DOCSLIB.ORG

Long-Run Uncertainty and Value

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Long-Run Uncertainty and Value Long-run Uncertainty and Value Lars Peter Hansen University of Chicago and NBER Thomas J. Sargent New York University and NBER Jose Scheinkman Princeton University and NBER July 9, 2009 Contents Contents vii List of Figures ix Preface xi Acknowledgements xiii 1 Stochastic Processes 1 1.1 Constructing a Stochastic Process: I . 1 1.2 StationaryStochasticProcesses . 3 1.3 Invariant Events and the Law of Large Numbers . 4 1.4 Constructing a Stochastic Process: II . 7 1.5 Stationarityreconsidered . 9 1.6 LimitingBehavior. .. .. 10 1.7 Finite-stateMarkovchain . 14 1.8 VectorAutoregression . 16 2 Additive Functionals 19 2.1 Construction .......................... 19 2.2 Limiteddependence. 20 2.3 MartingaleExtraction . 22 2.4 Cointegration .......................... 24 2.5 Identifying Shocks with Long-Run Consequences . 25 2.6 CentralLimitTheory. 30 2.7 Growth-rateRegimes . 33 2.8 AQuadraticModelofGrowth . 34 3 Multiplicative Functionals 37 vii viii CONTENTS 3.1 MultiplicativeGrowth . 37 3.2 FourGrowthSpecifications. 40 3.3 Multiplicative Martingale Extraction . 46 4 Likelihood-ratio Processes 49 4.1 LikelihoodProcesses . 49 4.2 Log-Likelihoods ......................... 52 4.3 Scoreprocess .......................... 53 4.4 Limiting Behavior of the Likelihood Ratio . 55 4.5 Twisted Probability Measures . 57 Bibliography 59 List of Figures 1.1 This figure depicts the state evolution as a function of sample point ω induced by the transformation S. The oval shaped region is the collection Ω of all sample points. 2 2.1 The top panel plots the logarithm of consumption (smooth blue series) and logarithm of corporate earnings (choppy red series). The bottom panel plots the difference in the logarithms of con- sumptionandcorporateearnings. 26 2.2 This figure plots the response of the logarithm of consumption to a permanent shock (solid) and and to a temporary shock (dashed line). The permanent shock is identified as the increment to the common martingale component of the logarithm of consumption and the logarithm of corporate earnings. The figure comes from Hansenetal.(2008). .. .. 28 2.3 This figure plots posterior histograms for the magnitudes of the short-term and long-term responses of the logarithm of consump- tion to the shocks. The magnitude is measured as the absolute value across the contributions from the two shocks. The up- per panel depicts the histogram for the immediate response and the lower panel depicts the histogram for the long-term limiting response. The figure comes from Hansen et al. (2008). 29 2.4 This figure plots the impulse responses for the logarithm of out- put obtained by estimating a three-variate vector autoregression. The shocks are identified as in Fisher (2006). The top panel gives the response to an investment-specific technology shock, the middle panel gives the response to neutral technology shock and the bottom panel gives the response to a transient shock that is uncorrelated with the other two shocks. 31 ix x List of Figures 2.5 This figure plots the impulse responses for the logarithm of hours obtained by estimating a three-variate vector autoregression. The shocks are identified as in Fisher (2006). The top panel gives the response to an investment-specific technology shock, the middle panel gives the response to neutral technology shock and the bottom panel gives the response to a transient shock that is uncorrelated with the other two shocks. 32 3.1 This figure illustrates a particular additive functional and its multiplicativecounterpart. 38 3.2 The top panel plots a simulation of the multiplicative process studied in example 3.2.2, while the bottom panel plots a simu- lation of the Cecchetti et al. (2000) process for annual aggregate U.S. mentioned in example 3.2.3. 43 4.1 The logarithmic function is the concave function in this plot. This function is zero when evaluated at unity. By forming aver- ages using the two endpoints of the straight line below the loga- rithmic function, we are lead to some point on the line segment depending upon the weights used in the averaging. Jensen’s in- equality in this case is just a statement that the line segment must lie below the logarithmic function. 52 4.2 The vertical axis is implied asymptotic bound on the decay rate for the mistake probabilities for distinguishing between two mod- els. The bound is scaled by 100 an thus expressed as a percent. This bound is reported for alternative choices of α as depicted on the horizontal axis. The difference between the two curves reflects the choice of models in the numerator and denominator of the likelihood ratio. The best bound is given by the maximum of the two curves. The maximizing value for each curve is the same and given a maximizing choice α∗ of one curve, the other curve is maximized at 1 α∗. ................... 