| It is known that the transition probabilities of the solution to a classical Ito stochastic differential equation (SDE) satisfy in the weak sense the associated Kolmogorov or Fokker-Planck equation. The Kolmogorov equation, which describes dynamics of the solution to the SDE, is a partial differential equation involving a first-order time derivative. In many applications, however, Kolmogorov type equations with fractional order time derivatives are employed to model complex phenomena. One of the main theorems in this thesis establishes that in the case of fractional order pseudo-differential equations, the associated class of SDEs is described within the framework of SDEs driven by a time-changed Levy process where the time-change is given by the inverse of a stable subordinator and is assumed independent of the Levy process. A similar correspondence between time-changed Gaussian processes and the associated Kolmogorov type equations with fractional order derivatives is obtained, where new classes of operators acting on the time variable are introduced. Generalization of time-changes to the inverses of mixtures of independent stable subordinators is also discussed. |
