DOCSLIB.ORG

Superprocesses and Mckean-Vlasov Equations with Creation of Mass

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Superprocesses and Mckean-Vlasov Equations with Creation of Mass Sup erpro cesses and McKean-Vlasov equations with creation of mass L. Overb eck Department of Statistics, University of California, Berkeley, 367, Evans Hall Berkeley, CA 94720, y U.S.A. Abstract Weak solutions of McKean-Vlasov equations with creation of mass are given in terms of sup erpro cesses. The solutions can b e approxi- mated by a sequence of non-interacting sup erpro cesses or by the mean- eld of multityp e sup erpro cesses with mean- eld interaction. The lat- ter approximation is asso ciated with a propagation of chaos statement for weakly interacting multityp e sup erpro cesses. Running title: Sup erpro cesses and McKean-Vlasov equations . 1 Intro duction Sup erpro cesses are useful in solving nonlinear partial di erential equation of 1+ the typ e f = f , 2 0; 1], cf. [Dy]. Wenowchange the p oint of view and showhowtheyprovide sto chastic solutions of nonlinear partial di erential Supp orted byanFellowship of the Deutsche Forschungsgemeinschaft. y On leave from the Universitat Bonn, Institut fur Angewandte Mathematik, Wegelerstr. 6, 53115 Bonn, Germany. 1 equation of McKean-Vlasovtyp e, i.e. wewant to nd weak solutions of d d 2 X X @ @ @ + d x; + bx; : 1.1 = a x; t i t t t t t ij t @t @x @x @x i j i i=1 i;j =1 d Aweak solution = 2 C [0;T];MIR satis es s Z 2 t X X @ @ a f = f + f + d f + b f ds: s ij s t 0 i s s @x @x @x 0 i j i Equation 1.1 generalizes McKean-Vlasov equations of twodi erenttyp es. Firstly,ifb = 0 it is the classical McKean-Vlasov equations asso ciated with d nonlinear IR -valued di usions and studied in many pap ers, e.g. [L,Oel,S1,S2]. If only b dep ends on then this equation is studied in [CR], [COR]. It is used in applications for some biological problems, like conduction of nerve impulse, cf. [GS,SP] and the references therein. Equations like 1.1 with non-linear drift in the di usion and nonlinear reaction term are sometimes called reaction-convection-di usion equation [Lo, p. 287] and can b e found in many biological applications, cf. [M, Section 11.4]. Our main result concerning equation 1.1 is Theorem 3.1 in Section 3 and claims that under Lipschitz conditions on the co ecients of 1.1 a unique solution of 1.1 can b e found as the limit of the interacting one-typ e su- P 1 N i;N 1;N N;N p erpro cess X as N !1where X ;:::;X is a N-typ e i=1 N sup erpro cess with mean- eld interaction. Obviously, this result should b e emb edded in a Propagation of Chaos statementforweakly interacting N- P 1 typ e sup erpro cesses including in particular tightness of i;N . This is X N the topic of Section 2. However, this result do es not necessarily implies that the limit pro cess gives a solution of 1.1, b ecause the application I from d d M M IR to M IR de ned by 1 Z I mf := f md 1.2 fails to b e continuous. Finally, in Prop osition 3.3 we construct a unique solution of 1.1 by a Picard-Lindelof approximation under stronger condition as in Theorem 3.1. This approximation gives a solution of 1.1 as a limit of intensity measures of sup erpro cesses with non-interactive immigration. 2 2 Weakly interacting and non-linear sup erpro cesses In this section we construct a non-linear sup erpro cess as an accumulation p oint of a sequence of weakly interacting multityp e sup erpro cesses. 