Monte Carlo and Empirical Methods for Stochastic Inference (MASM11/FMSN50)
|
|
- Claude Reynolds
- 5 years ago
- Views:
Transcription
1 Monte Carlo and Empirical Methods for Stochastic Inference (MASM11/FMSN50) Magnus Wiktorsson Centre for Mathematical Sciences Lund University, Sweden Lecture 6 Sequential Monte Carlo methods II February 1, 2018 M. Wiktorsson Monte Carlo and Empirical Methods for Stochastic Inference, L6 (1) 1 / 27
2 Plan of today's lecture 1 Last time: Sequential MC problems 2 3 M. Wiktorsson Monte Carlo and Empirical Methods for Stochastic Inference, L6 (2) 2 / 27
3 We are here 1 Last time: Sequential MC problems 2 3 M. Wiktorsson Monte Carlo and Empirical Methods for Stochastic Inference, L6 (3) 3 / 27
4 Last time: Sequential MC problems In the sequential MC framework, we aim at sequentially estimating sequences (τ n ) n 0 of expectations τ n = E fn (φ(x 0:n )) = φ(x 0:n )f n (x 0:n ) dx 0:n ( ) X n over spaces X n of increasing dimension, where the densities (f n ) are known up to normalizing constants only, i.e., for every n 0, where c n is an unknown constant. f n (x 0:n ) = z n(x 0:n ) c n, M. Wiktorsson Monte Carlo and Empirical Methods for Stochastic Inference, L6 (4) 4 / 27
5 Last time: Markov chains Some applications involved the notion of Markov chains: A Markov chain on X R d is a family of random variables (= stochastic process) (X k ) k 0 taking values in X such that P(X k+1 B X 0, X 1,..., X k ) = P(X k+1 B X k ). The density q of the distribution of X k+1 given X k = x k is called the transition density of (X k ). Consequently, P(X k+1 B X k = x k ) = q(x k+1 x k ) dx k+1. As a rst example we considered an AR(1) process: X 0 = 0, X k+1 = αx k + ɛ k+1, where α is a constant and (ɛ k ) are i.i.d. variables. M. Wiktorsson Monte Carlo and Empirical Methods for Stochastic Inference, L6 (5) 5 / 27 B
6 Last time: Markov chains (cont.) The following theorem provides the joint density f n (x 0, x 1,..., x n ) of X 0, X 1,..., X n. Theorem Let (X k ) be Markov with X 0 χ. Then for n > 0, n 1 f n (x 0, x 1,..., x n ) = χ(x 0 ) q(x k+1 x k ). k=0 Corollary (The Chapman-Kolmogorov equation) Let (X k ) be Markov. Then for n > 1, f n (x n x 0 ) = ( n 1 k=0 q(x k+1 x k ) ) dx 1 dx n 1. M. Wiktorsson Monte Carlo and Empirical Methods for Stochastic Inference, L6 (6) 6 / 27
7 Last time: Rare event analysis (REA) for Markov chains Let (X k ) be a Markov chain. Assume that we want to compute, for n = 0, 1, 2,... τ n = E(φ(X 0:n ) X 0:n B) = = B B f n (x 0:n ) φ(x 0:n ) P(X 0:n B) dx 0:n φ(x 0:n ) χ(x 0) n 1 k=0 q(x k+1 x k ) dx 0:n, P(X 0:n B) where B is a possibly rare event and P(X 0:n B) is generally unknown. We thus face a sequential MC problem ( ) with { z n (x 0:n ) χ(x 0 ) n 1 k=0 q(x k+1 x k ), c n P(X 0:n B). M. Wiktorsson Monte Carlo and Empirical Methods for Stochastic Inference, L6 (7) 7 / 27
8 Last time: Estimation in general HMMs Graphically: Y k 1 Y k Y k+1 (Observations)... X k 1 X k X k+1... (Markov chain) Y k X k = x k p(y k x k ) X k+1 X k = x k q(x k+1 x k ) X 0 χ(x 0 ) (Observation density) (Transition density) (Initial distribution) M. Wiktorsson Monte Carlo and Empirical Methods for Stochastic Inference, L6 (8) 8 / 27
9 Last time: Estimation in general HMMs In an HMM, the smoothing distribution f n (x 0:n y 0:n ) is the conditional distribution of a set X 0:n of hidden states given Y 0:n = y 0:n. Theorem (Smoothing distribution) where f n (x 0:n y 0:n ) = χ(x 0)p(y 0 x 0 ) n k=1 p(y k x k )q(x k x k 1 ), L n (y 0:n ) L n (y 0:n ) = density of the observations y 0:n n = χ(x 0 )p(y 0 x 0 ) p(y k x k )q(x k x k 1 ) dx 0 dx n. k=1 M. Wiktorsson Monte Carlo and Empirical Methods for Stochastic Inference, L6 (9) 9 / 27
10 Last time: Estimation in general HMMs Assume that we want to compute, online for n = 0, 1, 2,..., τ n = E(φ(X 0:n ) Y 0:n = y 0:n ) = φ(x 0:n )f n (x 0:n y 0:n ) dx 0 dx n = φ(x 0:n ) χ(x 0)p(y 0 x 0 ) n k=1 p(y k x k )q(x k x k 1 ) dx 0 dx n, L n (y 0:n ) where L n (y 0:n ) (= obscene integral) is generally unknown. We thus face a sequential MC problem ( ) with { z n (x 0:n ) χ(x 0 )p(y 0 x 0 ) n k=1 p(y k x k )q(x k x k 1 ), c n L n (y 0:n ). M. Wiktorsson Monte Carlo and Empirical Methods for Stochastic Inference, L6 (10) 10 / 27
