DOCSLIB.ORG

Lecture Notes 6 Random Processes • Definition and Simple Examples

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Lecture Notes 6 Random Processes • Definition and Simple Examples Lecture Notes 6 Random Processes • Definition and Simple Examples • Important Classes of Random Processes ◦ IID ◦ Random Walk Process ◦ Markov Processes ◦ Independent Increment Processes ◦ Counting processes and Poisson Process • Mean and Autocorrelation Function • Gaussian Random Processes ◦ Gauss–Markov Process EE 278: Random Processes Page 6–1 Random Process • A random process (RP) (or stochastic process) is an infinite indexed collection of random variables {X(t): t ∈ T }, defined over a common probability space • The index parameter t is typically time, but can also be a spatial dimension • Random processes are used to model random experiments that evolve in time: ◦ Received sequence/waveform at the output of a communication channel ◦ Packet arrival times at a node in a communication network ◦ Thermal noise in a resistor ◦ Scores of an NBA team in consecutive games ◦ Daily price of a stock ◦ Winnings or losses of a gambler EE 278: Random Processes Page 6–2 Questions Involving Random Processes • Dependencies of the random variables of the process ◦ How do future received values depend on past received values? ◦ How do future prices of a stock depend on its past values? • Long term averages ◦ What is the proportion of time a queue is empty? ◦ What is the average noise power at the output of a circuit? • Extreme or boundary events ◦ What is the probability that a link in a communication network is congested? ◦ What is the probability that the maximum power in a power distribution line is exceeded? ◦ What is the probability that a gambler will lose all his captial? • Estimation/detection of a signal from a noisy waveform EE 278: Random Processes Page 6–3 Two Ways to View a Random Process • A random process can be viewed as a function X(t, ω) of two variables, time t ∈T and the outcome of the underlying random experiment ω ∈ Ω ◦ For fixed t, X(t, ω) is a random variable over Ω ◦ For fixed ω, X(t, ω) is a deterministic function of t, called a sample function X(t, w1) t X(t, w2) t X(t, w3) t t1 t2 X(t1, w) X(t2, w) EE 278: Random Processes Page 6–4 Discrete Time Random Process • A random process is said to be discrete time if T is a countably infinite set, e.g., ◦ N = {0, 1, 2,...} ◦ Z = {..., −2, −1, 0, +1, +2,...} • In this case the process is denoted by Xn, for n ∈ N , a countably infinite set, and is simply an infinite sequence of random variables • A sample function for a discrete time process is called a sample sequence or sample path • A discrete-time process can comprise discrete, continuous, or mixed r.v.s EE 278: Random Processes Page 6–5 Example n • Let Z ∼ U[0, 1], and define the discrete time process Xn = Z for n ≥ 1 • Sample paths: x 1 n 2 1 1 Z = 2 4 1 8 1 16 n xn 1 1 Z = 4 4 1 16 1 64 n xn Z = 0 00 0 ... n 1234567 ... EE 278: Random Processes Page 6–6 n • First-order pdf of the process: For each n, Xn = Z is a r.v.; the sequence of pdfs of Xn is called the first-order pdf of the process xn 1 z 0 1 Since Xn is a differentiable function of the continuous r.v. Z, we can find its pdf as 1 1 1 f (x)= = xn−1 , 0 ≤ x ≤ 1 Xn nx(n−1)/n n EE 278: Random Processes Page 6–7 Continuous Time Random Process • A random process is continuous time if T is a continuous set • Example: Sinusoidal Signal with Random Phase X(t)= α cos(ωt + Θ) , t ≥ 0 where Θ ∼ U[0, 2π] and α and ω are constants • Sample functions: x(t) α Θ = 0 π π 3π 2π t 2ω ω 2ω ω x(t) Θ = π 4 t x(t) Θ = π 2 t EE 278: Random Processes Page 6–8 • The first-order pdf of the process is the pdf of X(t)= α cos(ωt + Θ). In an earlier homework exercise, we found it to be 1 fX(t)(x)= , −α<x< +α απ 1 − (x/α)2 The graph of the pdf is shownp below fX(t)(x) 1 απ x −α 0 +α Note that the pdf is independent of t. (The process is stationary) EE 278: Random Processes Page 6–9 Specifying a Random Process • In the above examples we specified the random process by describing the set of sample functions (sequences, paths) and explicitly providing a probability measure over the set of events (subsets of sample functions) • This way of specifying a random process has very limited applicability, and is suited only for