DOCSLIB.ORG

Exact Bayesian and Particle Filtering of Stochastic Hybrid Systemssystemssystems

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Exact Bayesian and Particle Filtering of Stochastic Hybrid Systemssystemssystems UNCLASSIFIED Nationaal Lucht--- en Ruimtevaartlaboratorium National Aerospace Laboratory NLR Executive summary Exact Bayesian and particle filtering of stochastic hybrid systemssystemssystems Problem area Description of work Report no. In literature on Bayesian filtering of This report considers filtering of NLR-TP-2006-691 stochastic hybrid systems most stochastic hybrid systems that go studies are limited to Markov jump beyond the well known Markov Author(s) systems. The main exceptions are jump system. First the non- H.A.P. Blom, E.A. Bloem approximate Bayesian filters for Markovian jump system studied in Report classification semi-Markov jump linear systems. this report is formally defined. Next Unclassified These studies showed that nonlinear the exact Bayesian filter recursion filtering becomes much more for this system is developed and the Date challenging under non-Markov implication of going beyond September 2006 jumps. This challenge however does Markov jumps is shown. not apply to particle filtering of Subsequently a novel particle filter Knowledge area(s) stochastic hybrid systems. In for jump nonlinear systems is Advanced (sensor-) information practice, non-Markov jumps rather developed which is referred to as processing are the rule, not the exception. For the Interacting Multiple Model example, on an airport, the particle filter (IMMPF). For probability at which a taxiing comparison Monte Carlo aircraft makes a maneuver depends simulations are performed. heavily on its position; e.g. when Descriptor(s) taxiing near a crossing on the Results and conclusions Stochastic hybrid systems airport, the probability of starting a Through Monte Carlo simulations Bayesian filtering turn is relatively high, whereas IMMPF has been tested and Particle filtering outside these areas this probability compared with standard Particle State dependent switching may be very small. A similar Filter (PF) and IMM. The results Jump-nonlinear systems difference applies to the probability show that the IMMPF performs Non-Markov jumps Maneuvering target tracking very well, even in cases where the of an aircraft making a turn when it is flying near a waypoint versus standard PF or IMM have problems. flying halfway two waypoints. Similar non-Markov jump behavior Applicability also applies to other traffic The applicability of the work modalities, and to any other comprises the implementation of intelligently controlled system. the resulting filtering algorithms in Nevertheless in target tracking, a multitarget tracker, in particular particle filtering studies have the Advanced suRveillance Tracker continued to focus on Markov jump And Server ARTAS. systems Nationaal Lucht- en Ruimtevaartlaboratorium, National Aerospace Laboratory NLR Anthony Fokkerweg 2, 1059 CM Amsterdam, P.O. Box 90502, 1006 BM Amsterdam, The Netherlands UNCLASSIFIED Telephone +31 20 511 31 13, Fax +31 20 511 32 10, Web site: www.nlr.nl Nationaal Lucht- en Ruimtevaartlaboratorium National Aerospace Laboratory NLR NLR-TP-2006-691 Exact Bayesian and particle filtering of stochastic hybrid systems H.A.P. Blom and E.A. Bloem This report contains an article to appear in IEEE Transaction on Aerospace and Electronic Systems. This report may be cited on condition that full credit is given to NLR and the authors. Customer: National Aerospace Laboratory NLR Working Plan number: 2005 AT.1.C Owner: National Aerospace Laboratory NLR Division: Air Transport Distribution: Unlimited Classification title: Unclassified September 2006 Approved by: Author Reviewer Managing department Anonymous peer reviewers Exact Bayesian and Particle Filtering of Stochastic Hybrid Systems Henk A.P. Blom Senior Member IEEE & Edwin A. Bloem challenging under non-Markov jumps. This challenge Abstract —The standard way of applying particle filtering to however does not apply to particle filtering of stochastic stochastic hybrid systems is to make use of hybrid particles, hybrid systems [5]. Hence there is no reason to continue where each particle consists of two components, one assuming focusing on Markov jump systems. In practice, non-Markov Euclidean values, and the other assuming discrete mode values. This paper develops a novel particle filter for a jumps rather are the rule, not the exception. For example, discrete-time stochastic hybrid system. The novelty lies in the on an airport, the probability at which a taxiing aircraft use of the exact Bayesian equations for the conditional mode makes a maneuver depends heavily on its position; e.g. probabilities given the observations. Therefore particles are when taxiing near a crossing on the airport, the probability needed for the Euclidean valued state component only. The of starting a turn is relatively high, whereas outside these novel particle filter is referred to as the Interacting Multiple areas this probability may be very small. A similar Model (IMM) particle filter because it incorporates a filter step which is of the same form as the interaction step of the difference applies