11 We are here 1 Last time: Sequential MC problems 2 3 M. Wiktorsson Monte Carlo and Empirical Methods for Stochastic Inference, L6 (11) 11 / 27
12 Conditional methods Say that we want to generate a random vector from a given bivariate density p(x, y). If we know how to draw from the conditional distribution p(y x) and the marginal p(x) this can be done naturally using the following scheme. draw Z 1 p(x) draw Z 2 p(y x = Z 1 ) return (Z 1, Z 2 ) M. Wiktorsson Monte Carlo and Empirical Methods for Stochastic Inference, L6 (12) 12 / 27
13 Conditional methods This can be naturally extended to n-variate densities p(x 1,..., x n ): draw Z 1 p(x 1 ) draw Z 2 p(x 2 x 1 = Z 1 ) draw Z 3 p(x 3 x 1 = Z 1, x 2 = Z 2 ). draw Z n 1 p(x n 1 x 1 = Z 1, x 2 = Z 2,..., x n 2 = Z n 2 ) draw Z n p(x n x 1 = Z 1, x 2 = Z 2,..., x n 1 = Z n 1 ) return (Z 1,..., Z n ) Theorem The vector (Z 1,..., Z n ) has indeed n-variate density function p(x 1,..., x n ). M. Wiktorsson Monte Carlo and Empirical Methods for Stochastic Inference, L6 (13) 13 / 27
14 We are here 1 Last time: Sequential MC problems 2 3 M. Wiktorsson Monte Carlo and Empirical Methods for Stochastic Inference, L6 (14) 14 / 27
15 We are here 1 Last time: Sequential MC problems 2 3 M. Wiktorsson Monte Carlo and Empirical Methods for Stochastic Inference, L6 (15) 15 / 27
16 It is natural to aim at solving the problem using usual self-normalized IS. However, the generated samples (Xi 0:n, ω n (Xi 0:n )) should be such that having (Xi 0:n, ω n (Xi 0:n )), the next sample (Xi 0:n+1, ω n+1 (Xi 0:n+1 )) is easily generated with a complexity that does not increase with n (online sampling). the approximation remains stable as n increases. We call each draw Xi 0:n = (Xi 0,..., Xn i ) a particle. Moreover, we denote importance weights by ωn i def = ω n (Xi 0:n ). M. Wiktorsson Monte Carlo and Empirical Methods for Stochastic Inference, L6 (16) 16 / 27
17 We are here 1 Last time: Sequential MC problems 2 3 M. Wiktorsson Monte Carlo and Empirical Methods for Stochastic Inference, L6 (17) 17 / 27
18 We proceed recursively. Assume that we have generated particles (X 0:n i ) from g n (x 0:n ) so that N i=1 ωn i N φ(x l=1 ωl i 0:n ) E fn (φ(x 0:n )), n where, as usual, ω i n = ω n (X 0:n i ) = z n (X 0:n i )/g n (X 0:n i ). Key trick: Choose an instrumental distribution satisfying g n+1 (x 0:n+1 ) = g n+1 (x n+1 x 0:n )g n+1 (x 0:n ) g n+1 (x 0:n+1 ) = g n+1 (x n+1 x 0:n )g n (x 0:n ). M. Wiktorsson Monte Carlo and Empirical Methods for Stochastic Inference, L6 (18) 18 / 27
19 SIS (cont.) Last time: Sequential MC problems Now assume that we have drawn X 0:n g n (x 0:n ). Then, as g n+1 (x 0:n+1 ) = g n+1 (x n+1 x 0:n )g n+1 (x 0:n ) = g n+1 (x n+1 x 0:n )g n (x 0:n ), the conditional method allows us to generate a draw X 0:n+1 from g n+1 (x 0:n+1 ) using the following procedure: draw X n+1 g n+1 (x n+1 x 0:n = X 0:n ) let X 0:n+1 (X 0:n, X n+1 ) This can be repeated recursively, yielding online sampling from the sequence (g n ). M. Wiktorsson Monte Carlo and Empirical Methods for Stochastic Inference, L6 (19) 19 / 27
20 SIS (cont.) Last time: Sequential MC problems Consequently, Xi 0:n+1 and ωn+1 i can be generated by keeping the previous Xi 0:n, simulating Xi n+1 g n+1 (x n+1 Xi 0:n ), setting Xi 0:n+1 = (Xi 0:n, Xi n+1 ), and computing ωn+1 i = z n+1(xi 0:n+1 ) g n+1 (Xi 0:n+1 ) z n+1 (Xi 0:n+1 ) = z n (Xi 0:n )g n+1 (X n+1 = z n+1 (X 0:n+1 i ) i Xi 0:n ) zn(x0:n i ) g n (X 0:n z n (X 0:n i )g n+1 (X n+1 i X 0:n i ) ωi n. i ) M. Wiktorsson Monte Carlo and Empirical Methods for Stochastic Inference, L6 (20) 20 / 27
21 SIS (cont.) Last time: Sequential MC problems Voilà, the sample (Xi 0:n+1, ωn+1 i ) can now be used to approximate E fn+1 (φ(x 0:n+1 ))! So, by running the SIS algorithm, we have updated an approximation to an approximation N i=1 N i=1 ωn i N φ(x l=1 ωl i 0:n ) E fn (φ(x 0:n )) n ωn+1 i N φ(x l=1 ωl i 0:n+1 ) E fn+1 (φ(x 0:n+1 )) n+1 by only adding a component Xi n+1 to Xi 0:n weights. and sequentially updating the M. Wiktorsson Monte Carlo and Empirical Methods for Stochastic Inference, L6 (21) 21 / 27
