Program Booklet
Total Page:16
File Type:pdf, Size:1020Kb

Load more
Recommended publications
-
A University of Sussex Phd Thesis Available Online Via Sussex
A University of Sussex PhD thesis Available online via Sussex Research Online: http://sro.sussex.ac.uk/ This thesis is protected by copyright which belongs to the author. This thesis cannot be reproduced or quoted extensively from without first obtaining permission in writing from the Author The content must not be changed in any way or sold commercially in any format or medium without the formal permission of the Author When referring to this work, full bibliographic details including the author, title, awarding institution and date of the thesis must be given Please visit Sussex Research Online for more information and further details NON-STATIONARY PROCESSES AND THEIR APPLICATION TO FINANCIAL HIGH-FREQUENCY DATA Mailan Trinh A thesis submitted for the degree of Doctor of Philosophy University of Sussex March 2018 UNIVERSITY OF SUSSEX MAILAN TRINH A THESIS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY NON-STATIONARY PROCESSES AND THEIR APPLICATION TO FINANCIAL HIGH-FREQUENCY DATA SUMMARY The thesis is devoted to non-stationary point process models as generalizations of the standard homogeneous Poisson process. The work can be divided in two parts. In the first part, we introduce a fractional non-homogeneous Poisson process (FNPP) by applying a random time change to the standard Poisson process. We character- ize the FNPP by deriving its non-local governing equation. We further compute moments and covariance of the process and discuss the distribution of the arrival times. Moreover, we give both finite-dimensional and functional limit theorems for the FNPP and the corresponding fractional non-homogeneous compound Poisson process. The limit theorems are derived by using martingale methods, regular vari- ation properties and Anscombe's theorem. -
STOCHASTIC COMPARISONS and AGING PROPERTIES of an EXTENDED GAMMA PROCESS Zeina Al Masry, Sophie Mercier, Ghislain Verdier
STOCHASTIC COMPARISONS AND AGING PROPERTIES OF AN EXTENDED GAMMA PROCESS Zeina Al Masry, Sophie Mercier, Ghislain Verdier To cite this version: Zeina Al Masry, Sophie Mercier, Ghislain Verdier. STOCHASTIC COMPARISONS AND AGING PROPERTIES OF AN EXTENDED GAMMA PROCESS. Journal of Applied Probability, Cambridge University press, 2021, 58 (1), pp.140-163. 10.1017/jpr.2020.74. hal-02894591 HAL Id: hal-02894591 https://hal.archives-ouvertes.fr/hal-02894591 Submitted on 9 Jul 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. Applied Probability Trust (4 July 2020) STOCHASTIC COMPARISONS AND AGING PROPERTIES OF AN EXTENDED GAMMA PROCESS ZEINA AL MASRY,∗ FEMTO-ST, Univ. Bourgogne Franche-Comt´e,CNRS, ENSMM SOPHIE MERCIER & GHISLAIN VERDIER,∗∗ Universite de Pau et des Pays de l'Adour, E2S UPPA, CNRS, LMAP, Pau, France Abstract Extended gamma processes have been seen to be a flexible extension of standard gamma processes in the recent reliability literature, for cumulative deterioration modeling purpose. The probabilistic properties of the standard gamma process have been well explored since the 1970's, whereas those of its extension remain largely unexplored. In particular, stochastic comparisons between degradation levels modeled by standard gamma processes and aging properties for the corresponding level-crossing times are nowadays well understood. -
Lecture Notes
Lecture Notes Lars Peter Hansen October 8, 2007 2 Contents 1 Approximation Results 5 1.1 One way to Build a Stochastic Process . 5 1.2 Stationary Stochastic Process . 6 1.3 Invariant Events and the Law of Large Numbers . 6 1.4 Another Way to Build a Stochastic Process . 8 1.4.1 Stationarity . 8 1.4.2 Limiting behavior . 9 1.4.3 Ergodicity . 10 1.5 Building Nonstationary Processes . 10 1.6 Martingale Approximation . 11 3 4 CONTENTS Chapter 1 Approximation Results 1.1 One way to Build a Stochastic Process • Consider a probability space (Ω, F, P r) where Ω is a set of sample points, F is an event collection (sigma algebra) and P r assigns proba- bilities to events . • Introduce a function S :Ω → Ω such that for any event Λ, S−1(Λ) = {ω ∈ Ω: S(ω) ∈ Λ} is an event. • Introduce a (Borel measurable) measurement function x :Ω → Rn. x is a random vector. • Construct a stochastic process {Xt : t = 1, 2, ...