very simple processes • A random process is typically specified (directly or indirectly) by specifying all its n-th order cdfs (pdfs, pmfs), i.e., the joint cdf (pdf, pmf) of the samples X(t1),X(t2),...,X(tn) for every order n and for every set of n points t1, t2,...,tn ∈T • The following examples of important random processes will be specified (directly or indirectly) in this manner EE 278: Random Processes Page 6–10 Important Classes of Random Processes • IID process: {Xn : n ∈ N } is an IID process if the r.v.s Xn are i.i.d. Examples: ◦ Bernoulli process: X1,X2,...,Xn,... i.i.d. ∼ Bern(p) ◦ Discrete-time white Gaussian noise (WGN): X1,...,Xn,... i.i.d. ∼ N (0, N) • Here we specified the n-th order pmfs (pdfs) of the processes by specifying the first-order pmf (pdf) and stating that the r.v.s are independent • It would be quite difficult to provide the specifications for an IID process by specifying the probability measure over the subsets of the sample space EE 278: Random Processes Page 6–11 The Random Walk Process • Let Z1,Z2,...,Zn,... be i.i.d., where 1 +1 with probability 2 Zn = 1 (−1 with probability 2 • The random walk process is defined by X0 = 0 n Xn = Zi , n ≥ 1 i=1 X • Again this process is specified by (indirectly) specifying all n-th order pmfs • Sample path: The sample path for a random walk is a sequence of integers, e.g., 0, +1, 0, −1, −2, −3, −4,... or 0, +1, +2, +3, +4, +3, +4, +3, +4,... EE 278: Random Processes Page 6–12 Example: xn 3 2 1 n 1 2 3 4 5 6 7 8 9 10 −1 −2 −3 zn : 1 1 1 −1 1 −1 −1 −1 −1 −1 • First-order pmf: The first-order pmf is P{Xn = k} as a function of n. Note that k ∈ {−n, −(n − 2),..., −2, 0, +2,..., +(n − 2), +n} for n even k ∈ {−n, −(n − 2),..., −1, +1, +3,..., +(n − 2), +n} for n odd Hence, P{Xn = k} = 0 if n + k is odd, or if k < −n or k > n EE 278: Random Processes Page 6–13 Now for n + k even, let a be the number of +1’s in n steps, then the number of −1’s is n − a, and we find that n + k k = a − (n − a) = 2a − n ⇒ a = 2 Thus 1 P{Xn = k} = P{2(n + k) heads in n independent coin tosses} n −n = n+k · 2 for n + k even and −n ≤ k ≤ n 2 P{Xn = k}, n even kk ... −6 −4 −2 0 +2 +4 +6 ... EE 278: Random Processes Page 6–14 Markov Processes • A discrete-time random process Xn is said to be a Markov process if the process future and past are conditionally independent given its present value • Mathematically this can be rephrased in several ways. For example, if the r.v.s {Xn : n ≥ 1} are discrete, then the process is Markov iff n−1 n x pXn+1|X (xn+1|xn, )= pXn+1|Xn(xn+1|xn) for every n • IID processes are Markov • The random walk process is Markov. To see this consider n n n n P{Xn+1 = xn+1 | X = x } = P{Xn + Zn+1 = xn+1 | X = x } = P{Xn + Zn+1 = xn+1 | Xn = xn} = P{Xn+1 = xn+1 | Xn = xn} EE 278: Random Processes Page 6–15 Independent Increment Processes • A discrete-time random process {Xn : n ≥ 0} is said to be independent increment if the increment random variables Xn1, Xn2 − Xn1, ..., Xnk − Xnk−1 are independent for all sequences of indices such that n1 < n2 < ··· < nk • Example: Random walk is an independent increment process because n1 n2 nk Xn1 = Zi , Xn2−Xn1 = Zi , ..., Xnk−Xnk−1 = Zi i=1 i=n +1 i=n +1 X X1 Xk−1 are independent because they are functions of independent random vectors • The independent increment property makes it easy to find the n-th order pmfs of a random walk process from knowledge only of the first-order pmf EE 278: Random Processes Page 6–16 • Example: Find P{X5 = 3, X10 = 6, X20 = 10} for random walk process {Xn} Solution: We use the independent increment property as follows P{X5 = 3, X10 = 6, X20 = 10} = P{X5 = 3, X10 − X5 = 3, X20 − X10 = 4} = P{X5 = 3}P{X5 = 3}P{X10 = 4} 5 5 10 = 2−5 2−5 2−10 = 3000 · 2−20 4 4 7 • In general if a process is independent increment, then it is also Markov. To see this let Xn be an independent increment process and define n T ∆X =[X1, X2 − X1,...,Xn − Xn−1] Then n n n n x X x pXn+1|X (xn+1 | ) = P{Xn+1 = xn+1 | = } n n = P{Xn+1 − Xn + Xn = xn+1 | ∆X = ∆x , Xn = xn} = P{Xn+1 = xn+1 | Xn = xn} • The converse is not necessarily true, e.g., IID processes are Markov but not independent increment EE 278: Random Processes Page 6–17 • The independent increment property can be extended to continuous-time processes: A process X(t), t ≥ 0, is said to be independent increment if X(t1), X(t2) − X(t1),..., X(tk) − X(tk−1) are independent for every 0 ≤ t1 < t2 <...