22 SIS: Pseudo code for i = 1 N do draw Xi 0 g 0 set ω0 i = z 0(Xi 0) g 0 (Xi 0) end for return (Xi 0, ωi 0 ) for k = 0, 1, 2,... do for i = 1 N do draw Xi k+1 g k+1 (x k+1 X 0:k set Xi 0:k+1 (Xi 0:k, Xi k+1 ) set ω i k+1 end for return (Xi 0:k+1, ωk+1 i ) end for i ) z k+1 (X 0:k+1 i ) z k (Xi 0:k )g k+1 (X k+1 i Xi 0:k ) ωi k M. Wiktorsson Monte Carlo and Empirical Methods for Stochastic Inference, L6 (22) 22 / 27
23 Example: REA reconsidered We consider again the example of REA for Markov chains (X = R, X 0 = x 0 = a): τ n = E(φ(X 0:n ) a X l, l = 0,..., n) = φ(x 0:n ) (a, ) n n 1 k=1 q(x k+1 x k ) P(a X l, l) } {{ } =z n(x 0:n )/c n dx 1:n. Choose g k+1 (x k+1 x 0:k ) to be the conditional density of X k+1 given X k and X k+1 a: g k+1 (x k+1 x 0:k ) = {cf. HA1, Problem 1} = q(x k+1 x k ) a q(z x k) dz. M. Wiktorsson Monte Carlo and Empirical Methods for Stochastic Inference, L6 (23) 23 / 27
24 Example: REA Last time: Sequential MC problems This implies that (recall that we have conditioned on X 0 = x 0 = a) g n (x 0:n ) = n 1 k=0 q(x k+1 x k ) a q(z x k) dz. In addition, the weights are updated according to ωk+1 i = z k+1 (Xi 0:k+1 ) z k (Xi 0:k )g k+1 (X k+1 = = k 1 l=0 q(xl+1 a i Xi 0:k k l=0 q(xl+1 i X l i ) ) ωi k i Xi l) q(xk+1 i Xi k) ωk i q(z Xk i ) dz q(z X k i ) dz ω i k. a M. Wiktorsson Monte Carlo and Empirical Methods for Stochastic Inference, L6 (24) 24 / 27
25 Example: REA; Matlab implementation for AR(1) process with Gaussian noise % design of instrumental distribution: int 1 normcdf(a,alpha*x,sigma); trunk_td_rnd =... % use e.g. HA1, Problem 1, to simulate % the conditional transition density; % SIS: part = a*ones(n,1); % initialization of all particles in a w = ones(n,1); for k = 1:(n 1), % main loop part_mut = trunk_td_rnd(part); w = w.*int(part); part = part_mut; end c = mean(w); % estimated probability M. Wiktorsson Monte Carlo and Empirical Methods for Stochastic Inference, L6 (25) 25 / 27
26 REA: Importance weight distribution Serious drawback of SIS: the importance weights degenerate! n = n = n = Importance weights (base 10 logarithm) M. Wiktorsson Monte Carlo and Empirical Methods for Stochastic Inference, L6 (26) 26 / 27
27 What's next? Last time: Sequential MC problems Weight degeneration is a universal problem with the SIS method and is due to the fact that the particle weights are generated through subsequent multiplications. This drawback preventedduring several decadesthe SIS method from being practically useful. Next week we will discuss an elegant solution to this problem: SIS with resampling (SISR). M. Wiktorsson Monte Carlo and Empirical Methods for Stochastic Inference, L6 (27) 27 / 27
Monte Carlo and Empirical Methods for Stochastic Inference (MASM11/FMSN50)
Monte Carlo and Empirical Methods for Stochastic Inference (MASM11/FMSN50) Magnus Wiktorsson Centre for Mathematical Sciences Lund University, Sweden Lecture 5 Sequential Monte Carlo methods I January
More informationMonte Carlo and Empirical Methods for Stochastic Inference (MASM11/FMSN50)
Monte Carlo and Empirical Methods for Stochastic Inference (MASM11/FMSN50) Magnus Wiktorsson Centre for Mathematical Sciences Lund University, Sweden Lecture 1 Introduction January 16, 2018 M. Wiktorsson
More informationMonte Carlo and Empirical Methods for Stochastic Inference (MASM11/FMSN50)
Monte Carlo and Empirical Methods for Stochastic Inference (MASM11/FMSN50) Magnus Wiktorsson Centre for Mathematical Sciences Lund University, Sweden Lecture 2 Random number generation January 18, 2018
More informationMonte Carlo and Empirical Methods for Stochastic Inference (MASM11/FMS091)
Monte Carlo and Empirical Methods for Stochastic Inference (MASM11/FMS091) Magnus Wiktorsson Centre for Mathematical Sciences Lund University, Sweden Lecture 3 Importance sampling January 27, 2015 M. Wiktorsson
More informationIntroduction to Sequential Monte Carlo Methods
Introduction to Sequential Monte Carlo Methods Arnaud Doucet NCSU, October 2008 Arnaud Doucet () Introduction to SMC NCSU, October 2008 1 / 36 Preliminary Remarks Sequential Monte Carlo (SMC) are a set
More informationSTAT/MATH 395 PROBABILITY II
STAT/MATH 395 PROBABILITY II Distribution of Random Samples & Limit Theorems Néhémy Lim University of Washington Winter 2017 Outline Distribution of i.i.d. Samples Convergence of random variables The Laws
More informationAnalysis of the Bitcoin Exchange Using Particle MCMC Methods