} via the formula: t Xt(ω) = X[S (ω)] or t Xt = X ◦ S . Example 1.1.1. Let Ω be a collection of infinite sequences of real numbers. Specifically, ω = (r0, r1, ...), S(ω) = (r1, r2, ...) and x(ω) = r0. Then Xt(ω) = rt. 5 6 CHAPTER 1. APPROXIMATION RESULTS 1.2 Stationary Stochastic Process Definition 1.2.1. The transformation S is measure-preserving if: P r(Λ) = P r{S−1(Λ)} for all Λ ∈ F. Proposition 1.2.2. When S is measure-preserving, the process {Xt : t = 1, 2, ...} has identical distributions for every t. -
Arbitrage Free Approximations to Candidate Volatility Surface Quotations
Journal of Risk and Financial Management Article Arbitrage Free Approximations to Candidate Volatility Surface Quotations Dilip B. Madan 1,* and Wim Schoutens 2,* 1 Robert H. Smith School of Business, University of Maryland, College Park, MD 20742, USA 2 Department of Mathematics, KU Leuven, 3000 Leuven, Belgium * Correspondence: [email protected] (D.B.M.); [email protected] (W.S.) Received: 19 February 2019; Accepted: 10 April 2019; Published: 21 April 2019 Abstract: It is argued that the growth in the breadth of option strikes traded after the financial crisis of 2008 poses difficulties for the use of Fourier inversion methodologies in volatility surface calibration. Continuous time Markov chain approximations are proposed as an alternative. They are shown to be adequate, competitive, and stable though slow for the moment. Further research can be devoted to speed enhancements. The Markov chain approximation is general and not constrained to processes with independent increments. Calibrations are illustrated for data on 2695 options across 28 maturities for SPY as at 8 February 2018. Keywords: bilateral gamma; fast Fourier transform; sato process; matrix exponentials JEL Classification: G10; G12; G13 1. Introduction A variety of arbitrage free models for approximating quoted option prices have been available for some time. The quotations are typically in terms of (Black and Scholes 1973; Merton 1973) implied volatilities for the traded strikes and maturities. Consequently, the set of such quotations is now popularly referred to as the volatility surface. Some of the models are nonparametric with the Dupire(1994) local volatility model being a popular example. The local volatility model is specified in terms of a deterministic function s(K, T) that expresses the local volatility for the stock if the stock were to be at level K at time T. -
Informs 2007 Proceedings
informs14th ® Applied Probability Conference July 9–11, 2007 Program Monday July 9, 2007 Track 1 Track 2 Track 3 Track 4 Track 5 Track 6 Track 7 Track 8 Track 9 Room CZ 4 CZ 5 CZ 10 CZ 11 CZ 12 CZ 13 CZ 14 CZ 15 CZ 16 9:00am - 9:15am Opening (Room: Blauwe Zaal) 9:15am - 10:15am Plenary - Peter Glynn (Room: Blauwe Zaal) MA Financial Random Fields Rare Event Asymptotic Scheduling Call Centers 1 MDP 1 Retrial Inventory 1 10:45am - 12:15pm Engineering 1 Simulation 1 Analysis 1 Queues Kou Kaj Dupuis Bassamboo / Borst / Koole Feinberg Artalejo Van Houtum Randhawa Wierman Keppo Scheffler Blanchet Lin Gupta Taylor Bispo Machihara Buyukkaramikli DeGuest Ruiz-Medina Glasserman Tezcan Ayesta Jongbloed Van der Laan Nobel Qiu Peng Kaj Juneja Gurvich Wierman Henderson Haijema Shin Timmer Weber Mahmoodi Dupuis Randhawa Winands Koole Feinberg Artalejo Van Houtum 12:45pm - 1.45pm Tutorial Philippe Robert MB Financial Percolation and Simulation 1 Stability of Stoch. Communication Many-server Games 1 Fluid Queues Search 2:00pm - 3:30pm Engineering 2 Related Topics Networks Systems 1 Models 1 Models Schoutens / Van den Berg Henderson Ramanan Choi Armony Economou Adan Klafter Valdivieso Werker Newman Chick Gamarnik Bae Tezcan Economou Dieker Benichou Koch Newman Haas Reiman Kim Jennings Amir Nazarathy Oshanin Scherer Meester Blanchet Williams Park Ward Dube Margolius Eliazar Valdivieso Kurtz Henderson Zachary Roubos Armony Economou Adan Metzler MC Exit Times Interacting Stoch. Prog. Stoch. Netw. & Flow-Level Markov Control Queueing Inventory 2 4:00pm - 5:30pm -
