Performance of Stochastic Programming Solutions
|
|
- Camron Hodge
- 6 years ago
- Views:
Transcription
1 Performance of Stochastic Programming Solutions Operations Research Anthony Papavasiliou 1 / 30
2 Performance of Stochastic Programming Solutions 1 The Expected Value of Perfect Information 2 The Value of the Stochastic Solution 3 Basic Inequalities 4 Estimating Performance 2 / 30
3 Two-Stage Stochastic Linear Programs min z = c T x + E ξ [min q(ω) T y(ω)] s.t. Ax = b T (ω)x + Wy(ω) = h(ω) x 0, y(ω) 0 First stage decisions x R n 1, c R n 1, b R m 1, A R m 1 n 1 For a given realization ω, second-stage data are q(ω) R n 2, h(ω) R m 2, T (ω) R m 2 n 1 All random variables of the problem are assembled in a single random vector ξ T (ω) = (q(ω) T, h(ω) T, T 1 (ω),..., T m2 (ω)) 3 / 30
4 Motivation Is it worth solving a stochastic program? How well could we do if we knew the future? How well could we do with a simpler model (e.g. expected value problem)? 4 / 30
5 Table of Contents 1 The Expected Value of Perfect Information 2 The Value of the Stochastic Solution 3 Basic Inequalities 4 Estimating Performance 5 / 30
6 Notation z(x, ξ) = c T x + Q(x, ξ) + δ(x K 1 ) Q(x, ξ) = min{q(ω) T y Wy = h(ω) T (ω)x} y What is the interpretation of z(x, ξ)? Recall, K 1 = {x Ax = b, x 0}, K 2 (ξ) = {x y : Wy = h(ω) T (ω)x} It can be that z(x, ξ) = + (if x / K 1 K 2 (ξ)) It can be that z(x, ξ) = (unbounded below) 6 / 30
7 Wait-and-See, Here-and-Now The wait-and-see value is the expected value of reacting with perfect foresight x(ξ) to ξ: WS = E ξ [min z(x, ξ)] x E ξ z( x(ξ), ξ) The here-and-now value is the expected value of the recourse problem: RP = min x E ξ z(x, ξ) We have swapped min and E. What s the difference? Which one is more difficult to compute? 7 / 30
8 Expected Value of Perfect Information (EVPI) The expected value of perfect information is the difference between the two solutions: EVPI = RP WS Interpretation: value of a perfect forecast for the future 8 / 30
9 Example: Capacity Expansion Planning We will solve this in class, on AMPL 9 / 30
10 Table of Contents 1 The Expected Value of Perfect Information 2 The Value of the Stochastic Solution 3 Basic Inequalities 4 Estimating Performance 10 / 30
11 Expected Value Problem Expected (or mean) value problem: EV = min z(x, ξ), ξ = E[ξ] x Expected value solution x( ξ): optimal solution of expected value problem 11 / 30
12 Value of the Stochastic Solution The expected result of using the EV solution measures the performance of x( ξ) (optimal second-stage reactions given x( ξ)): EEV = E ξ [z( x( ξ), ξ)] The value of the stochastic solution is VSS = EEV RP Which one is easier to compute: WS, RP, or EEV? Which one is harder? What can we say about VSS if x(ξ) is independent of ξ? 12 / 30
13 Table of Contents 1 The Expected Value of Perfect Information 2 The Value of the Stochastic Solution 3 Basic Inequalities 4 Estimating Performance 13 / 30
14 Crystal Ball For every ξ, we have z( x(ξ), ξ) z(x, ξ) where x is the optimal solution to the recourse problem Taking expectations on both sides yields WS RP Interpretation: we can do better if we have a crystal ball (i.e. we know the future in advance) 14 / 30
15 Lazy x is the optimal solution of {min x E ξ z(x, ξ)} x( ξ) is a feasible solution, therefore min x E ξ z(x, ξ) = RP EEV = E ξ z( x( ξ), ξ) Interpretation: we do worse when we are lazy (i.e. when we do not account for uncertainty explicitly) Would anything change if some of the x, y were integer? 15 / 30
16 Jensen s Inequality Suppose f is convex and ξ is a random variable, then f (E[ξ]) E[f (ξ)] 16 / 30
17 Lazy and a Liar! Suppose c, W, T are independent of ω (i.e., ξ = h): then EV WS We will show that z(x, h) = c T x + Q(x, h) + δ(x Ax = b, x 0) is jointly convex in (x, h) We will then show that f (ξ) = min x z(x, ξ) is convex in ξ From Jensen s inequality, we have Ef (ξ) f (Eξ) Interpretation: EV (the lazy solution) is a biased estimate of expected cost. Is it optimistic, or pessimistic? 17 / 30
