Loading...

Fractional Brownian Motion and Stochastic Differential Equations Driven by Fractional Noise

Naghshineh Arjmand, Omid | 2009

1217 Viewed
  1. Type of Document: Ph.D. Dissertation
  2. Language: Farsi
  3. Document No: 39986 (02)
  4. University: Sharif University of Technology
  5. Department: Mathematical Sciences
  6. Advisor(s): Zohori Zangeneh, Bijan
  7. Abstract:
  8. This thesis has been prepared in six chapters. In the first chapter, the necessary analytical preliminaries are revised. The second chapter is specified on the introducing of the fractional Brownian motion and the description of some of its properties. The subject of the third chapter is simulation. The practical utilization of stochastic models usually needs simulation; therefore the fractional Brownian motion and the processes derived from it are not exempted either. The fifth chapter is consisted of two major parts; the first part is the simulation of fractional Brownian motion, in which no new work has been done, and only one of the available methods has been explained. The second part is the conditional simulation, where the proposed subjects are not merely about the fractional Brownian motion and it contains all the Gaussian processes. The algorithms introduced in this part are new. The topic of the forth chapter is the fractional calculus. The importance of this chapter is its usage in the description of the stochastic integral and the stochastic differential equation driven by fractional Brownian motion. In the fifth chapter a modification of the Riemann-Stieltjes integral is given, using the content of chapter forth, and at the end of this chapter by using this modification we try to gain the goal of this thesis, which is the stochastic differential equation driven by fractional noise. The subject of chapter six is the description and proof of an existence theorem for the differential equations with fractional Brownian motion with the Hurst parameter . Usually, proofs of existence theorems for differential equations, either deterministic or stochastic, is dependant on a fixed point theorem in a suitable functional space. In such proofs the solution is explained as a fixed point of a functional map. Furthermore it is necessary to specify the Banach or Hilbert space with suitable norm, and then by using a fixed point theorem, the solution existence is proved. Such a behavior in differential equation although attractive but has a weak point, which is the overlooking of the revolutionary construction of a differential equation and the special role of the time variable. In a differential equation, roughly speaking, this is the past which justifies the future. The method used in chapter six, using this view point, is the approximation of the differential equation with a sequence of delay equations. This approach concludes an existence theorem assuming weaker hypothesis and a simpler proof in comparison to the work done by Nualart and Răşcanu.

  9. Keywords:
  10. Stochastic Differential Equation ; Stochastic Integral ; Fractional Brownian Motion ; Fractional Calculus ; Simulation ; Conditional Simulation

 Digital Object List

 Bookmark

...see more