Analysis of the Bitcoin Exchange Using Particle MCMC Methods by Michael Johnson M.Sc., University of British Columbia, 2013 B.Sc., University of Winnipeg, 2011 Project Submitted in Partial Fulfillment
More information15 : Approximate Inference: Monte Carlo Methods
10-708: Probabilistic Graphical Models 10-708, Spring 2016 15 : Approximate Inference: Monte Carlo Methods Lecturer: Eric P. Xing Scribes: Binxuan Huang, Yotam Hechtlinger, Fuchen Liu 1 Introduction to
More informationCPSC 540: Machine Learning
CPSC 540: Machine Learning Monte Carlo Methods Mark Schmidt University of British Columbia Winter 2018 Last Time: Markov Chains We can use Markov chains for density estimation, p(x) = p(x 1 ) }{{} d p(x
More informationCPSC 540: Machine Learning
CPSC 540: Machine Learning Monte Carlo Methods Mark Schmidt University of British Columbia Winter 2019 Last Time: Markov Chains We can use Markov chains for density estimation, d p(x) = p(x 1 ) p(x }{{}
More informationLecture 17: More on Markov Decision Processes. Reinforcement learning
Lecture 17: More on Markov Decision Processes. Reinforcement learning Learning a model: maximum likelihood Learning a value function directly Monte Carlo Temporal-difference (TD) learning COMP-424, Lecture
More informationOn Complexity of Multistage Stochastic Programs
On Complexity of Multistage Stochastic Programs Alexander Shapiro School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, Georgia 30332-0205, USA e-mail: ashapiro@isye.gatech.edu
More informationChapter 5 Univariate time-series analysis. () Chapter 5 Univariate time-series analysis 1 / 29
Chapter 5 Univariate time-series analysis () Chapter 5 Univariate time-series analysis 1 / 29 Time-Series Time-series is a sequence fx 1, x 2,..., x T g or fx t g, t = 1,..., T, where t is an index denoting
More informationLecture 22: Dynamic Filtering
ECE 830 Fall 2011 Statistical Signal Processing instructor: R. Nowak Lecture 22: Dynamic Filtering 1 Dynamic Filtering In many applications we want to track a time-varying (dynamic) phenomenon. Example
More informationELEMENTS OF MONTE CARLO SIMULATION
APPENDIX B ELEMENTS OF MONTE CARLO SIMULATION B. GENERAL CONCEPT The basic idea of Monte Carlo simulation is to create a series of experimental samples using a random number sequence. According to the
More informationThe Binomial Lattice Model for Stocks: Introduction to Option Pricing
1/33 The Binomial Lattice Model for Stocks: Introduction to Option Pricing Professor Karl Sigman Columbia University Dept. IEOR New York City USA 2/33 Outline The Binomial Lattice Model (BLM) as a Model
More informationFast Computation of the Economic Capital, the Value at Risk and the Greeks of a Loan Portfolio in the Gaussian Factor Model
arxiv:math/0507082v2 [math.st] 8 Jul 2005 Fast Computation of the Economic Capital, the Value at Risk and the Greeks of a Loan Portfolio in the Gaussian Factor Model Pavel Okunev Department of Mathematics
More informationStrategies for High Frequency FX Trading
Strategies for High Frequency FX Trading - The choice of bucket size Malin Lunsjö and Malin Riddarström Department of Mathematical Statistics Faculty of Engineering at Lund University June 2017 Abstract
More informationComputer Exercise 2 Simulation
Lund University with Lund Institute of Technology Valuation of Derivative Assets Centre for Mathematical Sciences, Mathematical Statistics Spring 2010 Computer Exercise 2 Simulation This lab deals with
More informationEE641 Digital Image Processing II: Purdue University VISE - October 29,
EE64 Digital Image Processing II: Purdue University VISE - October 9, 004 The EM Algorithm. Suffient Statistics and Exponential Distributions Let p(y θ) be a family of density functions parameterized by
More informationWrite legibly. Unreadable answers are worthless.
MMF 2021 Final Exam 1 December 2016. This is a closed-book exam: no books, no notes, no calculators, no phones, no tablets, no computers (of any kind) allowed. Do NOT turn this page over until you are
More informationThe Use of Importance Sampling to Speed Up Stochastic Volatility Simulations
The Use of Importance Sampling to Speed Up Stochastic Volatility Simulations Stan Stilger June 6, 1 Fouque and Tullie use importance sampling for variance reduction in stochastic volatility simulations.