to the probability of an aircraft making a IMM algorithm. Through Monte Carlo simulations, it is turn when it is flying near a way-point versus flying halfway shown that the IMM particle filter has significant advantage two way-points. Similar non-Markov jump behavior also over the standard particle filter, in particular for situations applies to other traffic modalities, and to any other where conditional switching rate or conditional mode intelligently controlled system. Nevertheless in target probabilities have small values. tracking, particle filtering studies have continued to focus on Markov jump systems [6] – [19]. Keywords: Stochastic hybrid systems, Bayesian filtering, Particle filtering, state dependent switching, jump-nonlinear systems, non-Markov jumps, maneuvering target tracking. In nonlinear filtering studies, the Sampling Importance Resampling (SIR) based approach [20] has surfaced as the I. INTRODUCTION baseline particle filter. It has shown to form an elegant and The paradigm of particle filtering has stimulated a general approach towards the numerical evaluation of the renewed interest in exact Bayesian filtering of non-linear conditional density solution of the Chapman-Kolmogorov- stochastic systems. In line with this, [1] recently developed Bayes (CKB) filter recursion [21] – [24]. For a large class the exact Bayesian filter for Markov jump non-linear of problems it has been shown that increasing the number of systems. The aim of the current paper is to extend the exact particles ensures weak sense convergence of the Bayesian filter characterization to the much larger class of approximation density to the exact conditional density at a non-Markov jump non-linear systems, and subsequently to rate that is independent of or increases linearly with the exploit this exact Bayesian characterization for the state dimension of the process to be estimated [25] – [27]. development of a novel particle filter for stochastic hybrid This means that the SIR particle filter significantly relaxes systems. the curse of dimensionality of CKB filtering. Practically, In literature on Bayesian filtering of stochastic hybrid the SIR particle filter has found a large variety of useful systems most studies are limited to Markov jump systems. applications, including some where established non-linear The main exceptions are approximate Bayesian filters for filtering approaches do not work at all, such as the visual semi-Markov jump linear systems [2] – [4]. These studies tracking example of [28] and the track-before-detect showed that nonlinear filtering becomes much more example of [29]. Because of its generality and theoretical validity, the SIR particle filter has become a useful reference in numerically approximating CKB filter Manuscript received October 25, 2004; revised October 7, 2005; performance [18]. released for publication March, 2006 This work was supported in part by the European Commission project HYBRIDGE, IST-2001-32460. The SIR particle filter is also capable in approximating Both authors are with the National Aerospace Laboratory NLR, P.O. Box the CKB equations of a stochastic hybrid Markov process 90502, 1006BM Amsterdam, The Netherlands (phone: +31 20 5113544; θ n θ fax: +31 20 5113210; e-mail: [email protected], [email protected]). {xt , t } , with xt assuming values in R , and t assuming 1 θ values in a finite set M of possible modes [5]. Let {xt , t } work well without the need of ordering the samples be the hidden state process to be estimated from noisy according to their weights. • Boosting diversity by regularization, i.e. prior to observations, then the SIR particle filter use NP particles, resampling, replace each Dirac measure in the jθ j particle j of which has two components (xt , t ) at moment empirical distribution by an absolutely continuous j θ j t, with xt assuming an Euclidean value and t assuming a distribution. This approach has largely been mode value. Each SIR particle filter cycle from t-1 to t developed for particle filtering of stochastic consists of three steps: processes that, without regularization, do not satisfy the regularity conditions under which the • Evolution: For each of the N particles at moment t-1, p weak convergence to the exact conditional density jθ j is satisfied [12]. A valuable alternative is to insert draw a new hybrid particle (xt , t ) according to the Chapman-Kolmogorov transition kernel; a Monte Carlo Markov Chain (MCMC) move in each SIR cycle [31], [11]. • Correction: For each of the N particles evaluate µ j p t • Importance density based resampling. This means as the likelihood of the measurement at moment t; that particle resampling anticipates how the jθ j µ j given (xt , t ) and normalize the resulting t ’s. weights per particle will evolve over the next prediction and correction steps. Each new particle • Resampling: Draw N independently identically p µ j sample receives a weight t that compensates for jθ j distributed (i.i.d.) hybrid particle values (xt , t ) , with this importance resampling. The optimal µ j importance density has been characterized by from the sum of t weighted Dirac measures at Doucet et al. [32]. Because exact implementation (x j ,θ j ) . t t is hard, approximations are needed.