Portfolio Optimization and Option Pricing Under Defaultable Lévy
Universit`adegli Studi di Padova Dipartimento di Matematica Scuola di Dottorato in Scienze Matematiche Indirizzo Matematica Computazionale XXVI◦ ciclo Portfolio optimization and option pricing under defaultable L´evy driven models Direttore della scuola: Ch.mo Prof. Paolo Dai Pra Coordinatore dell’indirizzo: Ch.ma Prof.ssa. Michela Redivo Zaglia Supervisore: Ch.mo Prof. Tiziano Vargiolu Universit`adegli Studi di Padova Corelatori: Ch.mo Prof. Agostino Capponi Ch.mo Prof. Andrea Pascucci Johns Hopkins University Universit`adegli Studi di Bologna Dottorando: Stefano Pagliarani To Carolina and to my family “An old man, who had spent his life looking for a winning formula (martingale), spent the last days of his life putting it into practice, and his last pennies to see it fail. The martingale is as elusive as the soul.” Alexandre Dumas Mille et un fantˆomes, 1849 Acknowledgements First and foremost, I would like to thank my supervisor Prof. Tiziano Vargiolu and my co-advisors Prof. Agostino Capponi and Prof. Andrea Pascucci, for coauthoring with me all the material appearing in this thesis. In particular, thank you to Tiziano Vargiolu for supporting me scientifically and finan- cially throughout my research and all the activities related to it. Thank you to Agostino Capponi for inviting me twice to work with him at Purdue University, and for his hospital- ity and kindness during my double stay in West Lafayette. Thank you to Andrea Pascucci for wisely leading me throughout my research during the last four years, and for giving me the chance to collaborate with him side by side, thus sharing with me his knowledge and his experience. -
CONDITIONAL ERGODICITY in INFINITE DIMENSION∗ by Xin
CONDITIONAL ERGODICITY IN INFINITE DIMENSION∗ By Xin Thomson Tong and Ramon van Handel Princeton University The goal of this paper is to develop a general method to establish conditional ergodicity of infinite-dimensional Markov chains. Given a Markov chain in a product space, we aim to understand the ergodic properties of its conditional distributions given one of the compo- nents. Such questions play a fundamental role in the ergodic theory of nonlinear filters. In the setting of Harris chains, conditional ergod- icity has been established under general nondegeneracy assumptions. Unfortunately, Markov chains in infinite-dimensional state spaces are rarely amenable to the classical theory of Harris chains due to the sin- gularity of their transition probabilities, while topological and func- tional methods that have been developed in the ergodic theory of infinite-dimensional Markov chains are not well suited to the inves- tigation of conditional distributions. We must therefore develop new measure-theoretic tools in the ergodic theory of Markov chains that enable the investigation of conditional ergodicity for infinite dimen- sional or weak-* ergodic processes. To this end, we first develop local counterparts of zero-two laws that arise in the theory of Harris chains. These results give rise to ergodic theorems for Markov chains that ad- mit asymptotic couplings or that are locally mixing in the sense of H. F¨ollmer,and to a non-Markovian ergodic theorem for stationary ab- solutely regular sequences. We proceed to show that local ergodicity is inherited by conditioning on a nondegenerate observation process. This is used to prove stability and unique ergodicity of the nonlinear filter. -
A Representation Free Quantum Stochastic Calculus