<tk and every k ≥ 2 • Markovity can also be extended to continuous-time processes: A process X(t) is said to be Markov if X(tk+1) and (X(t1),...,X(tk−1)) are conditionally independent given X(tk) for every 0 ≤ t1 < t2 <...<tk < tk+1 and every k ≥ 2 EE 278: Random Processes Page 6–18 Counting Processes and Poisson Process • A continuous-time random process N(t), t ≥ 0, is said to be a counting process if N(0) = 0 and N(t)= n, n ∈ {0, 1, 2,...}, is the number of events from 0 to t (hence N(t2) ≥ N(t1) for every t2 > t1 ≥ 0) Sample path of a counting process: n(t) 5 4 3 2 1 0 t 0 t1 t2 t3 t4 t5 t1, t2,..
Load more

Recommended publications
  • Online Discrimination of Nonlinear Dynamics with Switching Differential Equations
    Online Discrimination of Nonlinear Dynamics with Switching Differential Equations Sebastian Bitzer, Izzet B. Yildiz and Stefan J. Kiebel Max Planck Institute for Human Cognitive and Brain Sciences, Leipzig fbitzer,yildiz,[email protected] Abstract How to recognise whether an observed person walks or runs? We consider a dy- namic environment where observations (e.g. the posture of a person) are caused by different dynamic processes (walking or running) which are active one at a time and which may transition from one to another at any time. For this setup, switch- ing dynamic models have been suggested previously, mostly, for linear and non- linear dynamics in discrete time. Motivated by basic principles of computations in the brain (dynamic, internal models) we suggest a model for switching non- linear differential equations. The switching process in the model is implemented by a Hopfield network and we use parametric dynamic movement primitives to represent arbitrary rhythmic motions. The model generates observed dynamics by linearly interpolating the primitives weighted by the switching variables and it is constructed such that standard filtering algorithms can be applied. In two experi- ments with synthetic planar motion and a human motion capture data set we show that inference with the unscented Kalman filter can successfully discriminate sev- eral dynamic processes online. 1 Introduction Humans are extremely accurate when discriminating rapidly between different dynamic processes occurring in their environment. For example, for us it is a simple task to recognise whether an observed person is walking or running and we can use subtle cues in the structural and dynamic pattern of an observed movement to identify emotional state, gender and intent of a person.
    [Show full text]
  • CPY Document
    Lecture i: Overview 1 Diffusion by discontinous movements In 1827 Robert Brown, observed that suspended pollen grains are in an uninterrupted and irregular "swarming" motion. Brown was a botanist and at first he believed that only organic materials exhibited this agitation. But very soon he extended his observations to particles of inorganic material, such as a ground-up fragment of the Sphin. Through the nineteenth century there was a intermittent discussion concerning the cause of this Brownian motion, and in 1877 DelsailX suggested that the impact of molecules on a macroscopic particle produces observable displacements. In 1905, after nearly a century of debate, Einstein definitively explained this phenomenon (6,7). 1.1 Einstein's derivation of the diffusion equation Our interest here is in Einstein's derivation of the dision equation, which is very different from that of Fourier. We consider one-dimensional Brownian motion by projecting the location of the particle onto a straight line which we call the x-axs. Einstein's assumptions are the following: (i) the particles move independently of one another; (ii) we observe particle positions at time intervals 7 which are much greater than the time intervals between molecular collsions. As a result, the motion in one interval is independent of what happened in the previous interval. In the interval 7 each paricle has a random displacement ß along the x-axs. The probability density function (PDF) of ß is ø(ß). This means that if we observe N ).). 1 paricles for a time 7 then the number of particles which are displaced through a distance which lies between ß and ß + dß is dN = N ø(ß) dß.