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]
  • 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.
    [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]
  • 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]
  • An Infinite Hidden Markov Model with Similarity-Biased Transitions
    An Infinite Hidden Markov Model With Similarity-Biased Transitions Colin Reimer Dawson 1 Chaofan Huang 1 Clayton T. Morrison 2 Abstract ing a “stick” off of the remaining weight according to a Beta (1; γ) distribution. The parameter γ > 0 is known We describe a generalization of the Hierarchical as the concentration parameter and governs how quickly Dirichlet Process Hidden Markov Model (HDP- the weights tend to decay, with large γ corresponding to HMM) which is able to encode prior informa- slow decay, and hence more weights needed before a given tion that state transitions are more likely be- cumulative weight is reached. This stick-breaking process tween “nearby” states. This is accomplished is denoted by GEM (Ewens, 1990; Sethuraman, 1994) for by defining a similarity function on the state Griffiths, Engen and McCloskey. We thus have a discrete space and scaling transition probabilities by pair- probability measure, G , with weights β at locations θ , wise similarities, thereby inducing correlations 0 j j j = 1; 2;::: , defined by among the transition distributions. We present an augmented data representation of the model i.i.d. θj ∼ H β ∼ GEM(γ): (1) as a Markov Jump Process in which: (1) some jump attempts fail, and (2) the probability of suc- G0 drawn in this way is a Dirichlet Process (DP) random cess is proportional to the similarity between the measure with concentration γ and base measure H. source and destination states. This augmenta- The actual transition distribution, πj, from state j, is drawn tion restores conditional conjugacy and admits a from another DP with concentration α and base measure simple Gibbs sampler.
    [Show full text]
  • Final Report (PDF)
    Foundation of Stochastic Analysis Krzysztof Burdzy (University of Washington) Zhen-Qing Chen (University of Washington) Takashi Kumagai (Kyoto University) September 18-23, 2011 1 Scientific agenda of the conference Over the years, the foundations of stochastic analysis included various specific topics, such as the general theory of Markov processes, the general theory of stochastic integration, the theory of martingales, Malli- avin calculus, the martingale-problem approach to Markov processes, and the Dirichlet form approach to Markov processes. To create some focus for the very broad topic of the conference, we chose a few areas of concentration, including • Dirichlet forms • Analysis on fractals and percolation clusters • Jump type processes • Stochastic partial differential equations and measure-valued processes Dirichlet form theory provides a powerful tool that connects the probabilistic potential theory and ana- lytic potential theory. Recently Dirichlet forms found its use in effective study of fine properties of Markov processes on spaces with minimal smoothness, such as reflecting Brownian motion on non-smooth domains, Brownian motion and jump type processes on Euclidean spaces and fractals, and Markov processes on trees and graphs. It has been shown that Dirichlet form theory is an important tool in study of various invariance principles, such as the invariance principle for reflected Brownian motion on domains with non necessarily smooth boundaries and the invariance principle for Metropolis algorithm. Dirichlet form theory can also be used to study a certain type of SPDEs. Fractals are used as an approximation of disordered media. The analysis on fractals is motivated by the desire to understand properties of natural phenomena such as polymers, and growth of molds and crystals.
    [Show full text]
  • Poisson Processes and Jump Diffusion Model
    Mälardalen University MT 1410 Analytical Finance I Teacher: Jan Röman, IMA Seminar 2005-10-05 Poisson Processes and Jump Diffusion Model Authors: Ying Ni, Cecilia Isaksson 13/11/2005 Abstract Besides the Wiener process (Brownian motion), there’s an another stochastic process that is very useful in finance, the Poisson process. In a Jump diffusion model, the asset price has jumps superimposed upon the familiar geometric Brownian motion. The occurring of those jumps are modelled using a Poisson process. This paper introduces the definition and properties of Poisson process and thereafter the Jump diffusion process which consists two stochastic components, the “white noise” created by Wiener process and the jumps generated by Poisson processes. MT1410 Seminar Group: Cecilia Isaksson; Ying Ni - 1 - 13/11/2005 Table of contents 1. Introduction …..………………………………………………………………….3 2. Poisson Processes ……………………………………………………………….4 2.1 Lévy Processes, Wiener Processes & Poisson Processes ……………………….4 2.2 The Markov Property of Poisson Processes ……………………………………..6 2.2 Applications of Poisson Processes in Finance……………………………………..8 3. The Jump Diffusion Model …………………………………………………….9 3.1 The model…………………………………………………………………………..9 3.2 Practical problem………………………………………………………………….10 4. Conclusion ………………………………………………………………………11 5. References ………………………………………………………………………..12 MT1410 Seminar Group: Cecilia Isaksson; Ying Ni - 2 - 13/11/2005 1. Introduction In option pricing theory, the geometric Brownian Motion is the most frequently used model for the price dynamic of an underlying asset. However, it is argued that this model fails to capture properly the unexpected price changes, called jumps. Price jumps are important because they do exist in the market. A more realistic model should therefore also take jumps into account. Price jumps are in general infrequent events and therefore the number of those jumps can be modelled by a Poisson processes.