More informationHomework Assignments
Homework Assignments Week 1 (p. 57) #4.1, 4., 4.3 Week (pp 58 6) #4.5, 4.6, 4.8(a), 4.13, 4.0, 4.6(b), 4.8, 4.31, 4.34 Week 3 (pp 15 19) #1.9, 1.1, 1.13, 1.15, 1.18 (pp 9 31) #.,.6,.9 Week 4 (pp 36 37)
More informationRecent Advances in Fractional Stochastic Volatility Models
Recent Advances in Fractional Stochastic Volatility Models Alexandra Chronopoulou Industrial & Enterprise Systems Engineering University of Illinois at Urbana-Champaign IPAM National Meeting of Women in
More informationEstimating the Greeks
IEOR E4703: Monte-Carlo Simulation Columbia University Estimating the Greeks c 207 by Martin Haugh In these lecture notes we discuss the use of Monte-Carlo simulation for the estimation of sensitivities
More informationCourse information FN3142 Quantitative finance
Course information 015 16 FN314 Quantitative finance This course is aimed at students interested in obtaining a thorough grounding in market finance and related empirical methods. Prerequisite If taken
More informationEC316a: Advanced Scientific Computation, Fall Discrete time, continuous state dynamic models: solution methods
EC316a: Advanced Scientific Computation, Fall 2003 Notes Section 4 Discrete time, continuous state dynamic models: solution methods We consider now solution methods for discrete time models in which decisions
More informationKing s College London
King s College London University Of London This paper is part of an examination of the College counting towards the award of a degree. Examinations are governed by the College Regulations under the authority
More informationOn Solving Integral Equations using. Markov Chain Monte Carlo Methods
On Solving Integral quations using Markov Chain Monte Carlo Methods Arnaud Doucet Department of Statistics and Department of Computer Science, University of British Columbia, Vancouver, BC, Canada mail:
More informationAsymptotic results discrete time martingales and stochastic algorithms
Asymptotic results discrete time martingales and stochastic algorithms Bernard Bercu Bordeaux University, France IFCAM Summer School Bangalore, India, July 2015 Bernard Bercu Asymptotic results for discrete
More informationMonte Carlo Methods for Uncertainty Quantification
Monte Carlo Methods for Uncertainty Quantification Abdul-Lateef Haji-Ali Based on slides by: Mike Giles Mathematical Institute, University of Oxford Contemporary Numerical Techniques Haji-Ali (Oxford)
More informationModule 10:Application of stochastic processes in areas like finance Lecture 36:Black-Scholes Model. Stochastic Differential Equation.
Stochastic Differential Equation Consider. Moreover partition the interval into and define, where. Now by Rieman Integral we know that, where. Moreover. Using the fundamentals mentioned above we can easily
More informationCLIQUE OPTION PRICING
CLIQUE OPTION PRICING Mark Ioffe Abstract We show how can be calculated Clique option premium. If number of averaging dates enough great we use central limit theorem for stochastic variables and derived
More informationIEOR E4703: Monte-Carlo Simulation
IEOR E4703: Monte-Carlo Simulation Generating Random Variables and Stochastic Processes Martin Haugh Department of Industrial Engineering and Operations Research Columbia University Email: martin.b.haugh@gmail.com
More informationStatistical Computing (36-350)
Statistical Computing (36-350) Lecture 16: Simulation III: Monte Carlo Cosma Shalizi 21 October 2013 Agenda Monte Carlo Monte Carlo approximation of integrals and expectations The rejection method and
More informationUsing Agent Belief to Model Stock Returns
Using Agent Belief to Model Stock Returns America Holloway Department of Computer Science University of California, Irvine, Irvine, CA ahollowa@ics.uci.edu Introduction It is clear that movements in stock
More informationThe Binomial Lattice Model for Stocks: Introduction to Option Pricing
1/27 The Binomial Lattice Model for Stocks: Introduction to Option Pricing Professor Karl Sigman Columbia University Dept. IEOR New York City USA 2/27 Outline The Binomial Lattice Model (BLM) as a Model
More informationParticle methods and the pricing of American options
Particle methods and the pricing of American options Peng HU Oxford-Man Institute April 29, 2013 Joint works with P. Del Moral, N. Oudjane & B. Rémillard P. HU (OMI) University of Oxford 1 / 46 Outline
More informationMonte Carlo Based Reliability Analysis
Monte Carlo Based Reliability Analysis Martin Schwarz 15 May 2014 Martin Schwarz Monte Carlo Based Reliability Analysis 15 May 2014 1 / 19 Plan of Presentation Description of the problem Monte Carlo Simulation
More informationADVANCED OPERATIONAL RISK MODELLING IN BANKS AND INSURANCE COMPANIES
Small business banking and financing: a global perspective Cagliari, 25-26 May 2007 ADVANCED OPERATIONAL RISK MODELLING IN BANKS AND INSURANCE COMPANIES C. Angela, R. Bisignani, G. Masala, M. Micocci 1
More informationSummary Sampling Techniques
Summary Sampling Techniques MS&E 348 Prof. Gerd Infanger 2005/2006 Using Monte Carlo sampling for solving the problem Monte Carlo sampling works very well for estimating multiple integrals or multiple
More informationUnobserved Heterogeneity Revisited
Unobserved Heterogeneity Revisited Robert A. Miller Dynamic Discrete Choice March 2018 Miller (Dynamic Discrete Choice) cemmap 7 March 2018 1 / 24 Distributional Assumptions about the Unobserved Variables
More informationStock Loan Valuation Under Brownian-Motion Based and Markov Chain Stock Models