2.1 Weakly interacting superprocesses First wewant to describ e an N-typ e sup erpro cess with an interaction de- p ending only on the empirical pro cess. We start with real-valued resp. p ositive functions a ;d ; 1 i; j; k d; and b resp. c de ned on ij k d d [0; 1 IR M M IR , where M E denotes the space of probability 1 1 measures on the Polish space E equipp ed with the top ology of weak conver- d gence. De ne for every s 0, m 2 M M IR the time-inhomogeneous 1 2 d op erator Ls; mon C IR by 0 d d 2 X X @ f @f Ls; mf x:= a s; x; m x+ d s; x; m x: 2.1 ij i @x @x @x i j i i;j =1 i=1 The op erator serves as a description of the one-particle motion. Let us de- note the family Ls; m d by L.For the weakly interacting s2[0;1;m2M M IR 1 sup erpro cess the function bs; x; m describ es the mean branching rate and cs; x; m the variance in the branching rate while the empirical distribution of the pro cess equals m. In order to state now the basic de nition we need some notation. Let N 2 IN + d N d N ~ b e xed. If ~ = ;:::; 2 M IR ; f =f ;:::;f 2 B IR ; 1 N 1 N b + where B are b ounded measurable non-negative functions, then b N X 1 d 2 M M IR R~ := 1 i N i=1 R P N ~ fd: The exp onential function e f ; where f = and ~ f := ~ i i i=1 f ~ is de ned by e ~=exp~ f : Finally,we denote for a Polish space E ~ f the space of E -valued continuous paths by C . E N De nition 2.1 We cal l a measure P 2 M C d N an N-typeweakly 1 M IR 3 interacting measure-valued di usion starting at ~ if 0 Z N t X j ~ ~ ~ ~ + X e e X Ls; RX f + 2.2 X e 0 t ~ ~ s j s ~ s f f f 0 j =1 2 ~ ~ bs; RX f cs; RX f ds s j s j N 2 ~ ~ is a P martingale for al l f =f ;:::;f with non-negative f 2 C . X 1 N i 0 denotes the coordinate process on C d N . M IR 2.2 Propagation of Chaos One main issue in this section is the question under which conditions the N 1 1 sequence fP g is P -chaotic with a measure P 2 M C d . N 2IN 1 M IR 1 N De nition 2.2 Let P 2 M C d . We say that the sequence fP g 1 N 2IN M IR 1 is P -chaotic if for every k 2 IN N 1 k 1 P X ;:::;X = P as N k tends to in nity ; 2.3 1 k i=1 cf. [S1,S2]. Intuitively, this means that the interaction disapp ears if the number of typ es 1 N tends to in nity.Typically the measure P is a solution of a non-linear martingale problem. Roughly sp eaking, b ecause the law of large numb ers N 1 1 1 ~ implies RX ! P X , the measure P solves the martingale s s problem characterized by the martingale prop erty of the pro cess Z t 1 1 1 1 f e X e X + f + bs; P X e X X Ls; P X f t f 0 f s s s s 0 1 1 2 cs; P X f ds 2.4 s 2 d for every f 2 C IR . Such a measure is called non-linear sup erpro cess with 0 parameter L;b;c. N 1 According to [S2] an exchangeable sequence fP g is P -chaotic i the N 2IN P 1 N N N ~ j;N converges = distribution 2 M M C d of RX 1 1 X M IR j =1 : N N ~ 1 weakly to . Here fX g is a sequence of random variables on a P N 2IN N 1 N ~ probability space ; F ;P suchthatP X = P . 4 The next theorem gives general conditions under which there is an exchange- N N able solution P to 2.2 and the sequence f g is tight. Additionally N 2IN N we show that all limits p oints of f g are supp orted by the set of so- N 2IN lutions to the martingale problem formulated in 2.4. If E; disaPolish space with metric d we consider on M E theWasserstein metric 1 d ; := supfjf f j; jjf jj 1g 2.5 BL E;d where jjf jj = jjf jj ^ inf fK ; jf x f y jKdx; y 8x; y 2 E g: Set BL 1 = d d and = d d . 1 2 M IR ;d IR ;j:j d IR ;j:j Theorem 2.3 1. Let the functions a ;d ; 1 i; j; k d; c and b satisfy ij k d d the fol lowing assumptions for functions r on [0; 1 IR M M IR 1 jr s; x ;m r s; x ;m jK m ;m +jx x j2.6 1 1 2 2 r 1 1 2 1 2 d r s; x; m < 1 for each m 2 M M IR : 2.7 sup 1 d x2IR Additional ly we assume that one of the fol lowing growth conditions is satis ed: ZZ sup sup jr s; x; mjdxmd K < 1 2.8 0 d s2[0;1 m2M M IR 1 for al l functions b; c; a ;d ; 1 i; j; k d or ij k Z sup jr s; x; mjdx K 1 + K 2.9 0 1 d d IR m2M M IR 1 for al l functions b; c; a ;d ; 1 i; j; k d: ij k Then there exists an exchangeable solution of 2.2.
