CONDITIONAL EXPECTATION AND MARTINGALES
CONDITIONAL EXPECTATION AND MARTINGALES 1. INTRODUCTION Martingales play a role in stochastic processes roughly similar to that played by conserved quantities in dynamical systems. Unlike a conserved quantity in dynamics, which remains constant in time, a martingale’s value can change; however, its expectation remains constant in time.
Download CONDITIONAL EXPECTATION AND MARTINGALES
Information
Domain:
Source:
Link to this page:
Documents from same domain
Prologue - University of Chicago
galton.uchicago.eduLECTURE 5: BROWNIAN MOTION 1. Prologue We have seen in previous lectures that, for discrete multiperiod markets which admit no …
Bernoulli Distribution - University of Chicago
galton.uchicago.eduBernoulli Distribution Example: Toss of coin Deflne X = 1 if head comes up and X = 0 if tail comes up. Both realizations are equally likely: (X = 1) = (X = 0) = 1 2
A Review of Methods for Missing Data - University …
galton.uchicago.eduEducational Research and Evaluation 1380-3611/01/0704-353$16.00 2001, Vol. 7, No. 4, pp. 353–383 # Swets & Zeitlinger A Review of Methods for Missing Data …
Department of Statistics, University of Chicago
galton.uchicago.eduDepartment of Statistics, Columbia University PER A. MYKLAND Department of Statistics, University of Chicago We propose a methodology for evaluating the hedging errors of derivative securities due to the discreteness of trading times or the observation times of market prices, or
MARKOV CHAINS: BASIC THEORY - University of Chicago
galton.uchicago.eduMARKOV CHAINS: BASIC THEORY 3 Definition 2. A nonnegative matrix is a matrix with nonnegative entries. A stochastic matrix is a square nonnegative matrix all of whose row sums are 1. A substochastic matrix is a square nonnegative matrix all of whose row sums are 1.
CONVERGENCE RATES OF MARKOV CHAINS
galton.uchicago.eduMarkov chains for which the convergence rate is of particular interest: (1) the random-to-top shuffling model and (2) the Ehrenfest urn model. Along the way we will encounter a number of fundamental concepts and techniques, notably reversibility, total variation distance, and
Chapter 3. Multivariate Distributions.
galton.uchicago.edu3-1 Chapter 3. Multivariate Distributions. ... structure to include multivariate distributions, the probability distributions of pairs of random variables, triplets of random variables, and so forth. We will begin with the simplest such situation, that of pairs of ... describes a surface in 3-dimensional space, and the probability that (X;Y) ...
ONE-DIMENSIONAL RANDOM WALKS
galton.uchicago.edupost- y process is just an independent simple random walk started at y. But (10) (with the roles of x,y reversed) implies that this random walk must eventually visit x. When this happens, the random walk restarts again, so it will go back to y, and so on. Thus, by an easy induction argu-ment (see Corollary 14 below): Theorem 4.
BROWNIAN MOTION - Department of Statistics
galton.uchicago.eduMany stochastic processes behave, at least for long stretches of time, like random walks with small but frequent jumps. The argument above suggests that such processes will look, at least approximately, and on the appropriate time scale, like Brownian motion. Second, it suggests that many important “statistics” of the random walk will have lim-
CONDITIONAL EXPECTATION AND MARTINGALES
galton.uchicago.eduadapted sequence of integrable real-valued random variables, that is, a sequence with the prop-erty that for each n the random variable Xn is measurable relative to Fn and such that EjXnj˙ 1. The sequence X0,X1,... is said to be a martingale relative to the filtration {Fn}n‚0 if it is adapted and if for every n, (1) E(Xn¯1 jFn) ˘ Xn.
Related documents
UNITED STATES DISTRICT COURT WESTERN DISTRICT OF …
www.vawd.uscourts.govabsence, with the next available active distri ct judge in seniority, for direction as to . 4 the proper division for filing. Superseding indictments in any case shall be filed in the division in which the existing indictment is filed. Nothing in this rule shall
1 Sufficient statistics - University of Arizona
www.math.arizona.educonditional distribution. But then his random sample has the same distri-bution as a random sample drawn from the population (with its unknown value of θ). So statistician B can use his random sample X0 1,···,X0 n to com-pute whatever statistician A computes using his random sample X1,···,Xn, and he will (on average) do as well as ...
Reading 5b: Continuous Random Variables
ocw.mit.eduIn practice we often say ‘X has distribution F (x)’ rather than ‘X has cumulative distri bution function F (x).’ Example 5. Find the cumulative distribution function for the density in Example 2.
Reading 10b: Maximum Likelihood Estimates
ocw.mit.eduSuppose that the lifetime of Badger brand light bulbs is modeled by an exponential distri-bution with (unknown) parameter . We test 5 bulbs and nd they have lifetimes of 2, 3, 1, 3, and 4 years, respectively. What is the MLE for ? answer: We need to be careful with our notation. With ve di erent values it is best to
1 Inverse Transform Method - Columbia University
www.columbia.eduis the counting process of a Poisson process at rate , then N(1) has a Poisson distri-bution with mean . Thus if we can simulate N(1), then we can set X= N(1) and we are done. Let Y = N(1) + 1, and let t n = X 1 + + X n denote the nth point of the Poisson process; the X i are iid with an exponential distribution at rate . Note that Y = minfn 1 : t