Stock Loan Valuation Under Brownian-Motion Based and Markov Chain Stock Models David Prager 1 1 Associate Professor of Mathematics Anderson University (SC) Based on joint work with Professor Qing Zhang,
More informationIdentifying Long-Run Risks: A Bayesian Mixed-Frequency Approach
Identifying : A Bayesian Mixed-Frequency Approach Frank Schorfheide University of Pennsylvania CEPR and NBER Dongho Song University of Pennsylvania Amir Yaron University of Pennsylvania NBER February 12,
More informationComputer Exercise 2 Simulation
Lund University with Lund Institute of Technology Valuation of Derivative Assets Centre for Mathematical Sciences, Mathematical Statistics Fall 2017 Computer Exercise 2 Simulation This lab deals with pricing
More informationExam in TFY4275/FY8907 CLASSICAL TRANSPORT THEORY Feb 14, 2014
NTNU Page 1 of 5 Institutt for fysikk Contact during the exam: Professor Ingve Simonsen Exam in TFY4275/FY8907 CLASSICAL TRANSPORT THEORY Feb 14, 2014 Allowed help: Alternativ D All written material This
More informationFinancial Risk Management
Financial Risk Management Professor: Thierry Roncalli Evry University Assistant: Enareta Kurtbegu Evry University Tutorial exercices #4 1 Correlation and copulas 1. The bivariate Gaussian copula is given
More informationChapter 7: Estimation Sections
1 / 40 Chapter 7: Estimation Sections 7.1 Statistical Inference Bayesian Methods: Chapter 7 7.2 Prior and Posterior Distributions 7.3 Conjugate Prior Distributions 7.4 Bayes Estimators Frequentist Methods:
More informationMEASURING PORTFOLIO RISKS USING CONDITIONAL COPULA-AR-GARCH MODEL
MEASURING PORTFOLIO RISKS USING CONDITIONAL COPULA-AR-GARCH MODEL Isariya Suttakulpiboon MSc in Risk Management and Insurance Georgia State University, 30303 Atlanta, Georgia Email: suttakul.i@gmail.com,
More informationFinancial Time Series Volatility Analysis Using Gaussian Process State-Space Models
15 IEEE Global Conference on Signal and Information Processing (GlobalSIP) Financial Time Series Volatility Analysis Using Gaussian Process State-Space Models Jianan Han, Xiao-Ping Zhang Department of
More informationDrunken Birds, Brownian Motion, and Other Random Fun
Drunken Birds, Brownian Motion, and Other Random Fun Michael Perlmutter Department of Mathematics Purdue University 1 M. Perlmutter(Purdue) Brownian Motion and Martingales Outline Review of Basic Probability
More informationNotes on the EM Algorithm Michael Collins, September 24th 2005
Notes on the EM Algorithm Michael Collins, September 24th 2005 1 Hidden Markov Models A hidden Markov model (N, Σ, Θ) consists of the following elements: N is a positive integer specifying the number of
More informationComputer Vision Group Prof. Daniel Cremers. 7. Sequential Data
Group Prof. Daniel Cremers 7. Sequential Data Bayes Filter (Rep.) We can describe the overall process using a Dynamic Bayes Network: This incorporates the following Markov assumptions: (measurement) (state)!2
More informationA potentially useful approach to model nonlinearities in time series is to assume different behavior (structural break) in different subsamples
1.3 Regime switching models A potentially useful approach to model nonlinearities in time series is to assume different behavior (structural break) in different subsamples (or regimes). If the dates, the
More informationMartingales. by D. Cox December 2, 2009
Martingales by D. Cox December 2, 2009 1 Stochastic Processes. Definition 1.1 Let T be an arbitrary index set. A stochastic process indexed by T is a family of random variables (X t : t T) defined on a
More informationmay be of interest. That is, the average difference between the estimator and the truth. Estimators with Bias(ˆθ) = 0 are called unbiased.
1 Evaluating estimators Suppose you observe data X 1,..., X n that are iid observations with distribution F θ indexed by some parameter θ. When trying to estimate θ, one may be interested in determining
More informationWithout Replacement Sampling for Particle Methods on Finite State Spaces. May 6, 2017
Without Replacement Sampling for Particle Methods on Finite State Spaces Rohan Shah Dirk P. Kroese May 6, 2017 1 1 Introduction Importance sampling is a widely used Monte Carlo technique that involves
More informationWeek 1 Quantitative Analysis of Financial Markets Distributions B
Week 1 Quantitative Analysis of Financial Markets Distributions B Christopher Ting http://www.mysmu.edu/faculty/christophert/ Christopher Ting : christopherting@smu.edu.sg : 6828 0364 : LKCSB 5036 October
More informationBivariate Birnbaum-Saunders Distribution
Department of Mathematics & Statistics Indian Institute of Technology Kanpur January 2nd. 2013 Outline 1 Collaborators 2 3 Birnbaum-Saunders Distribution: Introduction & Properties 4 5 Outline 1 Collaborators
More informationLimit Theorems for the Empirical Distribution Function of Scaled Increments of Itô Semimartingales at high frequencies
Limit Theorems for the Empirical Distribution Function of Scaled Increments of Itô Semimartingales at high frequencies George Tauchen Duke University Viktor Todorov Northwestern University 2013 Motivation
More information4 Martingales in Discrete-Time
4 Martingales in Discrete-Time Suppose that (Ω, F, P is a probability space. Definition 4.1. A sequence F = {F n, n = 0, 1,...} is called a filtration if each F n is a sub-σ-algebra of F, and F n F n+1
More informationA GENERALIZED MARTINGALE BETTING STRATEGY
