Characteristic function (probability theory)

Abstract: This paper shows how the fast Fourier Transform may be used to value options when the characteristic function of the return is known analytically.

Abstract: This paper is devoted, in the main, to proving the asymptotic minimax character of the sample distribution function (d.f.) for estimating an unknown d.f. in $\mathscr{F}$ or $\mathscr{F}_c$ (defined in Section 1) for a wide variety of weight functions. Section 1 contains definitions and a discussion of measurability considerations. Lemma 2 of Section 2 is an essential tool in our proofs and seems to be of interest per se; for example, it implies the convergence of the moment generating function of $G_n$ to that of $G$ (definitions in (2.1)). In Section 3 the asymptotic minimax character is proved for a fundamental class of weight functions which are functions of the maximum deviation between estimating and true d.f. In Section 4 a device (of more general applicability in decision theory) is employed which yields the asymptotic minimax result for a wide class of weight functions of this character as a consequence of the results of Section 3 for weight functions of the fundamental class. In Section 5 the asymptotic minimax character is proved for a class of integrated weight functions. A more general class of weight functions for which the asymptotic minimax character holds is discussed in Section 6. This includes weight functions for which the risk function of the sample d.f. is not a constant over $\mathscr{F}_c.$ Most weight functions of practical interest are included in the considerations of Sections 3 to 6. Section 6 also includes a discussion of multinomial estimation problems for which the asymptotic minimax character of the classical estimator is contained in our results. Finally, Section 7 includes a general discussion of minimization of symmetric convex or monotone functionals of symmetric random elements, with special consideration of the "tied-down" Wiener process, and with a heuristic proof of the results of Sections 3, 4, 5, and much of Section 6.

Abstract: It is often required to approximate to the distribution of some statistic whose exact distribution cannot be conveniently obtained. When the first few moments are known, a common procedure is to fit a law of the Pearson or Edgeworth type having the same moments as far as they are given. Both these methods are often satisfactory in practice, but have the drawback that errors in the "tail" regions of the distribution are sometimes comparable with the frequencies themselves. The Edgeworth approximation in particular notoriously can assume negative values in such regions. The characteristic function of the statistic may be known, and the difficulty is then the analytical one of inverting a Fourier transform explicitly. In this paper we show that for a statistic such as the mean of a sample of size $n$, or the ratio of two such means, a satisfactory approximation to its probability density, when it exists, can be obtained nearly always by the method of steepest descents. This gives an asymptotic expansion in powers of $n^{-1}$ whose dominant term, called the saddlepoint approximation, has a number of desirable features. The error incurred by its use is $O(n^{-1})$ as against the more usual $O(n^{-1/2})$ associated with the normal approximation. Moreover it is shown that in an important class of cases the relative error of the approximation is uniformly $O(n^{-1})$ over the whole admissible range of the variable. The method of steepest descents was first used systematically by Debye for Bessel functions of large order (Watson [17]) and was introduced by Darwin and Fowler (Fowler [9]) into statistical mechanics, where it has remained an indispensable tool. Apart from the work of Jeffreys [12] and occasional isolated applications by other writers (e.g. Cox [2]), the technique has been largely ignored by writers on statistical theory. In the present paper, distributions having probability densities are discussed first, the saddlepoint approximation and its associated asymptotic expansion being obtained for the probability density of the mean $\bar{x}$ of a sample of $n$. It is shown how the steepest descents technique is related to an alternative method used by Khinchin [14] and, in a slightly different context, by Cramer [5]. General conditions are established under which the relative error of the saddlepoint approximation is $O(n^{-1})$ uniformly for all admissible $\bar{x}$, with a corresponding result for the asymptotic expansion. The case of discrete variables is briefly discussed, and finally the method is used for approximating to the distribution of ratios.

