简明Levy过程.pptx

OutlineIntroduction:Probability and Stochastic ProcessesThe Structure of Levy ProcessesApplications to Finance The talk is based on the paper by David Applebaum(University of Sheffield,UK),Notices of the AMS,Vol.51,No 11.Introduction:Probability Theory of Probability:aims to model and to measure the Chance The tools:Kolmogorovs theory of probability axioms(1930s)Probability can be rigorously founded on measure theoryIntroduction:Stochastic ProcessesTheory of Stochastic Processes:aims to model the interaction of Chance and TimeStochastic Processes:a family of random variables(X(t),t=0)defined on a probability space(Omega,F,P)and taking values in a measurable space(E,G)X(t)is a(E,G)measurable mapping from Omega to E:a random observation made on E at time tImportance of Stochastic ProcessesNot only mathematically rich objectsApplications:physics,engineering,ecology,economics,finance,etc.Examples:random walks,Markov processes,semimartingales,measure-valued diffusions,Levy Processes,etc.Importance of Levy ProcessesThere are many important examples:Brownian motion,Poisson Process,stable processes,subordinators,etc.Generalization of random walks to continuous timeThe simplest classes of jump-diffusion processesA natural models of noise to build stochastic integrals and to drive SDETheir structure is mathematically robustTheir structure contains many features that generalize naturally to much wider classes of processes,such as semimartingales,Feller-Markov processes,etc.Main Original Contributors to the Theory of Levy Processes:1930s-1940sPaul Levy(France)Alexander Khintchine(Russia)Kiyosi Ito(Japan)Paul Levy(1886-1971)Main Original PapersLevy P.Sur les integrales dont les elements sont des variables aleatoires independentes,Ann.R.Scuola Norm.Super.Pisa,Sei.Fis.e Mat.,Ser.2(1934),v.III,337-366;Ser.4(1935),217-218Khintchine A.A new derivation of one formula by Levy P.,Bull.Moscow State Univ.,1937,v.I,No 1,1-5Ito K.On stochastic processes,Japan J.Math.18(1942),261-301Definition of Levy Processes X(t)X(t)has independent and stationary incrementsEach X(0)=0 w.p.1X(t)is stochastically continuous,i.e,for all a0 and for all s=0,P(|X(t)-X(s)|a)-0 when t-sThe Structure of Levy Processes:The Levy-Khintchine FormulaIf X(t)is a Levy process,then its characteristic function equals to whereExamples of Levy ProcessesBrownian motion:characteristic(0,a,0)Brownian motion with drift (Gaussian processes):characteristic(b,a,0)Poisson process:characteristic(0,0,lambdaxdelta1),lambda-intensity,delta1-Dirac mass concentrated at 1The compound Poisson processInterlacing processes=Gaussian process+compound Poisson processStable processesSubordinatorsRelativistic processesSimulation of Standard Brownian MotionSimulation of the Poisson ProcessStable Levy ProcessesStable probability distributions arise as the possible weak limit of normalized sums of i.i.d.r.v.in the central limit theoremExample:Cauchy Process with density(index of stability is 1)Simulation of the Cauchy ProcessSubordinatorsA subordinator T(t)is a one-dimensional Levy process that is non-decreasingImportant application:time change of Levy process X(t):Y(t):=X(T(t)is also a new Levy processSimulation of the Gamma SubordinatorThe Levy-Ito Decomposition:Structure of the Sample Paths of Levy ProcessesApplication to Finance.I.Replace Brownian motion in BSM model with a more general Levy process(P.Carr,H.Geman,D.Madan and M.Yor)Idea:1)small jumps term describes the day-to-day jitter that causes minor fluctuations in stock prices;2)big jumps term describes large stock price movements caused by major market upsets arising from,e.g.,earthquakes,etc.Main Problems with Levy Processes in Finance.Market is incomplete,i.e.,there may be more than one possible pricing formulaOne of the methods to overcome it:entropy minimizationExample:hyperbolic Levy process(E.Eberlain)(with no Brownian motion part);a pricing formula have been developed that has minimum entropyHyperbolic Levy Process:Characteristic FunctionBessel Function of the Third Kind(!)The Bessel function of the third kind or Hankel function Hn(x)is a(complex)combination of the two solutions of Bessel DE:the real part is the Bessel function of the first kind,the complex part the Bessel function of the second kind(very complicated!)Bessel Differential EquationApplication of Levy Processes in Finance.II.BSM formula contains the constant of volatilityOne of the methods to improve it:stochastic volatility models(SDE for volatility)Example:stochastic volatility is an Ornstein-Uhlenbeck process driven by a subordinator T(t)(O.Barndorff-Nielsen and N.Shephard)Stochastic Volatility Model Using Levy ProcessReferences on Levy Processes(Books)D.Applebaum,Levy Processes and Stochastic Calculus,Cambridge University Press,2004O.E.Barndorff-Nielsen,T.Mikosch and S.Resnick(Eds.),Levy Processes:Theory and Applications,Birkhauser,2001J.Bertoin,Levy Processes,Cambridge University Press,1996W.Schoutens,Levy Processes in Finance:Pricing Financial Derivatives,Wiley,2003R.Cont and P Tankov,Financial Modelling with Jump Processes,Chapman&Hall/CRC,2004Thank you for your attention!