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      SDEs driven by a time-changed L\'evy process and their associated time-fractional order pseudo-differential equations

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          Abstract

          It is known that the transition probabilities of a solution to a classical It\^o stochastic differential equation (SDE) satisfy in the weak sense the associated Kolmogorov equation. The Kolmogorov equation is a partial differential equation with coeffcients determined by the corresponding SDE. Time-fractional Kolmogorov type equations are used to model complex processes in many fields. However, the class of SDEs that is associated with these equations is unknown except in a few special cases. The present paper shows that in the cases of either time-fractional order or more general time-distributed order differential equations, the associated class of SDEs can be described within the framework of SDEs driven by semimartingales. These semimartingales are time-changed L\'evy processes where the independent time-change is given respectively by the inverse of a single or mixture of independent stable subordinators. Examples are provided, including a fractional analogue of the Feynman-Kac formula.

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          Single-particle tracking: applications to membrane dynamics.

          Measurements of trajectories of individual proteins or lipids in the plasma membrane of cells show a variety of types of motion. Brownian motion is observed, but many of the particles undergo non-Brownian motion, including directed motion, confined motion, and anomalous diffusion. The variety of motion leads to significant effects on the kinetics of reactions among membrane-bound species and requires a revision of existing views of membrane structure and dynamics.
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            Fractional Calculus and Continuous-Time Finance III : the Diffusion Limit

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              Equivalence of the Fractional Fokker-Planck and Subordinated Langevin Equations: The Case of a Time-Dependent Force

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                Author and article information

                Journal
                01 July 2009
                2010-06-22
                Article
                10.1007/s10959-010-0289-4
                0907.0253
                2e0a59f2-9e0a-435c-979f-7924a7233628

                http://arxiv.org/licenses/nonexclusive-distrib/1.0/

                History
                Custom metadata
                60H10, 35S10, 60G51
                17 pages. Text rewritten in a succinct form; v2 contains a more detailed preliminary section (Section 2) as well as the proof of Lemma 3.2
                math.PR

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