Journal of Theoretical Probability

, Volume 29, Issue 2, pp 590–616 | Cite as

Pathwise Integrals and Itô–Tanaka Formula for Gaussian Processes

  • Tommi Sottinen
  • Lauri ViitasaariEmail author


We prove an Itô–Tanaka formula and existence of pathwise stochastic integrals for a wide class of Gaussian processes. Motivated by financial applications, we define the stochastic integrals as forward-type pathwise integrals introduced by Föllmer and as pathwise generalized Lebesgue–Stieltjes integrals introduced by Zähle. As an application, we illustrate the importance of the Itô–Tanaka formula for pricing and hedging of financial derivatives.


Föllmer integral Gaussian processes Generalized Lebesgue–Stieltjes integral Itô–Tanaka formula Mathematical finance Pathwise stochastic integral 

Mathematics Subject Classification 2010

60G15 60H05 91G20 


  1. 1.
    Azmoodeh, E.: Riemann-Stieltjes Integrals with Respect to Fractional Brownian Motion and Applications. PhD Thesis. Helsinki University of Technology Institute of Mathematic s Research Reports A590 (2010)Google Scholar
  2. 2.
    Azmoodeh, E., Mishura, Y., Valkeila, E.: On hedging European options in geometric fractional Brownian motion market model. Stat. Decis. 27, 129–143 (2010)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Azmoodeh, E., Viitasaari, L.: Rate of convergence for discretization of integrals with respect to fractional Brownian motion. J. Theor. Probab. (2013). doi: 10.1007/s10959-013-0495-y
  4. 4.
    Bender, C., Sottinen, T., Valkeila, E.: Arbitrage with fractional Brownian motion? Theory Stoch. Process. 13(29), 23–27 (2007)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Bender, C., Sottinen, T., Valkeila, E.: Pricing by hedging and no-arbitrage beyond semimartingales. Financ. Stoch. 12, 441–468 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Berman, S.M.: Local nondeterminism and local times of gaussian processes. Indiana Univ. Math. J. 23, 69–94 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Bertoin, J.: Temps locaux et intégration stochastique pour les processus de dirichlet. Séminaire de Probabilités 21, 191–205 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Coutin, L., Nualart, D., Tudor, C.: Tanaka formula for the fractional Brownian motion. Stoch. Process. Appl. 94(2), 301–315 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Föllmer, H.: Calcul d’ito sans probabilités. Séminaire de Probabilités 15, 143–150 (1981)zbMATHGoogle Scholar
  10. 10.
    Nualart, D., Răşcanu, A.: Differential equations driven by fractional Brownian motion. Collect. Math. 53, 55–81 (2002)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Revuz, D., Yor, M.: Continuous Martingales and Brownian Motion. Springer, Berlin (1999)CrossRefzbMATHGoogle Scholar
  12. 12.
    Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives, Theory and Applications. Gordon and Breach Science Publishers, Yvendon (1993)zbMATHGoogle Scholar
  13. 13.
    Sondermann, D.: Introduction to Stochastic Calculus for Finance: A New Didactic Approach. Springer, Berlin (2006)zbMATHGoogle Scholar
  14. 14.
    Sottinen, T., Valkeila, E.: On arbitrage and replication in the fractional Black–Scholes pricing model. Stat. Decis. 21, 93–107 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Young, L.C.: An inequality of the hölder type, connected with Stieltjes integration. Acta Math. 67, 251–282 (1936)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Zähle, M.: Integration with respect to fractal functions and stochastic calculus. Part I. Probab. Theory Relat. Fields 111, 333–372 (1998)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of VaasaVaasaFinland
  2. 2.Department of Mathematics and System AnalysisAalto University School of Science, HelsinkiAaltoFinland
  3. 3.Department of MathematicsSaarland University, SaarbrückenSaarbrückenGermany

Personalised recommendations