57 − Preface This manuscript started off as the Toulouse Lectures given by Lars Pe- ter Hansen. Our aim is to explore connections among topics that relate probability theory to the analysis of dynamic stochastic economic systems. Martingale methods have been a productive way to identify shocks with long-term consequences to economic growth and to characterize long-run dependence among macroeconomic time series. Typically they are applied by taking logarithms of time series such as output or consumption in order that growth can be modeled as accumulating linearly over time, albeit in a stochastic fashion. Martingale methods applied in this context have a long history in applied probability and applied time series analysis. We review these methods in the first part of this monograph. In the study of valu- ation, an alternative martingale approach provides a notion of long-term approximation. This approach borrows insights from large deviation the- ory, initiated in part to study the behavior of likelihood ratios of alternative time series models. We show how such methods provide characterizations of long-term model components and long-term contributions to valuation. Large deviation theory and the limiting behavior of likelihood ratios has also been central to some formulations of robust decision making. We de- velop this connection and build links to recursive utility theory in which investors care about the intertemporal composition of risk. Our interest in “robustness” and likelihood ratios is motivated by our conjecture that the modeling of the stochastic components to long-term growth is chal- lenging for both econometricians and the investors inside the models that econometricians build. More technical developments of some of these themes are given in Hansen and Scheinkman (1995), Anderson et al. (2003), Hansen and Scheinkman (2009) and Hansen (2008). xi Acknowledgements The author wishes to thank Jarda Borovicka, Adam Clemens, Serhiy Kokak, Junghoon Lee and Kenji Wada for their comments and assistance. These notes give some of the underlying support material for the Toulouse Lectures with the listed title. xiii Chapter 1 Stochastic Processes In this chapter, we describe two ways of constructing stochastic processes. The first is one that is especially convenient for stating and proving limit theorems. The second is more superficial in the sense that it directly speci- fies objects that are outcomes in the first construction. However, the second construction is the one that is most widely used in modeling economic time series. We shall use these constructions to characterize limiting behav- ior both for stationary environments and for environments with stochastic growth. 1.1 Constructing a Stochastic Process: I We begin with a method of constructing a stochastic process that is con- venient for characterizing the limit of points of time series averages.1 This construction works with a deterministic transformation S that maps a state of the world ω Ω today into a state of the world S(ω) Ω tomorrow. The state of the∈ world itself is not observed. Instead, a vect∈ or X(ω) that contains incomplete information about ω is observed. We assign probabili- ties over states of the world ω, then use the functions S and X to deduce a joint probability distribution for a sequence of X’s. In more detail: The probability space is a triple (Ω, F,Pr), where Ω is a set of sam- • ple points, F is an event collection (sigma algebra), and Pr assigns 1A good reference for the material in this section and the two that follow it is Breiman (1968). 1 2 CHAPTER 1. STOCHASTIC PROCESSES . . S3(2) S( 2) 2. S2(2) Figure 1.1: This figure depicts the state evolution as a function of sample point ω induced by the transformation S. The oval shaped region is the collection Ω of all sample points. probabilities to events. The (measurable) transformation S : Ω Ω used to model the evo- • lution over time has the property that for→ any event Λ F, ∈ S−1(Λ) = ω Ω : S(ω) Λ { ∈ ∈ } is an event. Notice that St(ω) is a deterministic sequence of states of the world in Ω. The vector-valued function X : Ω Rn used to model observations • → is Borel measurable. That is for any Borel set b in Rn, Λ= ω Ω : X(ω) b F. { ∈ ∈ } ∈ In other words, X is a random vector. 1.2. STATIONARY STOCHASTIC PROCESSES 3 The stochastic process Xt : t =1, 2, ... used to model a sequence of • observations is constructed{ via the formula:} t Xt(ω)= X[S (ω)] or X = X St. t ◦ The stochastic process X : t =1, 2, ... is a sequence of n-dimensional { t } random vectors, and the probability measure Pr allows us to make proba- bilistic statements about the joint distribution of successive components of this sequence. It will sometimes be convenient to extend this construction to date zero by letting X0 = X. While this construction of a stochastic process may at first sight appear special, it is not, as the following example illustrates.