Load more

Recommended publications
  • Numerical Solution of Jump-Diffusion LIBOR Market Models
    Finance Stochast. 7, 1–27 (2003) c Springer-Verlag 2003 Numerical solution of jump-diffusion LIBOR market models Paul Glasserman1, Nicolas Merener2 1 403 Uris Hall, Graduate School of Business, Columbia University, New York, NY 10027, USA (e-mail: [email protected]) 2 Department of Applied Physics and Applied Mathematics, Columbia University, New York, NY 10027, USA (e-mail: [email protected]) Abstract. This paper develops, analyzes, and tests computational procedures for the numerical solution of LIBOR market models with jumps. We consider, in par- ticular, a class of models in which jumps are driven by marked point processes with intensities that depend on the LIBOR rates themselves. While this formulation of- fers some attractive modeling features, it presents a challenge for computational work. As a first step, we therefore show how to reformulate a term structure model driven by marked point processes with suitably bounded state-dependent intensities into one driven by a Poisson random measure. This facilitates the development of discretization schemes because the Poisson random measure can be simulated with- out discretization error. Jumps in LIBOR rates are then thinned from the Poisson random measure using state-dependent thinning probabilities. Because of discon- tinuities inherent to the thinning process, this procedure falls outside the scope of existing convergence results; we provide some theoretical support for our method through a result establishing first and second order convergence of schemes that accommodates thinning but imposes stronger conditions on other problem data. The bias and computational efficiency of various schemes are compared through numerical experiments. Key words: Interest rate models, Monte Carlo simulation, market models, marked point processes JEL Classification: G13, E43 Mathematics Subject Classification (1991): 60G55, 60J75, 65C05, 90A09 The authors thank Professor Steve Kou for helpful comments and discussions.
    [Show full text]
  • Poisson Representations of Branching Markov and Measure-Valued
    The Annals of Probability 2011, Vol. 39, No. 3, 939–984 DOI: 10.1214/10-AOP574 c Institute of Mathematical Statistics, 2011 POISSON REPRESENTATIONS OF BRANCHING MARKOV AND MEASURE-VALUED BRANCHING PROCESSES By Thomas G. Kurtz1 and Eliane R. Rodrigues2 University of Wisconsin, Madison and UNAM Representations of branching Markov processes and their measure- valued limits in terms of countable systems of particles are con- structed for models with spatially varying birth and death rates. Each particle has a location and a “level,” but unlike earlier con- structions, the levels change with time. In fact, death of a particle occurs only when the level of the particle crosses a specified level r, or for the limiting models, hits infinity. For branching Markov pro- cesses, at each time t, conditioned on the state of the process, the levels are independent and uniformly distributed on [0,r]. For the limiting measure-valued process, at each time t, the joint distribu- tion of locations and levels is conditionally Poisson distributed with mean measure K(t) × Λ, where Λ denotes Lebesgue measure, and K is the desired measure-valued process. The representation simplifies or gives alternative proofs for a vari- ety of calculations and results including conditioning on extinction or nonextinction, Harris’s convergence theorem for supercritical branch- ing processes, and diffusion approximations for processes in random environments. 1. Introduction. Measure-valued processes arise naturally as infinite sys- tem limits of empirical measures of finite particle systems. A number of ap- proaches have been developed which preserve distinct particles in the limit and which give a representation of the measure-valued process as a transfor- mation of the limiting infinite particle system.
