Sasha Cyganowski

Maple for Stochastic Differential Equations

This chapter introduces the maple software package stochastic consisting of maple routines for stochastic calculus and stochastic differential equations and for constructing basic numerical methods for specific stochastic differential equations, with simple examples illustrating the use of the routines. A website address is given from which the software can be downloaded and where up to date information about installment, new developments and literature can be found.

MAPLE for Jump—Diffusion Stochastic Differential Equations in Finance

The occurrence of shocks in the financial market is well known and, since the 1976 paper of the Noble Prize laureate R.C. Merton, there have been numerous attempts to incorporated them into financial models. Such models often result in jump-diffusion stochastic differential equations. This chapter describes the use of MAPLE for such equations, in particular for the derivation of numerical schemes. It can be regarded as an addendum to the chapter in this book by [5], which can be referred to for general background and additional literature on stochastic differential equations and MAPLE. All the MAPLE code in this paper

Numerical Methods for SDEs

To begin we briefly recall some background material on the numerical solution of deterministic ordinary differential equations (ODEs) to provide an introduction to the corresponding ideas for SDEs. We then discuss the stochastic Euler scheme and its implementation in some detail before turning to higher order numerical schemes for scalar Ito SDEs. We also indicate how MAPLE can be used to automatically determine the coefficients of these numerical schemes, in particular for vector SDEs.

A Tour of Important Distributions

This chapter reviews a selection of the best known probability distributions, which are important from both a theoretical and a practical point of view. Two other probability distributions, needed for statistical inference, will be mentioned in Chapter 6.

Measure and Integral

This Chapter provides a short introduction to measure theory and the theory of the Lebesgue integral. Avoiding as much as possible technical details, we present the basic ideas and also list the basic properties of measure and integral that will be used in this book. Let us note here that probability P(A) defined in Chapter 1 is nothing else but the measure of event A. Another basic probability concept, the mathematical expectation, established later in the book is nothing else but the integral with respect to the probability measure. To some extent, calculus of probability, theory of stochastic processes (including stochastic differential equations) and mathematical statistics can be thought of as parts of the theory of measure and integration.

Random Variables and Distributions

Most quantities that we deal with every day are more or less of a random nature. The height of the person we first meet on leaving home in the morning, the grade we will earn in the next exam, the cost of a bottle of wine we will buy in the nearest shopping centre, and many more, serve as examples of so called random variables. Any such variable has its own specifications or characteristics. For example, the height of an adult male may take all values from 150 cm to 230 cm or even beyond this interval. Besides, the interval (175, 180) is more likely than other intervals of the same length, say (152, 157) or (216, 221). On the other hand, the grade we will earn during the coming exam will take finite many values: excellent, good, pass, fail, and for the reader some of these values are more likely than others.

Numerical Simulations and Statistical Inference

In this Chapter we present methods that are used for generating random or more precisely pseudo-random numbers. Then, we discuss briefly concepts and routines that are basic for statistical inference. Some of them are useful to test the quality of pseudo-random numbers and in statistical modeling.

Parameters of Probability Distributions

In most practical situations we are not able to fully determine a probability distribution of a random variable or vector under consideration. Nevertheless, we can often obtain partial information on the distribution that suffices for our purpose. The most basic information about the random variable is given by the mean or mathematical expectation and the variance or its square root, which is known as the standard deviation. Other parameters such as higher moments are also considered, but are not as significant so they will be not covered in this book. The substantial role of the mean and variation partially explains the Law of Large Numbers which states convergence of the averages of random variables to their common mean. We will thus discuss these laws as well as various types of convergence of random variables. We will complete this Chapter with a problem on the correlation of two or more random variables.