TABLE OF CONTENTS This appendix contains derivations of the partial derivatives stated in Chapter 9, Section I. In addition, a number of useful results applicable to probability densities are also derived. These results, which are provided here for ease of reference, are available in a variety other sources (e.g., Cox and Rubinstein, 1985, chap. 5; Stoll and Whalley, 1993, chap. 11). In the following, N[ x, y] refers to a normal density with mean x and variance y. N[ x] refers to a cumulative normal distribution function evaluated at x, and n[ x] ? N ?[ x] is the normal density function evaluated at X.
Chapter 8 makes use of a result concerning the standard deviation of a sum of standard normal random variables. More precisely, the discrete random walk was given the form:
where Z(1), Z(2), Z(3), form a stochastic process of independent random variables with the standard normal probability distribution: Z( t) ~ N[0,1]. This requires the Z( t) to be identically, independently distributed (iid) random variables. Over any time interval 0 to T, the variance of ? X( t) can be evaluated by determining the variance:
Hence, when ? t = T, the Z is N[0 ,T]. Now, consider what happens when the time interval ? t shrinks. Because Z
