SDEs Driven by a Time-Changed Lévy Process and Their Associated Time-Fractional Order Pseudo-Differential Equations

By Marjorie G. Hahn, Kei Kobayashi, Sabir Umarov. 2009

54 citations recorded by the source corpus.

It is known that the transition probabilities of a solution to a classical Itô stochastic differential equation (SDE) satisfy in the weak sense the associated Kolmogorov equation. The Kolmogorov equation is a partial differential equation with coefficients 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 Lévy 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.

Subfields: Applied Mathematics, Finance, Modeling and Simulation

View Paper: https://www.semanticscholar.org/paper/e43d2fb9367855fad777097c5c1504f0bd7c3bc2

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In Applied Mathematics

1. Palaniappan Vellaisamy
2. Erkan Nane
3. Mark M. Meerschaert
4. Kei Kobayashi
5. Anatoly N. Kochubei
6. Suleyman Ulusoy
7. Arun Kumar
8. Mohamed M. Khader

In Finance

1. Palaniappan Vellaisamy
2. Mark M. Meerschaert
3. Kei Kobayashi
4. Anatoly N. Kochubei
5. Arun Kumar

In Modeling and Simulation

1. Palaniappan Vellaisamy
2. Kei Kobayashi
3. Erkan Nane
4. Mark M. Meerschaert
5. Anatoly N. Kochubei
6. Charles Tadjeran
7. Fawang Liu
8. Ian L Turner

Citations in the corpus

• Distributed-order fractional diffusions on bounded domains (2009, background): “[15] studied the time-fractional equation ∂u/∂t = Lu in this case, as well as the distributed-order extension.”
• Stochastic Calculus for a Time-Changed Semimartingale and the Associated Stochastic Differential Equations (2009, background): “of order β with respect to the time variable t (see [4]), and δx is the Dirac delta function with mass at x. For the proof, see Hahn, Kobayashi and Umarov [ 5, Theorem 4.1 ].”