18 Proof that z(x, h) is convex in (x, h) Consider x 1, x 2 and λ (0, 1). Without loss of generality, assume Ax 1 = b, Ax 2 = b, x 1, x 2 0. z(x i, h i ) = c T x i + q T y i, where y i = min{q T y Wy = h i Tx i, y 0}, i = {1, 2} z(λx 1 + (1 λ)x 2, λh 1 + (1 λ)h 2 ) = c T (λx 1 + (1 λ)x 2 ) + q T y λ, where y λ = min{q T y Wy = λh 1 + (1 λ)h 2 T (λx 1 + (1 λ)x 2 ), y 0} λy 1 + (1 λ)y 2 is a feasible solution for min{q T y Wy = λh 1 + (1 λ)h 2 T (λx 1 + (1 λ)x 2 ), y 0}. Therefore, we have q T y λ λq T y 1 + (1 λ)q T y 2. It follows that z(λx 1 + (1 λ)x 2, λh 1 + (1 λ)h 2 ) λz(x 1, h 1 ) + (1 λ)z(x 2, h 2 ) 18 / 30
19 Proof that f (ξ) = min x z(x, ξ) is convex in ξ Consider ξ 1, ξ 2 with f (ξ 1 ) = z(x 1, ξ 1 ), f (ξ 2 ) = z(x 2, ξ 2 ): λf (ξ 1 ) + (1 λ)f (ξ 2 ) = λz(x 1, ξ 1 ) + (1 λ)z(x 2, ξ 2 ) z(λ(x 1, ξ 1 ) + (1 λ)(x 2, ξ 2 )) min x z(x, λξ 1 + (1 λ)ξ 2 ) = f (λξ 1 + (1 λ)ξ 2 ) by definition convexity of z by definition 19 / 30
20 Counter-Example: EV > WS Consider the following problem: min 2x + E ξ[ξ y] x 0 s.t. y 1 x y 0 where P(ξ = 1) = 3/4, P(ξ = 3) = 1/4 Does this problem satisfy the assumptions of slide 16? 20 / 30
21 Optimal second-stage decision: y = 1 x if 1 x 0, y = 0 otherwise Tradeoff: by increasing x we can push y to lower values, but incur certain cost 2x For ξ = = 3 2 we have {min 2x + 3 2y y 1 x, x 0, y 0} Optimal solution: x( ξ) = 0, y = 1 with EV = 3 2 To compute WS, note that for ξ = 1 the optimal first-stage decision is x = 0, with cost of 1, while for ξ = 3 the optimal first-stage decision is x = 1, with cost of 2: WS = = 5 4 < EV 21 / 30
22 Summary We have established that VSS 0, EVPI 0 VSS EEV EV, EVPI EEV EV If EEV EV = 0 then VSS = 0, EVPI = 0 (for example, if x(ξ) independent of ξ - this is rare) EVPI VSS EV WS RP EEV 22 / 30
23 Table of Contents 1 The Expected Value of Perfect Information 2 The Value of the Stochastic Solution 3 Basic Inequalities 4 Estimating Performance 23 / 30
24 Central Limit Theorem Suppose ξ(ω) is continuous, does this complicate the computation of EV, RP, EEV and WS? Central limit theorem: Suppose {X 1, X 2,...} is a sequence of i.i.d. random variables with E[X i ] = µ and Var[X i ] =? 2 <. Then as n approaches infinity, n(s n?µ) converge in distribution to a normal N(0, σ 2 ): n (( 1 n n i=1 ) ) X i µ d N(0, σ 2 ). Can we use the CLT? What would the X i be in our case? 24 / 30
25 Motivating Example The cost C of operating a facility is C(ξ 1 ) = 1 under normal operations, p(ξ 1 ) = 0.9 C(ξ 2 ) = 100 under emergency operations, p(ξ 2 ) = 0.1 µ = = 10.9 σ = 0.9 (1 10.9) ( ) 2 = / 30
26 Rare outcome (1 out of 10 samples) influences expected value calculation since it contributes by = 91.7% to expected value From central limit theorem, in order to get estimate of E[C] to be within 5% with 95.4% confidence, we need 2 σ n = 0.05µ, from which n = 11879! 26 / 30
27 Importance Sampling Suppose we wish to estimate E[C(ξ)], where ξ is a random variable distributed according to f (ξ) Monte Carlo pulls samples ξ i according to distribution f (ξ) and estimates E[C(ξ)] with N i=1 1 N C(ξ i) Importance sampling pulls samples ξ i according to distribution g(ξ) = f (ξ) C(ξ) E[C] and estimates E[ξ] with N i=1 1 f (ξ i ) C(ξ i ) N g(ξ i ) 27 / 30
28 Motivation of Importance Sampling Note that E[C(ξ)] = Ξ C(ξ) f (ξ)dξ = Ξ C(ξ) f (ξ) g(ξ) g(ξ)dξ C(ξ) f (ξ) The random variable g(ξ), which is distributed according to g(ξ), also has expectation E[C] Which g(ξ) minimizes the variance of this new random variable? g(ξ) = Any sample evaluates to E[C]! C(ξ) f (ξ) E[C] We cheated: g(ξ) requires knowledge of E[C], which is what we are estimating But we learned something: pull samples according to C(ξ) f (ξ) contribution to expected value, E[C]. Even if we do not know E[C], we can approximate it. 28 / 30
