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Integral Transformations and Anticipative Calculus for Fractional Brownian Motions
Yaozhong Hu, University of Kansas, Lawrence, KS

Memoirs of the American Mathematical Society
2005; 127 pp; softcover
Volume: 175
ISBN-10: 0-8218-3704-4
ISBN-13: 978-0-8218-3704-7
List Price: US$68
Individual Members: US$40.80
Institutional Members: US$54.40
Order Code: MEMO/175/825
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This paper studies two types of integral transformation associated with fractional Brownian motion. They are applied to construct approximation schemes for fractional Brownian motion by polygonal approximation of standard Brownian motion. This approximation is the best in the sense that it minimizes the mean square error. The rate of convergence for this approximation is obtained. The integral transformations are combined with the idea of probability structure preserving mapping introduced in [48] and are applied to develop a stochastic calculus for fractional Brownian motions of all Hurst parameter \(H\in (0, 1)\). In particular we obtain Radon-Nikodym derivative of nonlinear (random) translation of fractional Brownian motion over finite interval, extending the results of [48] to general case. We obtain an integration by parts formula for general stochastic integral and an Itô type formula for some stochastic integral. The conditioning, Clark derivative, continuity of stochastic integral are also studied. As an application we study a linear quadratic control problem, where the system is driven by fractional Brownian motion.

Table of Contents

  • Introduction
  • Representations
  • Induced transformation I
  • Approximation
  • Induced transformation II
  • Stochastic calculus of variation
  • Stochastic integration
  • Nonlinear translation (Absolute continuity)
  • Conditional expectation
  • Integration by parts
  • Composition (Itô formula)
  • Clark type representation
  • Continuation
  • Stochastic control
  • Appendix
  • Bibliography
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