Error Calculus for Finance and Physics : The Language of Dirichlet Forms (De Gruyter Expositions in Mathematics Vol.37) (Reprint XXXX. 2004. X, 234 S. 24,5 cm)


Error Calculus for Finance and Physics : The Language of Dirichlet Forms (De Gruyter Expositions in Mathematics Vol.37) (Reprint XXXX. 2004. X, 234 S. 24,5 cm)

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  • 製本 Hardcover:ハードカバー版/ページ数 240 p.
  • 商品コード 9783110180367

Full Description

Many recent advances in modelling within the applied sciences and engineering have focused on the increasing importance of sensitivity analyses. For a given physical, financial or environmental model, increased emphasis is now placed on assessing the consequences of changes in model outputs that result from small changes or errors in both the hypotheses and parameters. The approach proposed in this book is entirely new and features two main characteristics. Even when extremely small, errors possess biases and variances. The methods presented here are able, thanks to a specific differential calculus, to provide information about the correlation between errors in different parameters of the model, as well as information about the biases introduced by non-linearity. The approach makes use of very powerful mathematical tools (Dirichlet forms), which allow one to deal with errors in infinite dimensional spaces, such as spaces of functions or stochastic processes. The method is therefore applicable to non-elementary models along the lines of those encountered in modern physics and finance. This text has been drawn from presentations of research done over the past ten years and that is still ongoing. The work was presented in conjunction with a course taught jointly at the Universities of Paris 1 and Paris 6. The book is intended for students, researchers and engineers with good knowledge in probability theory.

Table of Contents

Preface                                            v
I Intuitive introduction to error structures 1 (16)
1 Error magnitude 1 (1)
2 Description of small errors by their biases 2 (6)
and variances
3 Intuitive notion of error structure 8 (2)
4 How to proceed with an error calculation 10 (2)
5 Application: Partial integration for a 12 (2)
Markov chain
Appendix. Historical comment: The benefit of 14 (2)
randomizing physical or natural quantities
Bibliography for Chapter I 16 (1)
II Strongly-continuous semigroups and Dirichlet 17 (15)
1 Strongly-continuous contraction semigroups 17 (3)
on a Banach space
2 The Ornstein-Uhlenbeck semigroup on R and 20 (8)
the associated Dirichlet form
Appendix. Determination of D for the 28 (3)
Omstein-Uhlenbeck semigroup
Bibliography for Chapter II 31 (1)
III Error structures 32 (19)
1 Main definition and initial examples 32 (5)
2 Performing calculations in error structures 37 (4)
3 Lipschitz functional calculus and existence 41 (3)
of densities
4 Closability of pre-structures and other 44 (6)
Bibliography for Chapter III 50 (1)
IV Images and products of error structures 51 (16)
1 Images 51 (5)
2 Finite products 56 (3)
3 Infinite products 59 (6)
Appendix. Comments on projective limits 65 (1)
Bibliography for Chapter IV 66 (1)
V Sensitivity analysis and error calculus 67 (26)
1 Simple examples and comments 67 (11)
2 The gradient and the sharp 78 (3)
3 Integration by parts formulae 81 (1)
4 Sensitivity of the solution of an ODE to a 82 (6)
functional coefficient
5 Substructures and projections 88 (4)
Bibliography for Chapter V 92 (1)
VI Error structures on fundamental spaces space 93 (44)
1 Error structures on the Monte Carlo space 93 (8)
2 Error structures on the Wiener space 101(21)
3 Error structures on the Poisson space 122(13)
Bibliography for Chapter VI 135(2)
VII Application to financial models 137(50)
1 Instantaneous error structure of a 137(6)
financial asset
2 From an instantaneous error structure to a 143(12)
pricing model
3 Error calculations on the Black-Scholes 155(10)
4 Error calculations for a diffusion model 165(20)
Bibliography for Chapter VII 185(2)
VIII Applications in the field of physics 187(44)
1 Drawing an ellipse (exercise) 187(3)
2 Repeated samples: Discussion 190(5)
3 Calculation of lengths using the 195(2)
Cauchy-Favard method (exercise)
4 Temperature equilibrium of a homogeneous 197(4)
solid (exercise)
5 Nonlinear oscillator subject to thermal 201(18)
interaction: The Gruneisen parameter
6 Natural error structures on dynamic systems 219(10)
Bibliography for Chapter VIII 229(2)
Index 231