DAVID K. NEAL AND MICHAEL D. RUSSELL Astract. A generalized martingale etting strategy is analyzed for which ets are increased y a factor of m 1 after each loss, ut return to the initial et amount after
More informationThe Vasicek Distribution
The Vasicek Distribution Dirk Tasche Lloyds TSB Bank Corporate Markets Rating Systems dirk.tasche@gmx.net Bristol / London, August 2008 The opinions expressed in this presentation are those of the author
More informationFinal exam solutions
EE365 Stochastic Control / MS&E251 Stochastic Decision Models Profs. S. Lall, S. Boyd June 5 6 or June 6 7, 2013 Final exam solutions This is a 24 hour take-home final. Please turn it in to one of the
More information2 Control variates. λe λti λe e λt i where R(t) = t Y 1 Y N(t) is the time from the last event to t. L t = e λr(t) e e λt(t) Exercises
96 ChapterVI. Variance Reduction Methods stochastic volatility ISExSoren5.9 Example.5 (compound poisson processes) Let X(t) = Y + + Y N(t) where {N(t)},Y, Y,... are independent, {N(t)} is Poisson(λ) with
More informationChapter 2 Uncertainty Analysis and Sampling Techniques
Chapter 2 Uncertainty Analysis and Sampling Techniques The probabilistic or stochastic modeling (Fig. 2.) iterative loop in the stochastic optimization procedure (Fig..4 in Chap. ) involves:. Specifying
More informationThe Correlation Smile Recovery
Fortis Bank Equity & Credit Derivatives Quantitative Research The Correlation Smile Recovery E. Vandenbrande, A. Vandendorpe, Y. Nesterov, P. Van Dooren draft version : March 2, 2009 1 Introduction Pricing
More informationPosterior Inference. , where should we start? Consider the following computational procedure: 1. draw samples. 2. convert. 3. compute properties
Posterior Inference Example. Consider a binomial model where we have a posterior distribution for the probability term, θ. Suppose we want to make inferences about the log-odds γ = log ( θ 1 θ), where
More informationValuation of derivative assets Lecture 8
Valuation of derivative assets Lecture 8 Magnus Wiktorsson September 27, 2018 Magnus Wiktorsson L8 September 27, 2018 1 / 14 The risk neutral valuation formula Let X be contingent claim with maturity T.
More information1 IEOR 4701: Notes on Brownian Motion
Copyright c 26 by Karl Sigman IEOR 47: Notes on Brownian Motion We present an introduction to Brownian motion, an important continuous-time stochastic process that serves as a continuous-time analog to
More informationOption Pricing Using Bayesian Neural Networks
Option Pricing Using Bayesian Neural Networks Michael Maio Pires, Tshilidzi Marwala School of Electrical and Information Engineering, University of the Witwatersrand, 2050, South Africa m.pires@ee.wits.ac.za,
More informationStatistical Inference and Methods
Department of Mathematics Imperial College London d.stephens@imperial.ac.uk http://stats.ma.ic.ac.uk/ das01/ 14th February 2006 Part VII Session 7: Volatility Modelling Session 7: Volatility Modelling
More informationStrategies for Improving the Efficiency of Monte-Carlo Methods
Strategies for Improving the Efficiency of Monte-Carlo Methods Paul J. Atzberger General comments or corrections should be sent to: paulatz@cims.nyu.edu Introduction The Monte-Carlo method is a useful
More informationA Markov Chain Monte Carlo Approach to Estimate the Risks of Extremely Large Insurance Claims
International Journal of Business and Economics, 007, Vol. 6, No. 3, 5-36 A Markov Chain Monte Carlo Approach to Estimate the Risks of Extremely Large Insurance Claims Wan-Kai Pang * Department of Applied
More informationPROBABILITY. Wiley. With Applications and R ROBERT P. DOBROW. Department of Mathematics. Carleton College Northfield, MN
PROBABILITY With Applications and R ROBERT P. DOBROW Department of Mathematics Carleton College Northfield, MN Wiley CONTENTS Preface Acknowledgments Introduction xi xiv xv 1 First Principles 1 1.1 Random
More informationInference of the Structural Credit Risk Model
Inference of the Structural Credit Risk Model using Yuxi Li, Li Cheng and Dale Schuurmans Abstract Credit risk analysis is not only an important research topic in finance, but also of interest in everyday
More informationAlgorithmic Trading using Reinforcement Learning augmented with Hidden Markov Model
Algorithmic Trading using Reinforcement Learning augmented with Hidden Markov Model Simerjot Kaur (sk3391) Stanford University Abstract This work presents a novel algorithmic trading system based on reinforcement
More informationComputational Finance Least Squares Monte Carlo
Computational Finance Least Squares Monte Carlo School of Mathematics 2019 Monte Carlo and Binomial Methods In the last two lectures we discussed the binomial tree method and convergence problems. One
More informationSTOCHASTIC CALCULUS AND BLACK-SCHOLES MODEL
STOCHASTIC CALCULUS AND BLACK-SCHOLES MODEL YOUNGGEUN YOO Abstract. Ito s lemma is often used in Ito calculus to find the differentials of a stochastic process that depends on time. This paper will introduce
More informationDerivatives Pricing and Stochastic Calculus
Derivatives Pricing and Stochastic Calculus Romuald Elie LAMA, CNRS UMR 85 Université Paris-Est Marne-La-Vallée elie @ ensae.fr Idris Kharroubi CEREMADE, CNRS UMR 7534, Université Paris Dauphine kharroubi
More informationM.Sc. ACTUARIAL SCIENCE. Term-End Examination
No. of Printed Pages : 15 LMJA-010 (F2F) M.Sc. ACTUARIAL SCIENCE Term-End Examination O CD December, 2011 MIA-010 (F2F) : STATISTICAL METHOD Time : 3 hours Maximum Marks : 100 SECTION - A Attempt any five
