Time-changed Stochastic Processes and Associated Fractional Order Partial Differential Equations.
Abstract: It is known that the transition probabilities of the solution to a
classical Itô 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, Ko... read morelmogorov 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 Lévy process where the time-change is
given by the inverse of a stable subordinator and is assumed independent of the
Lévy 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
Thesis (Ph.D.)--Tufts University, 2011.
Submitted to the Dept. of Mathematics.
Advisors: Marjorie Hahn, and Sabir Umarov.
Committee: Erkan Nane, and Christoph Borgers.
Keyword: Theoretical Mathematics.read less