Stochastic Viability and Comparison Theorems for Mixed Stochastic Differential Equations

  • Alexander Melnikov
  • Yuliya Mishura
  • Georgiy Shevchenko


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.


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 


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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

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