    [Show full text]
  • STRONGLY VERTEX-REINFORCED JUMP PROCESS on a COMPLETE GRAPH Olivier Raimond, Tuan-Minh Nguyen
    STRONGLY VERTEX-REINFORCED JUMP PROCESS ON A COMPLETE GRAPH Olivier Raimond, Tuan-Minh Nguyen To cite this version: Olivier Raimond, Tuan-Minh Nguyen. STRONGLY VERTEX-REINFORCED JUMP PROCESS ON A COMPLETE GRAPH. 2018. hal-01894369 HAL Id: hal-01894369 https://hal.archives-ouvertes.fr/hal-01894369 Preprint submitted on 12 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. STRONGLY VERTEX-REINFORCED JUMP PROCESS ON A COMPLETE GRAPH OLIVIER RAIMOND AND TUAN-MINH NGUYEN Abstract. The aim of our work is to study vertex-reinforced jump processes with super-linear weight function w(t) = tα, for some α > 1. On any complete graph G = (V, E), we prove that there is one vertex v ∈ V such that the total time spent at v almost surely tends to infinity while the total time spent at the remaining vertices is bounded. Contents 1. Introduction 1 2. Preliminary notations and remarks 3 3. Dynamics of the occupation measure process 4 4. Convergence to equilibria 10 5. Non convergence to unstable equilibria 14 5.1. Attraction towards the unstable manifold. 14 5.2. Avoiding repulsive traps 16 5.3.
    [Show full text]
  • Arxiv:1802.06520V2 [Q-Fin.PR] 15 Sep 2018 [ ] Value of Spectral Domain and Exclude the Information Outside the Value
    Pricing Options with Exponential Lévy Neural Network a, Jeonggyu Huh ∗ aSchool of Computational Sciences, Korea Institute for Advanced Study, Seoul 02455, Republic of Korea Abstract In this paper, we propose the exponential Lévy neural network (ELNN) for option pricing, which is a new non- parametric exponential Lévy model using artificial neural networks (ANN). The ELNN fully integrates the ANNs with the exponential Lévy model, a conventional pricing model. So, the ELNN can improve ANN-based models to avoid several essential issues such as unacceptable outcomes and inconsistent pricing of over-the-counter prod- ucts. Moreover, the ELNN is the first applicable non-parametric exponential Lévy model by virtue of outstanding researches on optimization in the field of ANN. The existing non-parametric models are rather not robust for application in practice. The empirical tests with S&P 500 option prices show that the ELNN outperforms two parametric models, the Merton and Kou models, in terms of fitting performance and stability of estimates. Keywords: exponential Lévy model; artificial neural network; non-parametric model; option pricing JEL classification: C45, G13 1. Introduction A Lévy process is a stochastic process with independent and stationary increments, roughly speaking, which is a jump-diffusion process generalized so that its sample paths allow an infinite number of jumps on a finite time interval. For a long time, it has been widely employed for option pricing to overcome limitations of the Black-Scholes model [7], which is inconsistent with several well-known facts such as volatility skews, heavy tails of return distributions and so on (cf. Hull and Basu [17]).