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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • Financial Modeling with Volterra Processes and Applications to Options, Interest Rates and Credit Risk Sami El Rahouli
    Financial modeling with Volterra processes and applications to options, interest rates and credit risk Sami El Rahouli To cite this version: Sami El Rahouli. Financial modeling with Volterra processes and applications to options, interest rates and credit risk. Probability [math.PR]. Université de Lorraine, Université du Luxembourg, 2014. English. tel-01303063 HAL Id: tel-01303063 https://hal.archives-ouvertes.fr/tel-01303063 Submitted on 15 Apr 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. Universit´ede Lorraine Universit´edu Luxembourg UFR S.T.M.I.A. et Facult´edes Sciences, de la Technologie Ecole´ Doctorale IAEM et de la Communication D.F.D. Math´ematiques These pr´esent´eele 28 f´evrier2014 pour l'obtention du doctorat de l'Universit´ede Lorraine l'Universit´edu Luxembourg Discipline Math´ematiques Discipline Math´ematiques par Sami El Rahouli Mod´elisationfinanci`ereavec des processus de Volterra et applications aux options, aux taux d'int´er^etet aux risques de cr´edit Membres du jury : Marco DOZZI Professeur `al'universit´ede Lorraine Directeur de th`ese Yuliya MISHURA Professeur `al'universit´ede Kyiv, Ukraine Rapporteur Ivan NOURDIN Professeur `al'universit´ede Lorraine Examinateur Giovanni PECCATI Professeur `al'universit´edu Luxembourg Examinateur Francesco RUSSO Professeur `al'ENSTA-ParisTech Rapporteur Anton THALMAIER Professeur `al'universit´edu Luxembourg Directeur de th`ese Contents Introduction vii 1 L´evyProcesses and Gaussian semimartingales 1 1.1 L´evyprocesses: definition .
    [Show full text]
  • A Malliavin-Skorohod Calculus in $ L^ 0$ and $ L^ 1$ for Additive And
    A Malliavin-Skorohod calculus in L0 and L1 for additive and Volterra-type processes Giulia Di Nunno∗ Josep Vives†‡ August 23, 2018 Abstract In this paper we develop a Malliavin-Skorohod type calculus for additive processes in the L0 and L1 settings, extending the probabilistic interpretation of the Malliavin-Skorohod operators to this context. We prove calculus rules and obtain a generalization of the Clark-Hausmann-Ocone formula for random variables in L1. Our theory is then applied to extend the stochastic integration with respect to volatility modulated L´evy-driven Volterra processes recently introduced in the literature. Our work yields to substantially weaker conditions that permit to cover integration with respect to e.g. Volterra processes driven by α-stable processes with α< 2. The presentation focuses on jump type processes. Keywords: Additive processes, L´evy processes, α-stable processes, Volterra processes, Malliavin- Skorohod calculus. Mathematical Subject Classification: 60H05, 60H07. 1 Introduction Malliavin-Skorohod calculus for square integrable functionals of an additive process is today a well established topic. K. Itˆoproved in [17] the so-called chaos representation property of square integrable functionals of the Brownian motion. A generalized version of this property in terms of a random measure associated to a L´evy process was proved by the same author in [18]. Later, a Malliavin- Skorohod calculus for Gaussian processes strongly based on the chaos representation property was developed. We refer the reader to [24] for the Gaussian Malliavin-Skorohod calculus. In [25] it was proved that an abstract Malliavin-Skorohod calculus could be established on any Hilbert space with Fock space structure.
