Radon-Nikodym Derivatives of Gaussian Measures

We give simple necessary and sufficient conditions on the mean and covariance for a Gaussian measure to be equivalent to Wiener measure. This was formerly an unsolved problem [26]. An unsolved problem is to obtain the Radom-Nikodym derivative dμ/dν where μ and ν are equivalent Gaussian measure [28]. We solve this problem for many cases of μ and ν, by writing dμ/dν in terms of Fredholm determinants and resolvents. The problem is thereby reduced to the calculation of these classical quantities, and explicit formulas can often be given. Our method uses Wiener measure μw as a catalyst; that is, we compute derivatives with respect to μw and then use the chain rule: dμ/dν = (dμ/dμw) / (dν/dμw). Wiener measure is singled out because it has a simple distinctive property--the Wiener process has a random Fourier-type expansion in the integrals of any complete orthonormal system. We show that any process equivalent to the Wiener process W can be realized by a linear transformation of W. This transformation necessarily involves stochastic integration and generalizes earlier nodulation transformations studied by Legal [21] and others [4], [27]. New variants of the Wiener process are introduced, both conditioned Wiener processes and free n-fold integrated Wiener processes. We given necessary and sufficient conditions for a Gaussian process to be equivalent to any one of the variants and also give the corresponding Radon-Niels (R-N) derivative. Last, some novel uses of R-N derivatives are given. We calculate explicitly: (i) the probability that W cross a slanted line in a finite time, (ii) the first passage probability for the process W (T + 1) − W(t), and (iii) a class of function space integrals. Using (iii) we prove a zero-one law for convergence of certain integrals on Wiener paths

First Passage Time for a Particular Gaussian Process

We find an explicit formula for the first passage probability, Qa(T|x) = Pr(S(t) \u3c a, 0 ≦ t ≦ T | S(0) = x), for all T \u3e 0, where S is the Gaussian process with mean zero and covariance ES(τ)S(t) = max (1-| t - τ|, 0). Previously, Qa(T | x) was known only for T ≦ 1. In particular for T = n an integer and - ∞ \u3c x \u3c a \u3c ∞, Qa(T | x) = 1⁄φ(x) ∫D . . . ∫ det φ(yi - yj+1 + a) dy2 . . . dyn+1, where the integral is a n-fold integral of y2, . . . , yn+1 over the region D given by D = {a - x \u3c y2 \u3c y1 \u3c . . . n+1} and the determinant is of size (n + 1)x(n + 1), 0 \u3c i, j ≦ n, with y0 ≡ 0, y1 ≡ a - x

A Model for Stock Price Fluctuations Based on Information

The author presents a new model for stock price fluctuations based on a concept of information. In contrast, the usual Black-Scholes-Merton-Samuelson (1965, 1973) model is based on the explicit assumption that information is uniformly held by everyone and plays no role in stock prices. The new model is based on the evident nonuniformity of information in the market and the evident time delay until new information becomes generally known. A second contribution of the paper is to present some problems with explicit solutions which are of value in obtaining insights. Several problems of mathematical interest are compared in order to better understand which optimal stopping problems have explicit solution

A View Of Reconstruction Tomography: XCT, ECT, NMRCT

I survey the present status of three subfields of reconstruction tomography, X-ray CT, emission CT, and magnetic resonance CT, and mention some new results and insights, as well as open problems. This is for my Frontier in Imaging Science lecture at the IEEE Nuclear Science International Workshop on Physics and Engineering of Computerized Multidimensional Imaging and Processing, April 2-4, 1986, Irvine, CA

The FKG Inequality and Some Monotonicity Properties of Partial Orders

Let (a1 , . . . , am, b1, . . . , bn) be a random permutation of 1, 2, . . ., m + n. Let P be a partial order on the a’s and b’s involving only inequalities of the form ai \u3c aj or bi \u3c bj, and let P\u27 be an extension of P to include inequalities of the form ai \u3c bj; i.e, P\u27 = P ∪ P\u27\u27, where P\u27\u27 involves only inequalities of the form ai \u3c bj. We prove the natural conjecture of R. L. Graham, A. C. Yao, and F. F. Yao [SIAM J. Alg. Discr. Meth. 1 (1980), pp. 251–258] that in particular (*) Pr (a1 \u3c b1|P\u27) ≥ Pr (a1 \u3c b1|P). We give a simple example to show that the more general inequality (*) where P is allowed to contain inequalities of the form ai \u3c bj is false. This is surprising because as Graham, Yao, and Yao proved, the general inequality (*) does hold if P totally orders both the a’s and the b’s separately. We give a new proof of the latter result. Our proofs are based on the FKG inequality

A Conversation With Yuri Vasilyevich Prokhorov

Yuri Vasilyevich Prokhorov was born on December 15, 1929. He graduated from Moscow University in 1949 and worked at the Mathematical Institute of the Academy of Sciences from 1952, and as a Professor on the faculty of Moscow University since 1957. He became a corresponding member of the Academy in 1966 and an Academician in 1972. He received the Lenin Prize in 1970. The basic directions of his research are the theory of probability and mathematical methods in theoretical physics. He developed asymptotic methods in the theory of probability. In the area of the classical limit theorems, he studied the conditions of applicability of the strong law of large numbers and the so-called local limit theorems for sums of independent random variables. He proposed new methods for studying limit theorems for random processes; these methods were based on studying the convergence of measures in function space. He applied these methods to establish the limiting transition from discrete processes to continuous ones. He found (in 1953 and 1956) necessary and sufficient conditions for weak convergence in function space. He has several papers on mathematical statistics, on queuing theory and also on the theory of stochastic control. This conversation took place at the Steklov Institute in early September 1990. It was taped in Russian and translated by Abram Kagan. The final version was edited by Ingram Olkin