    [Show full text]
  • CONFORMAL INVARIANCE and 2 D STATISTICAL PHYSICS −
    CONFORMAL INVARIANCE AND 2 d STATISTICAL PHYSICS − GREGORY F. LAWLER Abstract. A number of two-dimensional models in statistical physics are con- jectured to have scaling limits at criticality that are in some sense conformally invariant. In the last ten years, the rigorous understanding of such limits has increased significantly. I give an introduction to the models and one of the major new mathematical structures, the Schramm-Loewner Evolution (SLE). 1. Critical Phenomena Critical phenomena in statistical physics refers to the study of systems at or near the point at which a phase transition occurs. There are many models of such phenomena. We will discuss some discrete equilibrium models that are defined on a lattice. These are measures on paths or configurations where configurations are weighted by their energy with a preference for paths of smaller energy. These measures depend on at leastE one parameter. A standard parameter in physics β = c/T where c is a fixed constant, T stands for temperature and the measure given to a configuration is e−βE . Phase transitions occur at critical values of the temperature corresponding to, e.g., the transition from a gaseous to liquid state. We use β for the parameter although for understanding the literature it is useful to know that large values of β correspond to “low temperature” and small values of β correspond to “high temperature”. Small β (high temperature) systems have weaker correlations than large β (low temperature) systems. In a number of models, there is a critical βc such that qualitatively the system has three regimes β<βc (high temperature), β = βc (critical) and β>βc (low temperature).
    [Show full text]
  • Applied Probability
    ISSN 1050-5164 (print) ISSN 2168-8737 (online) THE ANNALS of APPLIED PROBABILITY AN OFFICIAL JOURNAL OF THE INSTITUTE OF MATHEMATICAL STATISTICS Articles Optimal control of branching diffusion processes: A finite horizon problem JULIEN CLAISSE 1 Change point detection in network models: Preferential attachment and long range dependence . SHANKAR BHAMIDI,JIMMY JIN AND ANDREW NOBEL 35 Ergodic theory for controlled Markov chains with stationary inputs YUE CHEN,ANA BUŠIC´ AND SEAN MEYN 79 Nash equilibria of threshold type for two-player nonzero-sum games of stopping TIZIANO DE ANGELIS,GIORGIO FERRARI AND JOHN MORIARTY 112 Local inhomogeneous circular law JOHANNES ALT,LÁSZLÓ ERDOS˝ AND TORBEN KRÜGER 148 Diffusion approximations for controlled weakly interacting large finite state systems withsimultaneousjumps.........AMARJIT BUDHIRAJA AND ERIC FRIEDLANDER 204 Duality and fixation in -Wright–Fisher processes with frequency-dependent selection ADRIÁN GONZÁLEZ CASANOVA AND DARIO SPANÒ 250 CombinatorialLévyprocesses.......................................HARRY CRANE 285 Eigenvalue versus perimeter in a shape theorem for self-interacting random walks MAREK BISKUP AND EVIATAR B. PROCACCIA 340 Volatility and arbitrage E. ROBERT FERNHOLZ,IOANNIS KARATZAS AND JOHANNES RUF 378 A Skorokhod map on measure-valued paths with applications to priority queues RAMI ATAR,ANUP BISWAS,HAYA KASPI AND KAV I TA RAMANAN 418 BSDEs with mean reflection . PHILIPPE BRIAND,ROMUALD ELIE AND YING HU 482 Limit theorems for integrated local empirical characteristic exponents from noisy high-frequency data with application to volatility and jump activity estimation JEAN JACOD AND VIKTOR TODOROV 511 Disorder and wetting transition: The pinned harmonic crystal indimensionthreeorlarger.....GIAMBATTISTA GIACOMIN AND HUBERT LACOIN 577 Law of large numbers for the largest component in a hyperbolic model ofcomplexnetworks..........NIKOLAOS FOUNTOULAKIS AND TOBIAS MÜLLER 607 Vol.
