Error Calculus for Finance and Physics

Abstract

Intuitive introduction to error structures * Error magnitude * Description of small errors by their biases and variances * Intuitive notion of error structure * How to proceed with an error calculation * Application: Partial integration for a Markov chain * Appendix. Historical comment: The benefit of randomizing physical or natural quantities * Bibliography for Chapter I Strongly-continuous semigroups and Dirichlet forms * Strongly-continuous contraction semigroups on a Banach space * The Ornstein-Uhlenbeck semigroup on R and the associated Dirichlet form * Appendix. Determination of D for the Ornstein-Uhlenbeck semigroup * Bibliography for Chapter II Error structures * Main definition and initial examples * Performing calculations in error structures * Lipschitz functional calculus and existence of densities * Closability of pre-structures and other examples * Bibliography for Chapter III Images and products of error structures * Images * Finite products * Infinite products * Appendix. Comments on projective limits * Bibliography for Chapter IV Sensitivity analysis and error calculus * Simple examples and comments * The gradient and the sharp * Integration by parts formulae * Sensitivity of the solution of an ODE to a functional coefficient * Substructures and projections * Bibliography for Chapter V Error structures on fundamental spaces * Error structures on the Monte Carlo space * Error structures on the Wiener space * Error structures on the Poisson space * Bibliography for Chapter VI Application to financial models * Instantaneous error structure of a financial asset * From an instantaneous error structure to a pricing model * Error calculations on the Black-Scholes model * Error calculations for a diffusion model * Bibliography for Chapter VII Applications in the field of physics * Drawing an ellipse (exercise) * Repeated samples: Discussion * Calculation of lengths using the Cauchy-Favard method (exercise) * Temperature equilibrium of a homogeneous solid (exercise) * Nonlinear oscillator subject to thermal interaction: The Gruneisen parameter * Natural error structures on dynamic systems * Bibliography for Chapter VIII