Abstract: 1. Goal.- 1.1 Order and Disorder: Some Typical Phenomena.- 1.2 Some Typical Problems and Difficulties.- 1.3 How We Shall Proceed.- 2. Probability.- 2.1 Object of Our Investigations: The Sample Space.- 2.2 Random Variables.- 2.3 Probability.- 2.4 Distribution.- 2.5 Random Variables with Densities.- 2.6 Joint Probability.- 2.7 Mathematical Expectation E(X), and Moments.- 2.8 Conditional Probabilities.- 2.9 Independent and Dependent Random Variables.- 2.10 Generating Functions and Characteristic Functions.- 2.11 A Special Probability Distribution: Binomial Distribution.- 2.12 The Poisson Distribution.- 2.13 The Normal Distribution (Gaussian Distribution).- 2.14 Stirling's Formula.- 2.15 Central Limit Theorem.- 3. Information.- 3.1 Some Basic Ideas.- 3.2 Information Gain: An Illustrative Derivation.- 3.3 Information Entropy and Constraints.- 3.4 An Example from Physics: Thermodynamics.- 3.5 An Approach to Irreversible Thermodynamics.- 3.6 Entropy-Curse of Statistical Mechanics?.- 4. Chance.- 4.1 A Model of Brownian Movement.- 4.2 The Random Walk Model and Its Master Equation.- 4.3 Joint Probability and Paths. Markov Processes. The Chapman-Kolmogorov Equation. Path Integrals.- Sections with an asterisk in the heading may be omitted during a first reading..- 4.4 How to Use Joint Probabilities. Moments. Characteristic Function. Gaussian Processes.- 4.5 The Master Equation.- 4.6 Exact Stationary Solution of the Master Equation for Systems in Detailed Balance.- 4.7 The Master Equation with Detailed Balance. Symmetrization, Eigenvalues and Eigenstates.- 4.8 Kirchhoff's Method of Solution of the Master Equation.- 4.9 Theorems about Solutions of the Master Equation.- 4.10 The Meaning of Random Processes, Stationary State, Fluctuations, Recurrence Time.- 4.11 Master Equation and Limitations of Irreversible Thermodynamics.- 5. Necessity.- 5.1 Dynamic Processes.- 5.2 Critical Points and Trajectories in a Phase Plane. Once Again Limit Cycles.- 5.3 Stability.- 5.4 Examples and Exercises on Bifurcation and Stability.- 5.5 Classification of Static Instabilities, or an Elementary Approach to Thorn's Theory of Catastrophes.- 6. Chance and Necessity.- 6.1 Langevin Equations: An Example.- 6.2 Reservoirs and Random Forces.- 6.3 The Fokker-Planck Equation.- 6.4 Some Properties and Stationary Solutions of the Fokker-Planck-Equation.- 6.6 Time-Dependent Solutions of the Fokker-Planck Equation.- 6.6 Solution of the Fokker-Planck Equation by Path Integrals.- 6.7 Phase Transition Analogy.- 6.8 Phase Transition Analogy in Continuous Media: Space-Dependent Order Parameter.- 7. Self-Organization.- 7.1 Organization.- 7.2 Self-Organization.- 7.3 The Role of Fluctuations: Reliability or Adaptibility? Switching.- 7.4 Adiabatic Elimination of Fast Relaxing Variables from the Fokker-Planck Equation.- 7.5 Adiabatic Elimination of Fast Relaxing Variables from the Master Equation.- 7.6 Self-Organization in Continuously Extended Media. An Outline of the Mathematical Approach.- 7.7 Generalized Ginzburg-Landau Equations for Nonequilibrium Phase Transitions.- 7.8 Higher-Order Contributions to Generalized Ginzburg-Landau Equations.- 7.9 Scaling Theory of Continuously Extended Nonequilibrium Systems.- 7.10 Soft-Mode Instability.- 7.11 Hard-Mode Instability.- 8. Physical Systems.- 8.1 Cooperative Effects in the Laser: Self-Organization and Phase Transition.- 8.2 The Laser Equations in the Mode Picture.- 8.3 The Order Parameter Concept.- 8.4 The Single-Mode Laser.- 8.5 The Multimode Laser.- 8.6 Laser with Continuously Many Modes. Analogy with Superconductivity.- 8.7 First-Order Phase Transitions of the Single-Mode Laser.- 8.8 Hierarchy of Laser Instabilities and Ultrashort Laser Pulses.- 8.9 Instabilities in Fluid Dynamics: The Benard and Taylor Problems.- 8.10 The Basic Equations.- 8.11 The Introduction of New Variables.- 8.12 Damped and Neutral Solutions (R ? Rc).- 8.13 Solution Near R = Rc (Nonlinear Domain). Effective Langevin Equations.- 8.14 The Fokker-Planck Equation and Its Stationary Solution.- 8.15 A Model for the Statistical Dynamics of the Gunn Instability Near Threshold.- 8.16 Elastic Stability: Outline of Some Basic Ideas.- 9. Chemical and Biochemical Systems.- 9.1 Chemical and Biochemical Reactions.- 9.2 Deterministic Processes, Without Diffusion, One Variable.- 9.3 Reaction and Diffusion Equations.- 9.4 Reaction-Diffusion Model with Two or Three Variables: The Brusselator and the Oregonator.- 9.5 Stochastic Model for a Chemical Reaction Without Diffusion. Birth and Death Processes. One Variable.- 9.6 Stochastic Model for a Chemical Reaction with Diffusion. One Variable.- 9.7 Stochastic Treatment of the Brusselator Close to Its Soft-Mode Instability.- 9.8 Chemical Networks.- 10. Applications to Biology.- 10.1 Ecology, Population-Dynamics.- 10.2 Stochastic Models for a Predator-Prey System.- 10.3 A Simple Mathematical Model for Evolutionary Processes.- 10.4 A Model for Morphogenesis.- 10.5 Order Parameters and Morphogenesis.- 10.6 Some Comments on Models of Morphogenesis.- 11. Sociology and Economics.- 11.1 A Stochastic Model for the Formation of Public Opinion.- 11.2 Phase Transitions in Economics.- 12. Chaos.- 12.1 What is Chaos?.- 12.2 The Lorenz Model. Motivation and Realization.- 12.3 How Chaos Occurs.- 12.4 Chaos and the Failure of the Slaving Principle.- 12.5 Correlation Function and Frequency Distribution.- 12.6 Discrete Maps, Period Doubling, Chaos, Intermittency.- 13. Some Historical Remarks and Outlook.- References, Further Reading, and Comments.