    [Show full text]
  • Statistical Parameter Inference of Bacterial Swimming Strategies
    Mathematisch-Naturwissenschaftliche Fakultät Maximilian Seyrich | Zahra Alirezaeizanjani | Carsten Beta | Holger Stark Statistical parameter inference of bacterial swimming strategies Suggested citation referring to the original publication: New Journal of Physics 20 (2018) 103033 DOI https://doi.org/10.1088/1367-2630/aae72c ISSN (online) 1367-2630 Postprint archived at the Institutional Repository of the Potsdam University in: Postprints der Universität Potsdam Mathematisch-Naturwissenschaftliche Reihe ; 914 ISSN 1866-8372 https://nbn-resolving.org/urn:nbn:de:kobv:517-opus4-446214 DOI https://doi.org/10.25932/publishup-44621 New J. Phys. 20 (2018) 103033 https://doi.org/10.1088/1367-2630/aae72c PAPER Statistical parameter inference of bacterial swimming strategies OPEN ACCESS Maximilian Seyrich1 , Zahra Alirezaeizanjani2 , Carsten Beta2 and Holger Stark1,3 RECEIVED 1 Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstrasse 36, D-10623 Berlin, Germany 22 May 2018 2 Institut für Physik und Astronomie, Universität Potsdam, Karl-Liebknecht-Strasse 24/25, D-14476 Potsdam, Germany REVISED 3 Author to whom any correspondence should be addressed. 10 September 2018 ACCEPTED FOR PUBLICATION E-mail: [email protected] and [email protected] 10 October 2018 Keywords: E.coli, run and tumble, chemotaxis, stochastic processes, bacterial swimming strategies, parameter inference PUBLISHED 25 October 2018 Original content from this Abstract work may be used under We provide a detailed stochastic description of the swimming motion of an E.coli bacterium in two the terms of the Creative Commons Attribution 3.0 dimension, where we resolve tumble events in time. For this purpose, we set up two Langevin licence. equations for the orientation angle and speed dynamics.
    [Show full text]
  • Long Time Behavior of Telegraph Processes Under Convex Potentials Joaquin Fontbona, Hélène Guérin, Florent Malrieu
    Long time behavior of telegraph processes under convex potentials Joaquin Fontbona, Hélène Guérin, Florent Malrieu To cite this version: Joaquin Fontbona, Hélène Guérin, Florent Malrieu. Long time behavior of telegraph processes under convex potentials. Stochastic Processes and their Applications, Elsevier, 2016, 126 (10), pp.3077-3101. 10.1016/j.spa.2016.04.002. hal-01176036 HAL Id: hal-01176036 https://hal.archives-ouvertes.fr/hal-01176036 Submitted on 14 Jul 2015 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Long time behavior of telegraph processes under convex potentials Joaquin Fontbona, H´el`ene Guerin´ , Florent Malrieu July 13, 2015 Contents 1 Introduction2 1.1 The model and main results...........................2 1.2 Preliminaries...................................5 2 Basic properties of the reflected process6 2.1 Distribution of the jump times.........................6 2.2 Hitting time of the origin............................ 10 3 The coupling time for the reflected process 13 3.1 The first crossing time.............................. 13 3.2 A way to stick the two paths.......................... 14 3.3 Coupling time from a crossing point...................... 15 3.4 The coupling time for the reflected process................... 20 4 The unreflected process 21 5 Diffusive scaling 22 References 24 Abstract We study the long-time behavior of variants of the telegraph process with position- dependent jump-rates, which result in a monotone gradient-like drift toward the ori- gin.