More informationContagion models with interacting default intensity processes
Contagion models with interacting default intensity processes Yue Kuen KWOK Hong Kong University of Science and Technology This is a joint work with Kwai Sun Leung. 1 Empirical facts Default of one firm
More informationHomework 1 posted, due Friday, September 30, 2 PM. Independence of random variables: We say that a collection of random variables
Generating Functions Tuesday, September 20, 2011 2:00 PM Homework 1 posted, due Friday, September 30, 2 PM. Independence of random variables: We say that a collection of random variables Is independent
More informationLecture outline. Monte Carlo Methods for Uncertainty Quantification. Importance Sampling. Importance Sampling
Lecture outline Monte Carlo Methods for Uncertainty Quantification Mike Giles Mathematical Institute, University of Oxford KU Leuven Summer School on Uncertainty Quantification Lecture 2: Variance reduction
More information1 Mar Review. Consumer s problem is. V (z, K, a; G, q z ) = max. subject to. c+ X q z. w(z, K) = zf 2 (K, H(K)) (4) K 0 = G(z, K) (5)
1 Mar 4 1.1 Review ² Stochastic RCE with and without state-contingent asset Consider the economy with shock to production. People are allowed to purchase statecontingent asset for next period. Consumer
More informationa 13 Notes on Hidden Markov Models Michael I. Jordan University of California at Berkeley Hidden Markov Models The model
Notes on Hidden Markov Models Michael I. Jordan University of California at Berkeley Hidden Markov Models This is a lightly edited version of a chapter in a book being written by Jordan. Since this is
More information1 Rare event simulation and importance sampling
Copyright c 2007 by Karl Sigman 1 Rare event simulation and importance sampling Suppose we wish to use Monte Carlo simulation to estimate a probability p = P (A) when the event A is rare (e.g., when p
More informationStratified Sampling in Monte Carlo Simulation: Motivation, Design, and Sampling Error
South Texas Project Risk- Informed GSI- 191 Evaluation Stratified Sampling in Monte Carlo Simulation: Motivation, Design, and Sampling Error Document: STP- RIGSI191- ARAI.03 Revision: 1 Date: September
More informationIntroduction Dickey-Fuller Test Option Pricing Bootstrapping. Simulation Methods. Chapter 13 of Chris Brook s Book.
Simulation Methods Chapter 13 of Chris Brook s Book Christopher Ting http://www.mysmu.edu/faculty/christophert/ Christopher Ting : christopherting@smu.edu.sg : 6828 0364 : LKCSB 5036 April 26, 2017 Christopher
More informationIEOR E4703: Monte-Carlo Simulation
IEOR E4703: Monte-Carlo Simulation Simulation Efficiency and an Introduction to Variance Reduction Methods Martin Haugh Department of Industrial Engineering and Operations Research Columbia University
More informationEE266 Homework 5 Solutions
EE, Spring 15-1 Professor S. Lall EE Homework 5 Solutions 1. A refined inventory model. In this problem we consider an inventory model that is more refined than the one you ve seen in the lectures. The
More informationOptimal Dam Management
Optimal Dam Management Michel De Lara et Vincent Leclère July 3, 2012 Contents 1 Problem statement 1 1.1 Dam dynamics.................................. 2 1.2 Intertemporal payoff criterion..........................
More informationReasoning with Uncertainty
Reasoning with Uncertainty Markov Decision Models Manfred Huber 2015 1 Markov Decision Process Models Markov models represent the behavior of a random process, including its internal state and the externally
More informationRisk Neutral Valuation
copyright 2012 Christian Fries 1 / 51 Risk Neutral Valuation Christian Fries Version 2.2 http://www.christian-fries.de/finmath April 19-20, 2012 copyright 2012 Christian Fries 2 / 51 Outline Notation Differential
More informationWeek 7 Quantitative Analysis of Financial Markets Simulation Methods
Week 7 Quantitative Analysis of Financial Markets Simulation Methods Christopher Ting http://www.mysmu.edu/faculty/christophert/ Christopher Ting : christopherting@smu.edu.sg : 6828 0364 : LKCSB 5036 November
More informationDYNAMIC CDO TERM STRUCTURE MODELLING
DYNAMIC CDO TERM STRUCTURE MODELLING Damir Filipović (joint with Ludger Overbeck and Thorsten Schmidt) Vienna Institute of Finance www.vif.ac.at PRisMa 2008 Workshop on Portfolio Risk Management TU Vienna,
More informationPoint Estimation. Stat 4570/5570 Material from Devore s book (Ed 8), and Cengage
6 Point Estimation Stat 4570/5570 Material from Devore s book (Ed 8), and Cengage Point Estimation Statistical inference: directed toward conclusions about one or more parameters. We will use the generic
More informationExact Sampling of Jump-Diffusion Processes
1 Exact Sampling of Jump-Diffusion Processes and Dmitry Smelov Management Science & Engineering Stanford University Exact Sampling of Jump-Diffusion Processes 2 Jump-Diffusion Processes Ubiquitous in finance
More informationArbitrages and pricing of stock options
Arbitrages and pricing of stock options Gonzalo Mateos Dept. of ECE and Goergen Institute for Data Science University of Rochester gmateosb@ece.rochester.edu http://www.ece.rochester.edu/~gmateosb/ November
More informationComputational Finance
Path Dependent Options Computational Finance School of Mathematics 2018 The Random Walk One of the main assumption of the Black-Scholes framework is that the underlying stock price follows a random walk
More information