    [Show full text]
  • A Switching Self-Exciting Jump Diffusion Process for Stock Prices Donatien Hainaut, Franck Moraux
    A switching self-exciting jump diffusion process for stock prices Donatien Hainaut, Franck Moraux To cite this version: Donatien Hainaut, Franck Moraux. A switching self-exciting jump diffusion process for stock prices. Annals of Finance, Springer Verlag, 2019, 15 (2), pp.267-306. 10.1007/s10436-018-0340-5. halshs- 01909772 HAL Id: halshs-01909772 https://halshs.archives-ouvertes.fr/halshs-01909772 Submitted on 31 Oct 2018 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. A switching self-exciting jump diusion process for stock prices Donatien Hainaut∗ ISBA, Université Catholique de Louvain, Belgium Franck Morauxy Univ. Rennes 1 and CREM May 25, 2018 Abstract This study proposes a new Markov switching process with clustering eects. In this approach, a hidden Markov chain with a nite number of states modulates the parameters of a self-excited jump process combined to a geometric Brownian motion. Each regime corresponds to a particular economic cycle determining the expected return, the diusion coecient and the long-run frequency of clustered jumps. We study rst the theoretical properties of this process and we propose a sequential Monte-Carlo method to lter the hidden state variables.
    [Show full text]
  • Birth and Death Process in Mean Field Type Interaction
    BIRTH AND DEATH PROCESS IN MEAN FIELD TYPE INTERACTION MARIE-NOÉMIE THAI ABSTRACT. Theaim ofthispaperis to study theasymptoticbehaviorofa system of birth and death processes in mean field type interaction in discrete space. We first establish the exponential convergence of the particle system to equilibrium for a suitable Wasserstein coupling distance. The approach provides an explicit quantitative estimate on the rate of convergence. We prove next a uniform propagation of chaos property. As a consequence, we show that the limit of the associated empirical distribution, which is the solution of a nonlinear differential equation, converges exponentially fast to equilibrium. This paper can be seen as a discrete version of the particle approximation of the McKean-Vlasov equations and is inspired from previous works of Malrieu and al and Caputo, Dai Pra and Posta. AMS 2000 Mathematical Subject Classification: 60K35, 60K25, 60J27, 60B10, 37A25. Keywords: Interacting particle system - mean field - coupling - Wasserstein distance - propagation of chaos. CONTENTS 1. Introduction 1 Long time behavior of the particle system 4 Propagation of chaos 5 Longtimebehaviorofthenonlinearprocess 7 2. Proof of Theorem1.1 7 3. Proof of Theorem1.2 12 4. Proof of Theorem1.5 15 5. Appendix 16 References 18 1. INTRODUCTION The concept of mean field interaction arised in statistical physics with Kac [17] and then arXiv:1510.03238v1 [math.PR] 12 Oct 2015 McKean [21] in order to describe the collisions between particles in a gas, and has later been applied in other areas such as biology or communication networks. A particle system is in mean field interaction when the system acts over one fixed particle through the em- pirical measure of the system.
    [Show full text]
  • A Stochastic Processes and Martingales
    A Stochastic Processes and Martingales A.1 Stochastic Processes Let I be either IINorIR+.Astochastic process on I with state space E is a family of E-valued random variables X = {Xt : t ∈ I}. We only consider examples where E is a Polish space. Suppose for the moment that I =IR+. A stochastic process is called cadlag if its paths t → Xt are right-continuous (a.s.) and its left limits exist at all points. In this book we assume that every stochastic process is cadlag. We say a process is continuous if its paths are continuous. The above conditions are meant to hold with probability 1 and not to hold pathwise. A.2 Filtration and Stopping Times The information available at time t is expressed by a σ-subalgebra Ft ⊂F.An {F ∈ } increasing family of σ-algebras t : t I is called a filtration.IfI =IR+, F F F we call a filtration right-continuous if t+ := s>t s = t. If not stated otherwise, we assume that all filtrations in this book are right-continuous. In many books it is also assumed that the filtration is complete, i.e., F0 contains all IIP-null sets. We do not assume this here because we want to be able to change the measure in Chapter 4. Because the changed measure and IIP will be singular, it would not be possible to extend the new measure to the whole σ-algebra F. A stochastic process X is called Ft-adapted if Xt is Ft-measurable for all t. If it is clear which filtration is used, we just call the process adapted.The {F X } natural filtration t is the smallest right-continuous filtration such that X is adapted.