    [Show full text]
  • Markovian Building Blocks for Individual-Based Modelling
    Downloaded from orbit.dtu.dk on: Sep 27, 2021 Markovian Building Blocks for Individual-Based Modelling Nilsson, Lars Anders Fredrik Publication date: 2007 Document Version Publisher's PDF, also known as Version of record Link back to DTU Orbit Citation (APA): Nilsson, L. A. F. (2007). Markovian Building Blocks for Individual-Based Modelling. IMM-PHD-2008-171 General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. Users may download and print one copy of any publication from the public portal for the purpose of private study or research. You may not further distribute the material or use it for any profit-making activity or commercial gain You may freely distribute the URL identifying the publication in the public portal If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim. Markovian Building Blocks for Individual-Based Modelling L. A. Fredrik Nilsson Kongens Lyngby 2006 IMM-PHD-2006-XX Technical University of Denmark Informatics and Mathematical Modelling Building 321, DK-2800 Kongens Lyngby, Denmark Phone +45 45253351, Fax +45 45882673 [email protected] www.imm.dtu.dk IMM-PHD: ISSN 0909-3192 Summary The present thesis consists of a summary report, four research articles, one technical report and one manuscript. The subject of the thesis is individual- based stochastic models.
    [Show full text]
  • Dissertation
    DISSERTATION Titel der Dissertation Consistent dynamic equity market code-books from a practical point of view Verfasser Sara Karlsson angestrebter akademischer Grad Doktorin der Naturwissenschaften (Dr.rer.nat) Wien, Mai 2011 Studienkennzahl lt. Studienblatt: A 791 405 Dissertationsgebiet lt. Studienblatt: Mathematik Betreuer: Univ.-Prof. Mag. Dr. Walter Schachermayer Abstract Gebr¨auchliche Aktienpreis- bzw. Martingalmodelle beschreiben die Dy- namik des Preises einer Aktie unter einem Martingalmaß; grundlegendes Beispiel ist das Black-Scholes Modell. Im Gegensatz dazu zielt ein Markt- modell darauf ab die Dynamik eines ganzen Marktes (d.h. Aktienpreis plus abgeleitete Optionen) zu beschreiben. In der vorliegenden Arbeit besch¨aftigen wir uns mit arbitragefreien Modellen die die Dynamik eines Aktienpreises sowie flussig¨ gehandelter Derivate beschreiben. (\equity market models" respektive \market models for stock options"). Die Motivation derartige Modelle zu betrachten, liegt darin, dass eu- rop¨aische Optionen flussig¨ gehandelt werden und daher Ruckschl¨ usse¨ auf die zugrundeliegende Stochastik erlauben. Marktmodelle werden auch fur¨ Anleihen angewandt; wir verweisen auf Heath, Jarrow, and Morton[1992]. In j ungerer¨ Zeit beeinflußt dieser Zugang auch Aktienpreismodelle (siehe beispielsweise Derman and Kani [1998], Dupire[1996]). Derman und Kani schlagen vor, die dynamische Entwicklung von M¨arkten mittels Differentialgleichungen fur¨ Aktienpreis und volatility surface zu modellieren. Ein anderer Zugang wird von Sch¨onbucher[1999] gew ¨ahlt; hier ist der Ausgangspunkt die gemeinsame Dynamik von Aktienpreis und impliziten Black-Scholes Volatilit¨aten. Carmona and Nadtochiy[2009] schlagen ein Marktmodell vor, in dem der Aktienpreis als exponentieller Levy-Prozeß gegeben ist. In diesem Fall wird eine zeitinhomogene Levy-Dichte verwen- det um die zus¨atzlichen, durch Optionspreise gegebenen Informationen miteinzubeziehen.
    [Show full text]