On the Propagation of the Weak Representation Property In
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Integral Representations of Martingales for Progressive Enlargements Of
INTEGRAL REPRESENTATIONS OF MARTINGALES FOR PROGRESSIVE ENLARGEMENTS OF FILTRATIONS ANNA AKSAMIT, MONIQUE JEANBLANC AND MAREK RUTKOWSKI Abstract. We work in the setting of the progressive enlargement G of a reference filtration F through the observation of a random time τ. We study an integral representation property for some classes of G-martingales stopped at τ. In the first part, we focus on the case where F is a Poisson filtration and we establish a predictable representation property with respect to three G-martingales. In the second part, we relax the assumption that F is a Poisson filtration and we assume that τ is an F-pseudo-stopping time. We establish integral representations with respect to some G-martingales built from F-martingales and, under additional hypotheses, we obtain a predictable representation property with respect to two G-martingales. Keywords: predictable representation property, Poisson process, random time, progressive enlargement, pseudo-stopping time Mathematics Subjects Classification (2010): 60H99 1. Introduction We are interested in the stability of the predictable representation property in a filtration en- largement setting: under the postulate that the predictable representation property holds for F and G is an enlargement of F, the goal is to find under which conditions the predictable rep- G G resentation property holds for . We focus on the progressive enlargement = (Gt)t∈R+ of a F reference filtration = (Ft)t∈R+ through the observation of the occurrence of a random time τ and we assume that the hypothesis (H′) is satisfied, i.e., any F-martingale is a G-semimartingale. It is worth noting that no general result on the existence of a G-semimartingale decomposition after τ of an F-martingale is available in the existing literature, while, for any τ, any F-martingale stopped at time τ is a G-semimartingale with an explicit decomposition depending on the Az´ema supermartingale of τ. -
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. -
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. -
PREDICTABLE REPRESENTATION PROPERTY for PROGRESSIVE ENLARGEMENTS of a POISSON FILTRATION Anna Aksamit, Monique Jeanblanc, Marek Rutkowski
PREDICTABLE REPRESENTATION PROPERTY FOR PROGRESSIVE ENLARGEMENTS OF A POISSON FILTRATION Anna Aksamit, Monique Jeanblanc, Marek Rutkowski To cite this version: Anna Aksamit, Monique Jeanblanc, Marek Rutkowski. PREDICTABLE REPRESENTATION PROPERTY FOR PROGRESSIVE ENLARGEMENTS OF A POISSON FILTRATION. 2015. hal- 01249662 HAL Id: hal-01249662 https://hal.archives-ouvertes.fr/hal-01249662 Preprint submitted on 2 Jan 2016 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. PREDICTABLE REPRESENTATION PROPERTY FOR PROGRESSIVE ENLARGEMENTS OF A POISSON FILTRATION Anna Aksamit Mathematical Institute, University of Oxford, Oxford OX2 6GG, United Kingdom Monique Jeanblanc∗ Laboratoire de Math´ematiques et Mod´elisation d’Evry´ (LaMME), Universit´ed’Evry-Val-d’Essonne,´ UMR CNRS 8071 91025 Evry´ Cedex, France Marek Rutkowski School of Mathematics and Statistics University of Sydney Sydney, NSW 2006, Australia 10 December 2015 Abstract We study problems related to the predictable representation property for a progressive en- largement G of a reference filtration F through observation of a finite random time τ. We focus on cases where the avoidance property and/or the continuity property for F-martingales do not hold and the reference filtration is generated by a Poisson process. -
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. -
Lecture 19 Semimartingales
Lecture 19:Semimartingales 1 of 10 Course: Theory of Probability II Term: Spring 2015 Instructor: Gordan Zitkovic Lecture 19 Semimartingales Continuous local martingales While tailor-made for the L2-theory of stochastic integration, martin- 2,c gales in M0 do not constitute a large enough class to be ultimately useful in stochastic analysis. It turns out that even the class of all mar- tingales is too small. When we restrict ourselves to processes with continuous paths, a naturally stable family turns out to be the class of so-called local martingales. Definition 19.1 (Continuous local martingales). A continuous adapted stochastic process fMtgt2[0,¥) is called a continuous local martingale if there exists a sequence ftngn2N of stopping times such that 1. t1 ≤ t2 ≤ . and tn ! ¥, a.s., and tn 2. fMt gt2[0,¥) is a uniformly integrable martingale for each n 2 N. In that case, the sequence ftngn2N is called the localizing sequence for (or is said to reduce) fMtgt2[0,¥). The set of all continuous local loc,c martingales M with M0 = 0 is denoted by M0 . Remark 19.2. 1. There is a nice theory of local martingales which are not neces- sarily continuous (RCLL), but, in these notes, we will focus solely on the continuous case. In particular, a “martingale” or a “local martingale” will always be assumed to be continuous. 2. While we will only consider local martingales with M0 = 0 in these notes, this is assumption is not standard, so we don’t put it into the definition of a local martingale. tn 3. -
