MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Grobler, J.J., Labuschagne, C.C.A.: The Itô integral for Brownian motion in vector lattices: Part 2. J. Math. Anal. Appl. 423, 820–833 (2015). MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Grobler, J.J., Labuschagne, C.C.A.: The quadratic variation of continuous time stochastic processes in vector lattices. J. Math. Anal. Appl. 450, 314–329 (2017). MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Grobler, J.J., Labuschagne, C.C.A.: The Itô integral for martingales in vector lattices. J. Math. Anal. Appl. 450, 1245–1274 (2017). MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Grobler, J.J., Labuschagne, C.C.A.: Itô’s rule and Lévy’s theorem in vector lattices. J. Math. Anal. Appl. (2017).
  22. 22.
    Karatzas, I., Shreve, S.E.: Brownian Motion and Stochastic Calculus. Graduate Texts in Mathematics, Springer, New York (1991)zbMATHGoogle Scholar
  23. 23.
    Kunita, H., Watanabe, S.: On square-integrable martingales. Nagoya Math. J. 30, 209–245 (1967)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Kuo, H.-H.: Introduction to Stochastic Integration. Springer, New York (2006)zbMATHGoogle Scholar
  25. 25.
    Kuo, W.-C., Labuschagne, C.C.A., Watson, B.A.: Discrete time stochastic processes on Riesz spaces. Indag. Math. 15, 435–451 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Kuo, W.-C., Labuschagne, C.C.A., Watson, B.A.: An Upcrossing Theorem for Martingales on Riesz Spaces. Soft Methodology and Random Information Systems, pp. 101–108. Springer, Berlin (2004)CrossRefGoogle Scholar
  27. 27.
    Kuo, W.-C., Labuschagne, C.C.A., Watson, B.A.: Conditional expectation on Riesz spaces. J. Math. Anal. Appl. 303, 509–521 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Kuo, W.-C., Labuschagne, C.C.A., Watson, B.A.: Convergence of Riesz space martingales. Indag. Math. 17, 271–283 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Kuo, W.-C., Labuschagne, C.C.A., Watson, B.A.: Ergodic theory and the strong law of large numbers on Riesz spaces. J. Math. Anal. Appl. 325, 422–437 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Loéve, M.: Probabiblity Theory I and II 4th Edition, Graduate Texts in Mathematics, vol. 45. Springer, New York (1977)Google Scholar
  31. 31.
    Luxemburg, W.A.J., Zaanen, A.C.: Riesz Spaces I. North-Holland Publishing Company, Amsterdam (1971)zbMATHGoogle Scholar
  32. 32.
    Meyer-Nieberg, P.: Banach Lattices. Springer, Berlin (1991)CrossRefzbMATHGoogle Scholar
  33. 33.
    Osuka, E.: Girsanov’s formula for \(G\)-Brownian motion. Stoch. Process. Appl. 123(4), 139–161 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Revuz, D., Yor, M.: Continuous Martingales and Brownian Motion. Springer, Berlin (1991)CrossRefzbMATHGoogle Scholar
  35. 35.
    Schaefer, H.H.: Banach Lattices and Positive Operators. Springer, Berlin (1974)CrossRefzbMATHGoogle Scholar
  36. 36.
    Vardy, J.J., Watson, B.A.: Markov processes on Riesz spaces. Positivity 16, 373–391 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Vulikh, B.Z.: Introduction to the Theory of Partially Ordered Spaces. Wolters-Noordhoff Scientific Publishers, Groningen (1967)zbMATHGoogle Scholar
  38. 38.
    Watson, B.A.: An Ândo–Douglas type theorem in Riesz spaces with a conditional expectation. Positivity 13(3), 543–558 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Zaanen, A.C.: Riesz Spaces II. North-Holland, Amsterdam (1983)zbMATHGoogle Scholar
  40. 40.
    Zaanen, A.C.: Introduction to Operator Theory in Riesz Spaces. Springer, Berlin (1991)zbMATHGoogle Scholar

Copyright information

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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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