    [Show full text]
  • Likelihood Evaluation of Jump-Diffusion Models Using
    Likelihood Evaluation of Jump-Diffusion Models Using Deterministic Nonlinear Filters∗ Jean-François Béginy;1 and Mathieu Boudreault2 1Simon Fraser University 2Université du Québec à Montréal This version: July 2, 2019 First draft: May 22, 2019 Abstract In this study, we develop a deterministic nonlinear filtering algorithm based on a high- dimensional version of Kitagawa (1987) to evaluate the likelihood function of models that allow for stochastic volatility and jumps whose arrival intensity is also stochastic. We show numerically that the deterministic filtering method is precise and much faster than the particle filter, in addition to yielding a smooth function over the parameter space. We then find the maximum likelihood estimates of various models that include stochastic volatility, jumps in the returns and variance, and also stochastic jump arrival intensity with the S&P 500 daily returns. During the Great Recession, the jump arrival intensity increases significantly and contributes to the clustering of volatility and negative returns. Keywords: Discrete nonlinear filtering; Particle filter; Sequential importance resampling; Stochastic volatility; Maximum likelihood estimation. arXiv:1906.04322v2 [q-fin.ST] 29 Jun 2019 ∗For their useful comments, the authors would like to thank Maciej Augustyniak, Geneviève Gauthier, Jonathan Grégoire, Frédéric Godin, as well as the seminar participants at the 2019 Quantact Workshop on Financial Mathemat- ics and the 23rd International Congress on Insurance: Mathematics and Economics. Bégin wishes to acknowledge the financial support of the National Science and Engineering Research Council of Canada (NSERC), Simon Fraser University and the NVIDIA Corporation. Boudreault would like to thank the financial support of NSERC. yCorresponding author. 1 Introduction and Motivation Over the last thirty years, jump-diffusion models (with stochastic volatility) have become in- creasingly popular for their ability to replicate important stylized facts such as heavy tails, no autocorrelation in the returns, volatility clustering, etc.
    [Show full text]
  • An Essay on the General Theory of Stochastic Processes
    Probability Surveys Vol. 3 (2006) 345–412 ISSN: 1549-5787 DOI: 10.1214/154957806000000104 An essay on the general theory of stochastic processes∗ Ashkan Nikeghbali ETHZ Departement Mathematik, R¨amistrasse 101, HG G16 Z¨urich 8092, Switzerland e-mail: [email protected] Abstract: This text is a survey of the general theory of stochastic pro- cesses, with a view towards random times and enlargements of filtrations. The first five chapters present standard materials, which were developed by the French probability school and which are usually written in French. The material presented in the last three chapters is less standard and takes into account some recent developments. AMS 2000 subject classifications: Primary 05C38, 15A15; secondary 05A15, 15A18. Keywords and phrases: General theory of stochastic processes, Enlarge- ments of filtrations, Random times, Submartingales, Stopping times, Hon- est times, Pseudo-stopping times. Received August 2005. Contents 1 Introduction................................. 346 Acknowledgements .. .. .. .. .. .. .. .. .. .. .. .. .. 347 2 Basicnotionsofthegeneraltheory . 347 2.1 Stoppingtimes ............................ 347 2.2 Progressive, Optional and Predictable σ-fields. 348 2.3 Classificationofstoppingtimes . 351 2.4 D´ebuttheorems. .. .. .. .. .. .. .. .. .. .. .. 353 3 Sectiontheorems .............................. 355 4 Projectiontheorems . .. .. .. .. .. .. .. .. .. .. .. 357 4.1 The optional and predictable projections . 357 4.2 Increasingprocessesandprojections . 360 4.3 Random measures on
    [Show full text]
  • Convergence of Jump Processes with Stochastic Intensity to Brownian Motion with Inert Drift
    Submitted to Bernoulli Convergence of jump processes with stochastic intensity to Brownian motion with inert drift CLAYTON BARNES1 1E-mail: [email protected] Consider a random walker on the nonnegative lattice, moving in continuous time, whose positive transition in- tensity is proportional to the time the walker spends at the origin. In this way, the walker is a jump process with a stochastic and adapted jump intensity. We show that, upon Brownian scaling, the sequence of such processes converges to Brownian motion with inert drift (BMID). BMID was introduced by Frank Knight in 2001 and generalized by White in 2007. This confirms a conjecture of Burdzy and White in 2008 in the one-dimensional setting. Keywords: Brownian motion; Discrete Approximation; Random Walk; Local Time 1. Introduction Brownian motion with inert drift (BMID) is a process X that satisfies the SDE dX = dB + dL + KLdt; (1) where K ≥ 0 is a constant and L is the local time of X at zero. This process behaves as a Brownian motion away from the origin but has a drift proportional to its local time at zero. Note that such a process is not Markovian because its drift depends on the past history. See Figure1 for sample path comparisons between reflected Brownian motion and BMID. BMID can be constructed path-by-path from a standard Brownian motion via the employment of a Skorohod map. This is discussed in more −n detail in Section2. We consider continuous time processes (Xn;Vn) on 2 N×R such that for K ≥ 0, n (i) Vn(t) := K2 · Leb(0 < s < t : Xn(s) = 0) is the scaled time Xn spends at the origin.
    [Show full text]