JOURNAL OF FUNCTIONAL ANALYSIS 104, 149-197 (1992) A Representation Free Quantum Stochastic Calculus L. ACCARDI Centro Matemarico V. Volterra, Diparfimento di Matematica, Universitci di Roma II, Rome, Italy F. FACNOLA Dipartimento di Matematica, Vniversild di Trento, Povo, Italy AND J. QUAEGEBEUR Department of Mathemarics, Universiteit Leuven, Louvain, Belgium Communicated by L. Gross Received January 2; revised March 1, 1991 We develop a representation free stochastic calculus based on three inequalities (semimartingale inequality, scalar forward derivative inequality, scalar conditional variance inequality). We prove that our scheme includes all the previously developed stochastic calculi and some new examples. The abstract theory is applied to prove a Boson Levy martingale representation theorem in bounded form and a general existence, uniqueness, and unitarity theorem for quantum stochastic differential equations. 0 1992 Academic Press. Inc. 0. INTRODUCTION Stochastic calculus is a powerful tool in classical probability theory. Recently various kinds of stochastic calculi have been introduced in quan- tum probability [18, 12, 10,211. The common features of these calculi are: - One starts from a representation of the canonical commutation (or anticommutation) relations (CCR or CAR) over a space of the form L2(R+, dr; X) (where 3? is a given Hilbert space). - One introduces a family of operator valued measures on R,, related to this representation, e.g., expressed in terms of creation or annihilation or number or field operators. 149 0022-1236192 $3.00 Copyright 0 1992 by Academic Press, Inc. All rights of reproduction in any form reserved. 150 QUANTUM STOCHASTIC CALCULUS -- One shows that it is possible to develop a theory of stochastic integration with respect to these operator valued measures sufficiently rich to allow one to solve some nontrivial stochastic differential equations. -
Normalized Random Measures Driven by Increasing Additive Processes
The Annals of Statistics 2004, Vol. 32, No. 6, 2343–2360 DOI: 10.1214/009053604000000625 c Institute of Mathematical Statistics, 2004 NORMALIZED RANDOM MEASURES DRIVEN BY INCREASING ADDITIVE PROCESSES By Luis E. Nieto-Barajas1, Igor Prunster¨ 2 and Stephen G. Walker3 ITAM-M´exico, Universit`adegli Studi di Pavia and University of Kent This paper introduces and studies a new class of nonparamet- ric prior distributions. Random probability distribution functions are constructed via normalization of random measures driven by increas- ing additive processes. In particular, we present results for the dis- tribution of means under both prior and posterior conditions and, via the use of strategic latent variables, undertake a full Bayesian analysis. Our class of priors includes the well-known and widely used mixture of a Dirichlet process. 1. Introduction. This paper considers the problem of constructing a stochastic process, defined on the real line, which has sample paths be- having almost surely (a.s.) as a probability distribution function (d.f.). The law governing the process acts as a prior in Bayesian nonparametric prob- lems. One popular idea is to take the random probability d.f. as a normal- ized increasing process F (t)= Z(t)/Z¯, where Z¯ = limt→∞ Z(t) < +∞ (a.s.). For example, the Dirichlet process [Ferguson (1973)] arises when Z is a suitably reparameterized gamma process. We consider the case of normal- ized random d.f.s driven by an increasing additive process (IAP) L, that is, Z(t)= k(t,x) dL(x) and provide regularity conditions on k and L to ensure F is a random probability d.f. -
A Note on Chaotic and Predictable Representations for It\^ O-Markov
A NOTE ON CHAOTIC AND PREDICTABLE REPRESENTATIONS FOR ITO-MARKOVˆ ADDITIVE PROCESSES ZBIGNIEW PALMOWSKI, ŁUKASZ STETTNER, AND ANNA SULIMA ABSTRACT. In this paper we provide predictable and chaotic representations for Itˆo-Markov additive processes X. Such a process is governed by a finite-state CTMC J which allows one to modify the pa- rameters of the Itˆo-jump process (in so-called regime switching manner). In addition, the transition of J triggers the jump of X distributed depending on the states of J just prior to the transition. This family of processes includes Markov modulated Itˆo-L´evy processes and Markov additive processes. The derived chaotic representation of a square-integrable random variable is given as a sum of stochastic integrals with respect to some explicitly constructed orthogonal martingales. We identify the predictable representation of a square-integrable martingale as a sum of stochastic integrals of predictable processes with respect to Brownian motion and power-jumps martingales related to all the jumps appearing in the model. This re- sult generalizes the seminal result of Jacod-Yor and is of importance in financial mathematics. The derived representation then allows one to enlarge the incomplete market by a series of power-jump assets and to price all market-derivatives. KEYWORDS. Markov additive processes ⋆ martingale representation ⋆ power-jump process ⋆ orthogo- nal polynomials ⋆ stochastic integral ⋆ Brownian motion ⋆ regime switching ⋆ complete market CONTENTS 1. Introduction 2 2. Main results 3 3. Proof of predictable representation 8 3.1. Polynomial representation 8 4. Proofs of main results 13 4.1. Proof of Theorem 2 13 arXiv:1612.09216v2 [math.PR] 25 Aug 2017 4.2. -