    [Show full text]
  • Local Conditioning in Dawson–Watanabe Superprocesses
    The Annals of Probability 2013, Vol. 41, No. 1, 385–443 DOI: 10.1214/11-AOP702 c Institute of Mathematical Statistics, 2013 LOCAL CONDITIONING IN DAWSON–WATANABE SUPERPROCESSES By Olav Kallenberg Auburn University Consider a locally finite Dawson–Watanabe superprocess ξ =(ξt) in Rd with d ≥ 2. Our main results include some recursive formulas for the moment measures of ξ, with connections to the uniform Brown- ian tree, a Brownian snake representation of Palm measures, continu- ity properties of conditional moment densities, leading by duality to strongly continuous versions of the multivariate Palm distributions, and a local approximation of ξt by a stationary clusterη ˜ with nice continuity and scaling properties. This all leads up to an asymptotic description of the conditional distribution of ξt for a fixed t> 0, given d that ξt charges the ε-neighborhoods of some points x1,...,xn ∈ R . In the limit as ε → 0, the restrictions to those sets are conditionally in- dependent and given by the pseudo-random measures ξ˜ orη ˜, whereas the contribution to the exterior is given by the Palm distribution of ξt at x1,...,xn. Our proofs are based on the Cox cluster representa- tions of the historical process and involve some delicate estimates of moment densities. 1. Introduction. This paper may be regarded as a continuation of [19], where we considered some local properties of a Dawson–Watanabe super- process (henceforth referred to as a DW-process) at a fixed time t> 0. Recall that a DW-process ξ = (ξt) is a vaguely continuous, measure-valued diffu- d ξtf µvt sion process in R with Laplace functionals Eµe− = e− for suitable functions f 0, where v = (vt) is the unique solution to the evolution equa- 1 ≥ 2 tion v˙ = 2 ∆v v with initial condition v0 = f.
    [Show full text]
  • Particle Filtering-Based Maximum Likelihood Estimation for Financial Parameter Estimation
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Spiral - Imperial College Digital Repository Particle Filtering-based Maximum Likelihood Estimation for Financial Parameter Estimation Jinzhe Yang Binghuan Lin Wayne Luk Terence Nahar Imperial College & SWIP Techila Technologies Ltd Imperial College SWIP London & Edinburgh, UK Tampere University of Technology London, UK London, UK [email protected] Tampere, Finland [email protected] [email protected] [email protected] Abstract—This paper presents a novel method for estimating result, which cannot meet runtime requirement in practice. In parameters of financial models with jump diffusions. It is a addition, high performance computing (HPC) technologies are Particle Filter based Maximum Likelihood Estimation process, still new to the finance industry. We provide a PF based MLE which uses particle streams to enable efficient evaluation of con- straints and weights. We also provide a CPU-FPGA collaborative solution to estimate parameters of jump diffusion models, and design for parameter estimation of Stochastic Volatility with we utilise HPC technology to speedup the estimation scheme Correlated and Contemporaneous Jumps model as a case study. so that it would be acceptable for practical use. The result is evaluated by comparing with a CPU and a cloud This paper presents the algorithm development, as well computing platform. We show 14 times speed up for the FPGA as the pipeline design solution in FPGA technology, of PF design compared with the CPU, and similar speedup but better convergence compared with an alternative parallelisation scheme based MLE for parameter estimation of Stochastic Volatility using Techila Middleware on a multi-CPU environment.
    [Show full text]
  • 1 Introduction
    The analysis of marked and weighted empirical processes of estimated residuals Vanessa Berenguer-Rico Department of Economics; University of Oxford; Oxford; OX1 3UQ; UK and Mansfield College; Oxford; OX1 3TF [email protected] Søren Johansen Department of Economics; University of Copenhagen and CREATES; Department of Economics and Business; Aarhus University [email protected] Bent Nielsen1 Department of Economics; University of Oxford; Oxford; OX1 3UQ; UK and Nuffield College; Oxford; OX1 1NF; U.K. bent.nielsen@nuffield.ox.ac.uk 29 April 2019 Summary An extended and improved theory is presented for marked and weighted empirical processes of residuals of time series regressions. The theory is motivated by 1- step Huber-skip estimators, where a set of good observations are selected using an initial estimator and an updated estimator is found by applying least squares to the selected observations. In this case, the weights and marks represent powers of the regressors and the regression errors, respectively. The inclusion of marks is a non-trivial extention to previous theory and requires refined martingale arguments. Keywords 1-step Huber-skip; Non-stationarity; Robust Statistics; Stationarity. 1 Introduction We consider marked and weighted empirical processes of residuals from a linear time series regression. Such processes are sums of products of an adapted weight, a mark that is a power of the innovations and an indicator for the residuals belonging to a half line. They have previously been studied by Johansen & Nielsen (2016a) - JN16 henceforth - generalising results by Koul & Ossiander (1994) and Koul (2002) for processes without marks. The results presented extend and improve upon expansions previously given in JN16, while correcting a mistake in the argument, simplifying proofs and allowing weaker conditions on the innovation distribution and regressors.