Fclts for the Quadratic Variation of a CTRW and for Certain Stochastic Integrals
FCLTs for the Quadratic Variation of a CTRW and for certain stochastic integrals No`eliaViles Cuadros (joint work with Enrico Scalas) Universitat de Barcelona Sevilla, 17 de Septiembre 2013 Damped harmonic oscillator subject to a random force The equation of motion is informally given by x¨(t) + γx_(t) + kx(t) = ξ(t); (1) where x(t) is the position of the oscillating particle with unit mass at time t, γ > 0 is the damping coefficient, k > 0 is the spring constant and ξ(t) represents white L´evynoise. I. M. Sokolov, Harmonic oscillator under L´evynoise: Unexpected properties in the phase space. Phys. Rev. E. Stat. Nonlin Soft Matter Phys 83, 041118 (2011). 2 of 27 The formal solution is Z t x(t) = F (t) + G(t − t0)ξ(t0)dt0; (2) −∞ where G(t) is the Green function for the homogeneous equation. The solution for the velocity component can be written as Z t 0 0 0 v(t) = Fv (t) + Gv (t − t )ξ(t )dt ; (3) −∞ d d where Fv (t) = dt F (t) and Gv (t) = dt G(t). 3 of 27 • Replace the white noise with a sequence of instantaneous shots of random amplitude at random times. • They can be expressed in terms of the formal derivative of compound renewal process, a random walk subordinated to a counting process called continuous-time random walk. A continuous time random walk (CTRW) is a pure jump process given by a sum of i.i.d. random jumps fYi gi2N separated by i.i.d. random waiting times (positive random variables) fJi gi2N. -
Solving Black-Schole Equation Using Standard Fractional Brownian Motion
http://jmr.ccsenet.org Journal of Mathematics Research Vol. 11, No. 2; 2019 Solving Black-Schole Equation Using Standard Fractional Brownian Motion Didier Alain Njamen Njomen1 & Eric Djeutcha2 1 Department of Mathematics and Computer’s Science, Faculty of Science, University of Maroua, Cameroon 2 Department of Mathematics, Higher Teachers’ Training College, University of Maroua, Cameroon Correspondence: Didier Alain Njamen Njomen, Department of Mathematics and Computer’s Science, Faculty of Science, University of Maroua, Po-Box 814, Cameroon. Received: May 2, 2017 Accepted: June 5, 2017 Online Published: March 24, 2019 doi:10.5539/jmr.v11n2p142 URL: https://doi.org/10.5539/jmr.v11n2p142 Abstract In this paper, we emphasize the Black-Scholes equation using standard fractional Brownian motion BH with the hurst index H 2 [0; 1]. N. Ciprian (Necula, C. (2002)) and Bright and Angela (Bright, O., Angela, I., & Chukwunezu (2014)) get the same formula for the evaluation of a Call and Put of a fractional European with the different approaches. We propose a formula by adapting the non-fractional Black-Scholes model using a λH factor to evaluate the european option. The price of the option at time t 2]0; T[ depends on λH(T − t), and the cost of the action S t, but not only from t − T as in the classical model. At the end, we propose the formula giving the implied volatility of sensitivities of the option and indicators of the financial market. Keywords: stock price, black-Scholes model, fractional Brownian motion, options, volatility 1. Introduction Among the first systematic treatments of the option pricing problem has been the pioneering work of Black, Scholes and Merton (see Black, F. -
On the Convergence of Quadratic Variation for Compound Fractional Poisson Processes in “Fcaa” Journal
ON THE CONVERGENCE OF QUADRATIC VARIATION FOR COMPOUND FRACTIONAL POISSON PROCESSES IN \FCAA" JOURNAL Enrico Scalas 1, No`eliaViles 2 Abstract The relationship between quadratic variation for compound renewal processes and M-Wright functions is discussed. The convergence of qua- dratic variation is investigated both as a random variable (for given t) and as a stochastic process. Key Words and Phrases: Continuous time random walk; Inverse stable subordinator; Mittag-Leffler waiting time; Fractional Poisson process; M- Wright functions; Quadratic variation; Compound renewal process. 1. Introduction This paper discusses the relationship between quadratic variation for pure jump processes (usually called continuous-time random walks by physi- cists) and M-Wright functions, often appearing in fractional calculus. Some results highlighted in this short paper were already published in [6]. How- ever, the main focus of that paper was defining and studying the properties of stochastic integrals driven by pure jump processes. Moreover, in this paper, we study the convergence of quadratic variation as a stochastic pro- cess. 1.1. Continuous-Time Random Walks 1 Let fJigi=1 be a sequence of independent and identically distributed (i.i.d.) positive random variables with the meaning of durations between c Year Diogenes Co., Sofia pp. xxx{xxx 2 Enrico Scalas, N. Viles events in a point process. They define a renewal process. Let n X T0 = 0;Tn = Ji; for n > 1; (1.1) i=1 be the epoch of the n-th event. Then the process Nt counting the number of events up to time t is Nt = maxfn : Tn ≤ tg; (1.2) where it is assumed that the number of events occurring up to t is a finite, but arbitrary, non-negative integer. -