    [Show full text]
  • Exact and Approximate Hidden Markov Chain Filters Based on Discrete Observations
    Exact and Approximate Hidden Markov Chain Filters Based on Discrete Observations Nicole B¨auerlea, Igor Gilitschenskib, Uwe D. Hanebeckb aInstitute of Stochastics, Karlsruhe Institute of Technology, D-76131 Karlsruhe, Germany bInstitute for Anthropomatics and Robotics, Karlsruhe Institute of Technology, D-76131 Karlsruhe, Germany Abstract We consider a Hidden Markov Model (HMM) where the integrated continuous-time Markov chain can be observed at discrete time points perturbed by a Brownian motion. The aim is to derive a filter for the underlying continuous-time Markov chain. The recursion formula for the discrete-time filter is easy to derive, however involves densities which are very hard to obtain. In this paper we derive exact formulas for the necessary densities in the case the state space of the HMM consists of two elements only. This is done by relating the underlying integrated continuous-time Markov chain to the so-called asymmetric telegraph process and by using recent results on this process. In case the state space consists of more than two elements we present three different ways to approximate the densities for the filter. The first approach is based on the continuous filter problem. The second approach is to derive a PDE for the densities and solve it numerically and the third approach is a crude discrete time approximation of the Markov chain. All three approaches are compared in a numerical study. Keywords: Hidden Markov Model, Discrete Bayesian Filter, Wonham Filter, Asymmetric Telegraph process 1. Introduction We consider a Hidden Markov Model (HMM) where the integrated continuous-time Markov chain can be observed at discrete time points perturbed by a Brownian motion.
    [Show full text]
  • Exactly Solvable Models of Stochastic Gene Expression Lucy Ham,1, A) David Schnoerr,2, B) Rowan D
    bioRxiv preprint doi: https://doi.org/10.1101/2020.01.05.895359; this version posted January 6, 2020. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY-NC-ND 4.0 International license. Exactly solvable models of stochastic gene expression Lucy Ham,1, a) David Schnoerr,2, b) Rowan D. Brackston,2, c) and Michael P. H. Stumpf1, 2, d) 1)School of BioSciences and School of Mathematics and Statistics, University of Melbourne, Parkville VIC 3010, Australia 2)Dept. Life Sciences, Imperial College London, SW7 2AZ, UK (Dated: 6 January 2020) Stochastic models are key to understanding the intricate dynamics of gene expression. But the simplest models which only account for e.g. active and inactive states of a gene fail to capture common observations in both prokaryotic and eukaryotic organisms. Here we consider multistate models of gene expression which generalise the canonical Telegraph process, and are capable of capturing the joint effects of e.g. transcription factors, heterochromatin state and DNA accessibility (or, in prokaryotes, Sigma-factor activity) on transcript abundance. We propose two approaches for solving classes of these generalised systems. The first approach offers a fresh perspective on a general class of multistate models, and allows us to “decompose" more complicated systems into simpler processes, each of which can be solved analytically. This enables us to obtain a solution of any model from this class. We further show that these models cannot have a heavy-tailed distribution in the absence of extrinsic noise.
    [Show full text]
  • Wolfgang Stadje, Forty Years of Research in Probability and OR
    Wolfgang Stadje, forty years of research in probability and OR Osnabrück, Germany October 20-21, 2016 Scientific Program Thursday, October 20 09.30: Welcome 09.40: S.K. Bar-Lev (Haifa): Group Testing Procedures with Incomplete Identification and their Applications in Medical and Industrial Problems 10.25: I. Kleiner (Haifa): Recycled Incomplete Identification Procedures for Blood Screening 10.50: Coffee Break 11.15: E. v. Doorn (Twente) Asymptotic Aperiodicity and the Strong Ratio Limit Property of Markov Chains 11.40: P. Moyal (Compiègne) Stability of the General Stochastic Matching Model 12.05: O. Kella (Jerusalem) Wolfgang and I, the Small Collaboration Part of Over Two Decades of Our Personal Friendship 12.30: Lunch break 13.45: R. Ravid (Braude) A New Look on the Shortest Queue with Jockeying 14.10: A. Löpker (Dresden) Time Reversal and Perfect Simulation for the INAR(1) Autoregressive Process 14.35: A. Gnedin (London) Cycle Counts in Random Permutations and the Infinite-Server Queues 15.00: Coffee Break 15.30 M. Haviv (Jerusalem) Self-Regulation in Three Queueing Scenarios 15.55: E. Schulte-Geers (BSI) Probability in a Calculus Problem Visit at the museum (16.45-18.00) followed by workshop dinner Sponsored by Friday, October 21 09:30: I. Adan (Eindhoven) A Rate Balance Principle 09.55: K. Avrachenkov (INRIA) Hitting Times in Markov Chains with Restart and their Applications to Ranking 10.20: Y. Baryshnikov (Illinois) Search on the Brink of Chaos 10.50: Coffee Break 11.15: J.A.C Resing (Eindhoven) Some Two-Dimensional Queues and Risk Models 11.40: S.
