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STOCHASTIC SET-BACKS

David Stirzaker

in Combinatorics, Complexity, and Chance: A Tribute to Dominic Welsh

This chapter examines a random process (X(t):t ≥ 0) taking values in R, that is governed by the events of an independent renewal process N(t), as follows: whenever an event of N(t) occurs, the ... More

This chapter examines a random process (X(t):t ≥ 0) taking values in R, that is governed by the events of an independent renewal process N(t), as follows: whenever an event of N(t) occurs, the process X(t) is restarted and runs independently of the past with initial value that has the same distribution as X(0). The case when each segment of the process between consecutive events of N(t) is a diffusion is studied, and expressions for the characteristic function of X(t) and its stationary distribution as t → ∞ are presented. An expression is derived for the expected first-passage time of X(t) to any value a, and several explicit examples of interest are considered. The chapter presents two approaches: first, it uses Wald's equation which supplies the mean in quite general circumstances; second, it explores possibilities for use of the moment-generating function of the first-passage time.Less

The Mathematics of the Martingale Approach

Tomas Björk

in Arbitrage Theory in Continuous Time

This chapter presents the two main workhorses of the martingale approach to arbitrage theory: the Martingale Representation Theorem and the Girsanov Theorem. The Martingale Representation Theorem ... More

This chapter presents the two main workhorses of the martingale approach to arbitrage theory: the Martingale Representation Theorem and the Girsanov Theorem. The Martingale Representation Theorem shows that in a Wiener world, every martingale can be written as a stochastic integral w.r.t, the underlying Wiener process. The Girsanov Theorem gives complete control of all absolutely continuous measure transformations in a Wiener world. Practice exercises are included.Less

Introduction and Overview

Anindya Banerjee, Juan J. Dolado, John W. Galbraith, and David F. Hendry

in Co-integration, Error Correction, and the Econometric Analysis of Non-Stationary Data

Serves as an introductory overview for the rest of the book, and outlines its main aims. As a basis for the following chapters, an overview and clarification of equilibrium relationships in economic ... More

Serves as an introductory overview for the rest of the book, and outlines its main aims. As a basis for the following chapters, an overview and clarification of equilibrium relationships in economic theory is presented. A preliminary discussion of testing for orders of integration and the estimation of long‐run relationships is provided. The chapter summarizes key concepts from time‐series analysis and the theory of stochastic processes and, in particular, the theory of Brownian motion processes. Several empirical examples are offered as illustration of these concepts.Less

Markov processes in continuous time and space

Eric Renshaw

in Stochastic Population Processes: Analysis, Approximations, Simulations

Having dealt with processes in discrete space, and discrete and continuous time, this chapter investigates Markov processes in continuous space and time. It follows a natural progression through the ... More

Having dealt with processes in discrete space, and discrete and continuous time, this chapter investigates Markov processes in continuous space and time. It follows a natural progression through the Wiener process, the Fokker–Planck diffusion equation and the Ornstein–Uhlenbeck process, and tracks the relationships between them. When barriers are introduced theoretical development is not straightforward, so care needs to be taken when constructing solutions.Less

Testing for a Unit Root

Anindya Banerjee, Juan J. Dolado, John W. Galbraith, and David F. Hendry

in Co-integration, Error Correction, and the Econometric Analysis of Non-Stationary Data

Methods of testing for a unit root in an observed series are described in this chapter. Both parametric regression tests and non‐parametric adjustments to these test statistics are considered, and ... More

Methods of testing for a unit root in an observed series are described in this chapter. Both parametric regression tests and non‐parametric adjustments to these test statistics are considered, and tables of critical values for commonly used tests are given. The chapter also uses functionals of Wiener processes to describe the asymptotic distributions of important test statistics.Less

Models in Finance

GREGORY C. CHOW

in Dynamic Economics: Optimization by the Lagrange Method

Using stochastic differential equations instead of utilizing stochastic difference equations, most of the models involved in finance follow Merton’s work and are developed in continuous time. In this ... More