Long Time Behaviour and Mean-Field Limit of Atlas Models Julien Reygner
Long time behaviour and mean-field limit of Atlas models Julien Reygner To cite this version: Julien Reygner. Long time behaviour and mean-field limit of Atlas models. ESAIM: Proceedings and Surveys, EDP Sciences, 2017, Journées MAS 2016 de la SMAI – Phénomènes complexes et hétérogènes, 60, pp.132-143. 10.1051/proc/201760132. hal-01525360 HAL Id: hal-01525360 https://hal.archives-ouvertes.fr/hal-01525360 Submitted on 19 May 2017 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 behaviour and mean-field limit of Atlas models Julien Reygner ABSTRACT. This article reviews a few basic features of systems of one-dimensional diffusions with rank-based characteristics. Such systems arise in particular in the modelling of financial mar- kets, where they go by the name of Atlas models. We mostly describe their long time and large scale behaviour, and lay a particular emphasis on the case of mean-field interactions. We finally present an application of the reviewed results to the modelling of capital distribution in systems with a large number of agents. 1. Introduction The term Atlas model was originally introduced in Fernholz’ monograph on Stochastic Portfolio Theory [13] to describe a stock market in which the asset prices evolve according to independent and identically distributed processes, except the smallest one which undergoes an additional up- ward push. -
Martingale Theory
CHAPTER 1 Martingale Theory We review basic facts from martingale theory. We start with discrete- time parameter martingales and proceed to explain what modifications are needed in order to extend the results from discrete-time to continuous-time. The Doob-Meyer decomposition theorem for continuous semimartingales is stated but the proof is omitted. At the end of the chapter we discuss the quadratic variation process of a local martingale, a key concept in martin- gale theory based stochastic analysis. 1. Conditional expectation and conditional probability In this section, we review basic properties of conditional expectation. Let (W, F , P) be a probability space and G a s-algebra of measurable events contained in F . Suppose that X 2 L1(W, F , P), an integrable ran- dom variable. There exists a unique random variable Y which have the following two properties: (1) Y 2 L1(W, G , P), i.e., Y is measurable with respect to the s-algebra G and is integrable; (2) for any C 2 G , we have E fX; Cg = E fY; Cg . This random variable Y is called the conditional expectation of X with re- spect to G and is denoted by E fXjG g. The existence and uniqueness of conditional expectation is an easy con- sequence of the Radon-Nikodym theorem in real analysis. Define two mea- sures on (W, G ) by m fCg = E fX; Cg , n fCg = P fCg , C 2 G . It is clear that m is absolutely continuous with respect to n. The conditional expectation E fXjG g is precisely the Radon-Nikodym derivative dm/dn. -
Conformally Invariant Scaling Limits in Planar Critical Percolation∗
Probability Surveys Vol. 8 (2011) 155{209 ISSN: 1549-5787 DOI: 10.1214/11-PS180 Conformally invariant scaling limits in planar critical percolation∗ Nike Suny Department of Statistics, Stanford University 390 Serra Mall Stanford, CA 94305 e-mail: [email protected] Abstract: This is an introductory account of the emergence of confor- mal invariance in the scaling limit of planar critical percolation. We give an exposition of Smirnov's theorem (2001) on the conformal invariance of crossing probabilities in site percolation on the triangular lattice. We also give an introductory account of Schramm-Loewner evolutions (SLEκ), a one-parameter family of conformally invariant random curves discovered by Schramm (2000). The article is organized around the aim of proving the result, due to Smirnov (2001) and to Camia and Newman (2007), that the percolation exploration path converges in the scaling limit to chordal SLE6. No prior knowledge is assumed beyond some general complex analysis and probability theory. AMS 2000 subject classifications: Primary 60K35, 30C35; secondary 60J65. Keywords and phrases: Conformally invariant scaling limits, perco- lation, Schramm-Loewner evolutions, preharmonicity, preholomorphicity, percolation exploration path. Received August 2011. Contents 1 Introduction . 156 1.1 Critical percolation . 157 1.2 Conformal invariance . 159 1.3 The percolation scaling limit . 160 1.3.1 Smirnov's theorem on crossing probabilities . 160 1.3.2 Schramm-Loewner evolutions . 161 1.3.3 Percolation exploration path and convergence to SLE .. 163 arXiv:0911.0063v2 [math.PR] 21 Oct 2011 6 1.3.4 Outline of remaining sections . 163 2 Brownian motion . 165 2.1 Conformal invariance of planar Brownian motion .