    [Show full text]
  • Lecture 19: Polymer Models: Persistence and Self-Avoidance
    Lecture 19: Polymer Models: Persistence and Self-Avoidance Scribe: Allison Ferguson (and Martin Z. Bazant) Martin Fisher School of Physics, Brandeis University April 14, 2005 Polymers are high molecular weight molecules formed by combining a large number of smaller molecules (monomers) in a regular pattern. Polymers in solution (i.e. uncrystallized) can be characterized as long flexible chains, and thus may be well described by random walk models of various complexity. 1 Simple Random Walk (IID Steps) Consider a Rayleigh random walk (i.e. a fixed step size a with a random angle θ). Recall (from Lecture 1) that the probability distribution function (PDF) of finding the walker at position R~ after N steps is given by −3R2 2 ~ e 2a N PN (R) ∼ d (1) (2πa2N/d) 2 Define RN = |R~| as the end-to-end distance of the polymer. Then from previous results we q √ 2 2 ¯ 2 know hRN i = Na and RN = hRN i = a N. We can define an additional quantity, the radius of gyration GN as the radius from the centre of mass (assuming the polymer has an approximately 1 2 spherical shape ). Hughes [1] gives an expression for this quantity in terms of hRN i as follows: 1 1 hG2 i = 1 − hR2 i (2) N 6 N 2 N As an example of the type of prediction which can be made using this type of simple model, consider the following experiment: Pull on both ends of a polymer and measure the force f required to stretch it out2. Note that the force will be given by f = −(∂F/∂R) where F = U − TS is the Helmholtz free energy (T is the temperature, U is the internal energy, and S is the entropy).
    [Show full text]
  • Semimartingale Properties of a Generalized Fractional Brownian Motion and Its Mixtures with Applications in finance
    Semimartingale properties of a generalized fractional Brownian motion and its mixtures with applications in finance TOMOYUKI ICHIBA, GUODONG PANG, AND MURAD S. TAQQU ABSTRACT. We study the semimartingale properties for the generalized fractional Brownian motion (GFBM) introduced by Pang and Taqqu (2019) and discuss the applications of the GFBM and its mixtures to financial models, including stock price and rough volatility. The GFBM is self-similar and has non-stationary increments, whose Hurst index H 2 (0; 1) is determined by two parameters. We identify the region of these two parameter values where the GFBM is a semimartingale. Specifically, in one region resulting in H 2 (1=2; 1), it is in fact a process of finite variation and differentiable, and in another region also resulting in H 2 (1=2; 1) it is not a semimartingale. For regions resulting in H 2 (0; 1=2] except the Brownian motion case, the GFBM is also not a semimartingale. We also establish p-variation results of the GFBM, which are used to provide an alternative proof of the non-semimartingale property when H < 1=2. We then study the semimartingale properties of the mixed process made up of an independent Brownian motion and a GFBM with a Hurst parameter H 2 (1=2; 1), and derive the associated equivalent Brownian measure. We use the GFBM and its mixture with a BM to study financial asset models. The first application involves stock price models with long range dependence that generalize those using shot noise processes and FBMs. When the GFBM is a process of finite variation (resulting in H 2 (1=2; 1)), the mixed GFBM process as a stock price model is a Brownian motion with a random drift of finite variation.
    [Show full text]