The Generalized Join the Shortest Orbit Queue System: Stability, Exact Tail Asymptotics and Stationary Approximations
The generalized join the shortest orbit queue system: Stability, exact tail asymptotics and stationary approximations Ioannis Dimitriou∗ Department of Mathematics, University of Patras, 26504 Patras, Greece April 19, 2021 Abstract We introduce the generalized join the shortest queue model with retrials and two infinite capac- ity orbit queues. Three independent Poisson streams of jobs, namely a smart, and two dedicated streams, flow into a single server system, which can hold at most one job. Arriving jobs that find the server occupied are routed to the orbits as follows: Blocked jobs from the smart stream are routed to the shortest orbit queue, and in case of a tie, they choose an orbit randomly. Blocked jobs from the dedicated streams are routed directly to their orbits. Orbiting jobs retry to connect with the server at different retrial rates, i.e., heterogeneous orbit queues. Applications of such a system are found in the modelling of wireless cooperative networks. We are interested in the asymptotic behaviour of the stationary distribution of this model, provided that the system is stable. More precisely, we investigate the conditions under which the tail asymptotic of the minimum orbit queue length is exactly geometric. Moreover, we apply a heuristic asymptotic approach to obtain approx- imations of the steady-state joint orbit queue-length distribution. Useful numerical examples are presented and shown that the results obtained through the asymptotic analysis and the heuristic approach agreed. Keywords: Generalized Join the Shortest Orbit Queue; Retrials; Exact tail asymptotics; Station- ary approximations; Stability 1 Introduction In this work, we focus on the asymptotic stationary behaviour of the generalized join the shortest orbit queue (GJSOQ) policy with retrials. -
Markov-Modulated and Feedback Fluid Queues
Markov-modulated and feedback fluid queues Werner Scheinhardt Faculty of Mathematical Sciences University of Twente P.O. Box 217 7500 AE Enschede The Netherlands ISBN 90-3651248-4 MARKOV-MODULATED AND FEEDBACK FLUID QUEUES PROEFSCHRIFT ter verkrijging van de graad van doctor aan de Universiteit Twente, op gezag van de rector magnificus, prof. dr. F.A. van Vught, volgens besluit van het College voor Promoties in het openbaar te verdedigen op vrijdag 4 december 1998 te 15.00 uur. door Willem Richard Werner Scheinhardt geboren op 24 februari 1969 te Santiago Dit proefschrift is goedgekeurd door de promotor en de assistent promotor, prof. dr. ir. J.H.A. de Smit dr. ir. E.A. van Doorn Voorwoord Aan het eind van dit proefschrift gekomen, rest nog het schrijven van het begin ervan, het voorwoord. Gebruikelijk is om daarin allen te bedanken die op ´e´en of andere wijze aan de totstandkoming van het proefschrift hebben bijgedragen. Graag houd ik deze traditie in ere, en wel omdat de volgende personen dit ten volle verdienen. Uiteraard wil ik beginnen met mijn dagelijks begeleider Erik van Doorn hartelijk te bedanken voor zijn inzet en enthousiasme. Van v´o´or het eerste sollicitatiegesprek tot na het laatste 2-teken was hij intensief betrokken bij mijn doen en laten. Een belangrijk deel van het onderzoek in dit proefschrift is in samenwerking met hem tot stand gekomen. Ook de samenwerking met Dick Kroese was aangenaam en productief. Hij wist vaak antwoorden die ik zelf niet zou hebben gevonden, en leerde mij omgaan met het symbolisch manipulatie-pakket \Mathematica", dat heel wat \tiresome analysis" heeft uitgevoerd.