    [Show full text]
  • Stochastic Processes and Models 0000.Pdf
    Stochastic Processes and Models This page intentionally left blank Stochastic Processes and Models David Stirzaker St John’s College, Oxford 1 3 Great Clarendon Street, Oxford OX2 6DP Oxford University Press is a department of the University of Oxford. It furthers the University’s objective of excellence in research, scholarship, and education by publishing worldwide in Oxford New York Auckland Cape Town Dar es Salaam Hong Kong Karachi Kuala Lumpur Madrid Melbourne Mexico City Nairobi New Delhi Shanghai Taipei Toronto With offices in Argentina Austria Brazil Chile Czech Republic France Greece Guatemala Hungary Italy Japan Poland Portugal Singapore South Korea Switzerland Thailand Turkey Ukraine Vietnam Oxford is a registered trade mark of Oxford University Press in the UK and in certain other countries Published in the United States by Oxford University Press Inc., New York © Oxford University Press, 2005 The moral rights of the author have been asserted Database right Oxford University Press (maker) First published 2005 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, without the prior permission in writing of Oxford University Press, or as expressly permitted by law, or under terms agreed with the appropriate reprographics rights organization. Enquiries concerning reproduction outside the scope of the above should be sent to the Rights Department, Oxford University Press, at the address above You must not circulate this book in any other binding
    [Show full text]
  • Estimation for the Discretely Observed Telegraph Process Udc 519.21
    Teor Imovr. ta Matem. Statist. Theor. Probability and Math. Statist. Vip. 78, 2008 No. 78, 2009, Pages 37–47 S 0094-9000(09)00760-1 Article electronically published on August 4, 2009 ESTIMATION FOR THE DISCRETELY OBSERVED TELEGRAPH PROCESS UDC 519.21 S. M. IACUS AND N. YOSHIDA Abstract. The telegraph process {X(t),t>0} is supposed to be observed at n +1 equidistant time points ti = i∆n, i =0, 1,...,n. The unknown value of λ, the un- derlying rate of the Poisson process, is a parameter to be estimated. The asymptotic framework considered is the following: ∆n → 0, n∆n = T →∞as n →∞.We show that previously proposed moment type estimators are consistent and asymp- totically normal but not efficient. We study further an approximated moment type estimator which is still not efficient but comes in explicit form. For this estimator the 3 → additional assumption n∆n 0 is required in order to obtain asymptotic normality. Finally, we propose a new estimator which is consistent, asymptotically normal and asymptotically efficient under no additional hypotheses. 1. Introduction The telegraph process (see Goldstein [8] and Kac [12]) models a random motion with finite velocity, and it is usually proposed as an alternative to diffusion models. The process describes the position of a particle moving on the real line, alternatively with constant velocity +v or −v. The changes of direction are governed by a homogeneous Poisson process with rate λ>0. The telegraph process or telegrapher’s process is defined as t N(s) (1) X(t)=x0 + V (0) (−1) ds, t > 0, 0 where V (0) is the initial velocity taking values ±v with equal probability and inde- pendently of the Poisson process {N(t),t > 0}.
    [Show full text]
  • A General Multitype Branching Process with Age, Memory and Population Dependence Christine Jacob
    A general multitype branching process with age, memory and population dependence Christine Jacob To cite this version: Christine Jacob. A general multitype branching process with age, memory and population dependence. IWAP 2010 V-th. International Workshop on Applied Probability, Jul 2010, Madrid, Spain. hal- 02755517 HAL Id: hal-02755517 https://hal.inrae.fr/hal-02755517 Submitted on 3 Jun 2020 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. IWAP 2010 5th International Workshop on Applied Probability BOOK OF ABSTRACTS AND DETAILED PROGRAMME 5-8 July, 2010 Colmenarejo, Madrid, Spain Universidad Carlos III de Madrid, Spain Preface The 5th International Workshop in Applied Probability, IWAP 2010, held in Colmenarejo, Madrid, Spain, has been built on the success of previous meetings. These meetings took place at Sim´onBol´ıvar Univer- sity (Venezuela, 2002), University of Piraeus (Greece, 2004), University of Connecticut (USA, 2006) and University of Technology of Compi`egne(France, 2008). Over 300 researchers in applied probability from all over the world are attending this workshop, making IWAP 2010 one of the largest IWAPs. Currently, IWAP is one of the major international meetings for researchers in Applied Probability.
