# Integral Transformations and Anticipative Calculus for Fractional Brownian Motions

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*Yaozhong Hu*

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

# Table of Contents

## Integral Transformations and Anticipative Calculus for Fractional Brownian Motions

- Contents v6 free
- Abstract vii8 free
- Chapter 1. Introduction 110 free
- Chapter 2. Representations 716 free
- Chapter 3. Induced Transformation I 1524
- Chapter 4. Approximation 1928
- Chapter 5. Induced Transformation II 2736
- 5.1. Operators Associated With Z[sub(H)](t,s) 2736
- 5.2. Inverse Operator of T[sub(H,T)] 2938
- 5.3. B[sub(H,T)]T[sub(H,T)] when 1/2 < H < 1 2938
- 5.4. T[sub(H,T)]B[sub(H,T)] for 1/2 < H < 1 3039
- 5.5. B[sub(H,T)]T[sub(H,T)] for 0 < H < 1/2 3342
- 5.6. T[sub(H,T)]B[sub(H,T)] for 0 < H < 1/2 3544
- 5.7. Transpose of T[sub(H,T)] 3645
- 5.8. The Expression for T[sub(H,T)]T*[sub(H,T)] 3847
- 5.9. The transpose of B[sub(H,T)] 3948
- 5.10. The Expression of B*[sub(H,T)]B[sub(H,T)] 4150
- 5.11. Extension of T*[sub(H,T)] and B*[sub(H,T)] 4352
- 5.12. Representation of Brownian motion by fractional Brownian motion 4453

- Chapter 6. Stochastic Calculus of Variation 4554
- Chapter 7. Stochastic Integration 6170
- Chapter 8. Nonlinear Translation (Absolute Continuity) 7180
- Chapter 9. Conditional Expectation 7988
- Chapter 10. Integration By Parts 8796
- Chapter 11. Composition (ltô Formula) 95104
- Chapter 12. Clark Type Representation 105114
- Chapter 13. Continuation 109118
- Chapter 14. Stochastic Control 117126
- Chapter 15. Appendix 121130
- Bibliography 123132