Using stochastic differential equations instead of utilizing stochastic difference equations, most of the models involved in finance follow Merton’s work and are developed in continuous time. In this chapter, an alternative stochastic differential equation is introduced to replace the vector for state variables. The chapter introduces the Wiener process in which a change in the time variable is perceived to be normally distributed and with zero mean. After illustrating how dynamic programming is employed in a model that involves continuous time, we look into the illustration included in this chapter about how to solve such problems using the method of Lagrange multipliers. We also attempt to examine the optimal control function, optimum consumption, and other issues such as capital asset pricing in the event of shifts in investments.Less

Stochastic Integrals

Tomas Björk

in Arbitrage Theory in Continuous Time

We introduce the Wiener process, the Itô stochastic integral, and derive the Itô formula. The connection with martingale theory is discussed, and there are several worked-out examples

Large Deviations Principle for Diffusions

Gopinath Kallianpur and P. Sundar

in Stochastic Analysis and Diffusion Processes

Large deviations theory formulated by Varadhan has made a tremendous impact in a variety of fields such as mathematical physics, control theory, and statistics, to name a few. After a brief ... More

Large deviations theory formulated by Varadhan has made a tremendous impact in a variety of fields such as mathematical physics, control theory, and statistics, to name a few. After a brief discussion of the general theory and examples, the large deviations principle (LDP) is shown to be equivalent to the Laplace principle in our context. The rate function for the LVP is obtained, in general, via relative entropy. Next, the Boué-Dupuis representation theorem for positive functionals of a Wiener process is established. Using the representation theorem, the Laplace principle is proved for diffusions.Less

The Pricing of Stock Options

Lisa Borland

in Nonextensive Entropy: Interdisciplinary Applications

We describe how a stock price model based on nonextensive statistics can be used to derive a generalized theory for pricing stock options. A review of theoretical and empirical results is ... More

We describe how a stock price model based on nonextensive statistics can be used to derive a generalized theory for pricing stock options. A review of theoretical and empirical results is presented…. In 1973, Black and Scholes [1] and Merton [12] published their seminal papers which developed a theory of the fair price of options. Scholes and Merton were later to receive the 1997 Nobel prize for this famous work (Fisher Black had unfortunately passed away two years earlier). Options are important financial instruments which are traded in a huge volume all around the world on a variety of exchanges. There are options on underlying assets ranging from orange juice to gold, stocks to currency. In principle, an option is simply the right—but not the obligation—to execute some previously agreed upon action, for example, the right to buy or sell the underlying asset at some predetermined price, called the strike. It is not difficult to understand that the existence of such instruments could be extremely useful—for example, the right to buy an asset at a certain price protects against unforeseen events which could lead to huge price rises and thereby losses to someone who knows that they will need the asset at some time in the future. Similarly, the right to sell the asset at a certain price will protect against unforeseen drops in its value. These examples illustrate the use of options to hedge oneself against possible future events. Another use is more speculative: If a trader believes that the price of a stock will rise above a certain price at some date in the future, then it is in his interest to secure an option to buy the stock at some fixed lower price. Then, if the price of the stock does rise above that price, the trader can execute his option, just to turn around and resell the stock again at the higher market price. Less

Stochastic Equations Of Motion

Abraham Nitzan

in Chemical Dynamics in Condensed Phases: Relaxation, Transfer and Reactions in Condensed Molecular Systems

We have already observed that the full phase space description of a system of N particles (taking all 6N coordinates and velocities into account) requires the solution of the deterministic Newton ... More

