**Memoirs of the American Mathematical Society**

2005;
127 pp;
Softcover

MSC: Primary 60; 26; 44;

Print ISBN: 978-0-8218-3704-7

Product Code: MEMO/175/825

List Price: $68.00

Individual Member Price: $40.80

**Electronic ISBN: 978-1-4704-0426-0
Product Code: MEMO/175/825.E**

List Price: $68.00

Individual Member Price: $40.80

# 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 [

#### 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