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Stochastic Viability and Comparison Theorems for Mixed Stochastic Differential Equations

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.

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Correspondence to Georgiy Shevchenko.

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Melnikov, A., Mishura, Y. & Shevchenko, G. Stochastic Viability and Comparison Theorems for Mixed Stochastic Differential Equations. Methodol Comput Appl Probab 17, 169–188 (2015). https://doi.org/10.1007/s11009-013-9336-9

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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