We have already observed that the full phase space description of a system of N particles (taking all 6N coordinates and velocities into account) requires the solution of the deterministic Newton (or Schrödinger) equations of motion, while the time evolution of a small subsystem is stochastic in nature. Focusing on the latter, we would like to derive or construct appropriate equations of motion that will describe this stochastic motion. This chapter discusses some methodologies used for this purpose, focusing on classical mechanics as the underlying dynamical theory. In Chapter 10 we will address similar issues in quantum mechanics. The time evolution of stochastic processes can be described in two ways: 1. Time evolution in probability space. In this approach we seek an equation (or equations) for the time evolution of relevant probability distributions. In the most general case we deal with an infinite hierarchy of functions, P(zntn; zn−1tn−1; . . . ; z1t1) as discussed in Section 7.4.1, but simpler cases exist, for example, for Markov processes the evolution of a single function, P(z, t; z0t0), fully characterizes the stochastic dynamics. Note that the stochastic variable z stands in general for all the variables that determine the state of our system. 2. Time evolution in variable space. In this approach we seek an equation of motion that describes the evolution of the stochastic variable z(t) itself (or equations of motion for several such variables). Such equations of motions will yield stochastic trajectories z(t) that are realizations of the stochastic process under study. The stochastic nature of these equations is expressed by the fact that for any initial condition z0 at t = t0 they yield infinitely many such realizations in the same way that measurements of z(t) in the laboratory will yield different such realizations. Two routes can be taken to obtain such stochastic equations of motions, of either kind: 1. Derive such equations from first principles. In this approach, we start with the deterministic equations of motion for the entire system, and derive equations of motion for the subsystem of interest. The stochastic nature of the latter stems from the fact that the state of the complementary system, “the rest of the world,” is not known precisely, and is given only in probabilistic terms. Less

Distributions of High-Frequency Stock Market Observables

Roberto Osorio, Lisa Borland, and Constantino Tsallis

in Nonextensive Entropy: Interdisciplinary Applications

Power laws and scaling are two features that have been known for some time in the distribution of returns (i.e., price fluctuations), and, more recently, in the distribution of volumes (i.e., ... More

Power laws and scaling are two features that have been known for some time in the distribution of returns (i.e., price fluctuations), and, more recently, in the distribution of volumes (i.e., numbers of shares traded) of financial assets. As in numerous examples in physics, these power laws can be understood as the asymptotic behavior of distributions that derive from nonextensive thermostatistics. Recent applications of the (Q-Gaussian distribution to returns of exchange rates and stock indices are extended here for individual U.S. stocks over very small time intervals and explained in terms of a feedback mechanism in the dynamics of price formation. In addition, we discuss some new empirical findings for the probability density of low volumes and show how the overall volume distribution is described by a function derived from q-exponentials. In March 1900 at the Sorbonne, a 30-year-old student—who had studied under Poincaré—submitted a doctoral thesis [2] that demonstrated an intimate knowledge of trading operations in the Paris Bourse. He proposed a probabilistic method to value some options on rentes, which were then the standard French government bonds. His work was based on the idea that rente prices evolved according to a random-walk process that resulted in a Gaussian distribution of price differences with a dispersion proportional to the square root of time. Although the importance of Louis Bachelier's accomplishment was not recognized by his contemporaries [24], it preceded by five years Einstein's famous independent, but mathematically equivalent, description of diffusion under Brownian motion. The idea of a Gaussian random-walk process (later preferably applied to logarithmic prices) eventually became one of the basic tenets of most twentieth-century quantitative works in finance, including the Black-Scholes [3] complete solution to the option-valuation problem—of which a special case had been solved by Bachelier in his thesis. In the times of the celebrated Black-Scholes solution, however, a change in perspective was already under way. Starting with the groundbreaking works of Mandelbrot [18] and Fama [11], it gradually became apparent that probability distribution functions of price changes of assets (including commodities, stocks, and bonds), indices, and exchange rates do not follow Bachelier's principle of Gaussian (or "normal") behavior. Less

Manifolds, Active Models, And Deformable Templates

Ulf Grenander and Michael I. Miller

in Pattern Theory: From representation to inference

To study shape we introduce manifolds and submanifolds examined in the continuum as the generators. Transformations are constructed which are built from the matrix ... More

To study shape we introduce manifolds and submanifolds examined in the continuum as the generators. Transformations are constructed which are built from the matrix groups and infinite products. This gives rise to many of the widely used structural models in image analysis often termed active models, essentially the deformable templates. These deformations are studied as both diffeomorphisms as well as immersions. A calculus is introduced based on transport theory for activating these deformable shapes by taking variations with respect to the matrix groups parameterizing them. Segmentation based on activating these manifolds is examined based on Gaussian random fields and variations with respect to the parameterizations. Less