Stochastic Viability and Comparison Theorems for Mixed Stochastic Differential Equations

  • Alexander Melnikov
  • Yuliya Mishura
  • Georgiy Shevchenko
Article

Abstract

For a mixed stochastic differential equation containing both Wiener process and a Hölder continuous process with exponent γ > 1/2, we prove a stochastic viability theorem. As a consequence, we get a result about positivity of solution and a pathwise comparison theorem. An application to option price estimation is given.

Keywords

Mixed stochastic differential equation Pathwise integral Stochastic viability Comparison theorem Long-range dependence fractional Brownian motion Stochastic differential equation with random drift 

AMS 2000 Subject Classifications

60G22 60G15 60H10 26A33 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Aubin J-P, Doss H (2003) Characterization of stochastic viability of any nonsmooth set involving its generalized contingent curvature. Stoch Anal Appl 21(5):955–981CrossRefMATHMathSciNetGoogle Scholar
  2. Bender C, Sottinen T, Valkeila E (2011) Fractional processes as models in stochastic finance. In: Advanced mathematical methods for finance. Springer, Berlin, pp 75–104CrossRefGoogle Scholar
  3. Cheridito P (2001) Regularizing fractional Brownian motion with a view towards stock price modelling. PhD thesis, Swiss Federal Institute Of Technology, ZurichGoogle Scholar
  4. Ciotir I, Răşcanu A (2009) Viability for differential equations driven by fractional Brownian motion. J Differ Equ 247(5):1505–1528CrossRefMATHGoogle Scholar
  5. Cont R (2005) Long range dependence in financial markets. In: Fractals in engineering. Springer, London, pp 159–179CrossRefGoogle Scholar
  6. Cox JC, Ingersoll JE, Ross SA (1985) A theory of the term structure of interest rates. Econometrica 53(2):385–407CrossRefMATHMathSciNetGoogle Scholar
  7. Doss H (1977) Liens entre équations différentielles stochastiques et ordinaires. Ann Inst Henri Poincaré, Nouv Sér, Sect B 13:99–125MATHMathSciNetGoogle Scholar
  8. Doss H, Lenglart E (1977) Sur le comportement asymptotique des solutions d’équations différentielles stochastiques. C R Acad Sci, Paris, Sér A 284:971–974MATHMathSciNetGoogle Scholar
  9. Feyel D, de la Pradelle A (2001) The FBM Itô’s formula through analytic continuation. Electron J Probab 6:1–22CrossRefMathSciNetGoogle Scholar
  10. Filipović D (2000) Invariant manifolds for weak solutions to stochastic equations. Probab Theory Relat Fields 118(3):323–341CrossRefMATHGoogle Scholar
  11. Guerra J, Nualart D (2008) Stochastic differential equations driven by fractional Brownian motion and standard Brownian motion. Stoch Anal Appl 26(5):1053–1075CrossRefMATHMathSciNetGoogle Scholar
  12. Ikeda N, Watanabe S (1977) A comparison theorem for solutions of stochastic differential equations and its applications. Osaka J Math 14(3):619–633MATHMathSciNetGoogle Scholar
  13. Ikeda N, Watanabe S (1989) Stochastic differential equations and diffusion processes. In: North-Holland Mathematical Library, vol 24, 2nd edn. North-Holland, AmsterdamGoogle Scholar
  14. Ioffe M (2010) Probability distribution of Cox–Ingersoll–Ross process. Working paper, Egar Technology, New YorkGoogle Scholar
  15. Jarrow RA, Protter P, Sayit H (2009) No arbitrage without semimartingales. Ann Appl Probab 19(2):596–616CrossRefMATHMathSciNetGoogle Scholar
  16. Krasin VY, Melnikov AV (2009) On comparison theorem and its applications to finance. In: Optimality and risk—modern trends in mathematical finance. Springer, Berlin, pp 171–181CrossRefGoogle Scholar
  17. Kubilius K (2002) The existence and uniqueness of the solution of an integral equation driven by a p-semimartingale of special type. Stoch Process Appl 98(2):289–315CrossRefMATHMathSciNetGoogle Scholar
  18. Milian A (1995) Stochastic viability and a comparison theorem. Colloq Math 68(2):297–316MATHMathSciNetGoogle Scholar
  19. Mishura YS (2008) Stochastic calculus for fractional Brownian motion and related processes. Springer, BerlinCrossRefMATHGoogle Scholar
  20. Mishura YS, Shevchenko GM (2012) Mixed stochastic differential equations with long-range dependence: existence, uniqueness and convergence of solutions. Comput Math Appl 64:3217–3227. arXiv:1112.2332 CrossRefMATHMathSciNetGoogle Scholar
  21. Samko SG, Kilbas AA, Marichev OI (1993) Fractional integrals and derivatives. Theory and applications. Gordon and Breach Science, New YorkGoogle Scholar
  22. Skorokhod AV (1965) Studies in the theory of random processes. Addison-Wesley, ReadingMATHGoogle Scholar
  23. Willinger W, Taqqu MS, Teverovsky V (1999) Stock market prices and long-range dependence. Finance and Stochastics 3(1):1–13CrossRefMATHMathSciNetGoogle Scholar
  24. Yamada T (1973) On a comparison theorem for solutions of stochastic differential equations and its applications. J Math Kyoto Univ 13:497–512MATHMathSciNetGoogle Scholar
  25. Zabczyk J (2000) Stochastic invariance and consistency of financial models. Atti Accad Naz Lincei, Cl Sci Fis Mat Nat, IX Ser, Rend Lincei, Mat Appl 11(2):67–80MATHMathSciNetGoogle Scholar
  26. Zähle M (1999) On the link between fractional and stochastic calculus. In: Grauel H, Gundlach M (eds) Stochastic dynamics. Springer, New York, pp 305–325CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Alexander Melnikov
    • 1
  • Yuliya Mishura
    • 2
  • Georgiy Shevchenko
    • 2
  1. 1.Department of Mathematical and Statistical SciencesUniversity of AlbertaEdmontonCanada
  2. 2.Faculty of Mechanics and Mathematics, Department of Probability, Statistics and Actuarial MathematicsKyiv National Taras Shevchenko UniversityKyivUkraine

Personalised recommendations