    [Show full text]
  • Jump-Telegraph Models for the Short Rate
    Jump-telegraph models for the short rate: pricing and convexity adjustments of zero coupon bonds Oscar L´opez∗1, Gerardo E. Oleaga†2, and Alejandra S´anchez‡1 1Universidad Nacional de Colombia, Carrera 45 No. 26-85, Bogot´a, Colombia 2Departamento de An´alisis Matem´atico y Matem´atica Aplicada & Instituto de Matem´atica Interdisciplinar, Universidad Complutense de Madrid, Spain January 11, 2019 Abstract In this article, we consider a Markov-modulated model with jumps for short rate dynamics. We obtain closed formulas for the term structure and forward rates using the properties of the jump-telegraph process and the expectation hypothesis. The results are compared with the numerical solution of the corresponding partial differential equation. Keywords: Jump-telegraph process, Markov-modulated model, term structure, zero coupon bond, short rate model, convexity adjustment. 1 Introduction Regime changes associated to unexpected events in the economy provide an important field of study in quantitative finance. In the dynamics of the interest rates, the impact of those changes appears in the form of stochastic jumps. A well-known mathematical tool to model these types of events is provided by the family of Markov-modulated processes (see [7], [10]). Among them, the so-called telegraphic processes with jumps are of particular importance for a number of reasons (see [6], [8]). First, they capture some of the stylized facts reported in arXiv:1901.02995v1 [q-fin.MF] 10 Jan 2019 the literature (see [13], [5] and the references therein); next, closed formulas for the primary descriptors (mean, variance, moment generating functions) are available; therefore, they are good candidates to model interest rate dynamics, providing explicit results for fixed income instruments.
    [Show full text]
  • Option Pricing in Market Models Driven by Telegraph Processes with Jumps
    Option Pricing in Market Models Driven by Telegraph Processes with Jumps Ph.D. Thesis Oscar Javier L´opez Alfonso Universidad del Rosario August 2014 Option Pricing in Market Models Driven by Telegraph Processes with Jumps Ph.D. Thesis Oscar Javier L´opez Alfonso Faculty of Economics Universidad del Rosario Bogot´a, Colombia Thesis advisor: Nikita Ratanov, Universidad del Rosario Thesis committee: Enzo Orsingher, Sapienza Universit`adi Roma Adolfo Quiroz, Universidad de los Andes Rafael Serrano, Universidad del Rosario A mis padres Patricia y Jorge Contents Introduction v I Processes related to telegraph processes 1 1 Telegraph processes 3 1.1 Two-state continuous-time Markov chain . ..... 3 1.2 Definition of the telegraph process . .... 4 1.3 Distribution .................................. 5 1.4 Meanandvariance .............................. 12 1.5 Momentgeneratingfunction . 14 1.6 Notesandreferences ............................. 16 2 Poissonprocesseswithtelegraphcompensator 17 2.1 Distribution .................................. 18 2.2 Meanandvariance .............................. 22 2.3 Probability generating function . ..... 24 2.4 CompensatedPoissonprocess . 25 2.5 CompoundPoissonprocesses. 26 2.5.1 Examples of Compound Poisson Processes . 28 2.6 Mean and characteristic function . 29 2.7 Compensated compound Poisson process . 30 2.8 Randommeasure ............................... 31 2.9 Notesandreferences ............................. 32 3 Jump-Telegraph processes 33 3.1 Distribution .................................. 34 3.2 Meanandvariance .............................. 38 3.3 Momentgeneratingfunction . 40 3.4 Martingaleproperties . 41 i ii CONTENTS 3.5 Itˆo’sformula.................................. 43 3.6 Generalized jump-telegraph process with random jumps . .......... 44 3.7 Distributionandproperties . 45 3.8 Martingaleproperties . 48 3.9 Notesandreferences ............................. 49 II Market models 51 4 Basic concepts in mathematical finance 53 4.1 No-arbitrage and martingale measures . ..... 53 4.2 Fundamentaltheorems ...........................
    [Show full text]