29 Back to the Example Problem: rare bad outcome had the greatest influence on expected value Remedy: Redefine distribution so that we observe bad outcome earlier, then adjust our expected value calculations in order to unbias result q(ξ 1 ) = p(ξ 1) C(ξ 1 ) E[C] q(ξ 2 ) = p(ξ 2) C(ξ 2 ) E[C] = = = Estimates for ξ 1, ξ 2 are constant and equal to E[C]: = C(ξ 1 ) p(ξ 1) q(ξ 1 ) = = C(ξ 2 ) p(ξ 2) 0.1 = 100 = 10.9 q(ξ 2 ) / 30
30 Further Reading 4.1 BL: the expected value of perfect information 4.2 BL: the value of the stochastic solution 4.3 BL: basic inequalities 4.4 BL: the relationship between EVPI and VSS 4.5 BL: examples 4.6 BL: bounds on EVPI and VSS 30 / 30
The Values of Information and Solution in Stochastic Programming
The Values of Information and Solution in Stochastic Programming John R. Birge The University of Chicago Booth School of Business JRBirge ICSP, Bergamo, July 2013 1 Themes The values of information and
More informationStochastic Optimization
Stochastic Optimization Introduction and Examples Alireza Ghaffari-Hadigheh Azarbaijan Shahid Madani University (ASMU) hadigheha@azaruniv.edu Fall 2017 Alireza Ghaffari-Hadigheh (ASMU) Stochastic Optimization
More informationStochastic Dual Dynamic Programming
1 / 43 Stochastic Dual Dynamic Programming Operations Research Anthony Papavasiliou 2 / 43 Contents [ 10.4 of BL], [Pereira, 1991] 1 Recalling the Nested L-Shaped Decomposition 2 Drawbacks of Nested Decomposition
More informationIE 495 Lecture 11. The LShaped Method. Prof. Jeff Linderoth. February 19, February 19, 2003 Stochastic Programming Lecture 11 Slide 1
IE 495 Lecture 11 The LShaped Method Prof. Jeff Linderoth February 19, 2003 February 19, 2003 Stochastic Programming Lecture 11 Slide 1 Before We Begin HW#2 $300 $0 http://www.unizh.ch/ior/pages/deutsch/mitglieder/kall/bib/ka-wal-94.pdf
More informationChapter 6. Importance sampling. 6.1 The basics
Chapter 6 Importance sampling 6.1 The basics To movtivate our discussion consider the following situation. We want to use Monte Carlo to compute µ E[X]. There is an event E such that P(E) is small but
More informationStochastic Programming: introduction and examples
Stochastic Programming: introduction and examples Amina Lamghari COSMO Stochastic Mine Planning Laboratory Department of Mining and Materials Engineering Outline What is Stochastic Programming? Why should
More informationStochastic Programming and Financial Analysis IE447. Midterm Review. Dr. Ted Ralphs
Stochastic Programming and Financial Analysis IE447 Midterm Review Dr. Ted Ralphs IE447 Midterm Review 1 Forming a Mathematical Programming Model The general form of a mathematical programming model is:
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 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 informationScenario Generation and Sampling Methods
Scenario Generation and Sampling Methods Güzin Bayraksan Tito Homem-de-Mello SVAN 2016 IMPA May 9th, 2016 Bayraksan (OSU) & Homem-de-Mello (UAI) Scenario Generation and Sampling SVAN IMPA May 9 1 / 30
More informationSampling and sampling distribution
Sampling and sampling distribution September 12, 2017 STAT 101 Class 5 Slide 1 Outline of Topics 1 Sampling 2 Sampling distribution of a mean 3 Sampling distribution of a proportion STAT 101 Class 5 Slide
More informationChapter 8: Sampling distributions of estimators Sections
Chapter 8: Sampling distributions of estimators Sections 8.1 Sampling distribution of a statistic 8.2 The Chi-square distributions 8.3 Joint Distribution of the sample mean and sample variance Skip: p.
More informationUniversal Portfolios
CS28B/Stat24B (Spring 2008) Statistical Learning Theory Lecture: 27 Universal Portfolios Lecturer: Peter Bartlett Scribes: Boriska Toth and Oriol Vinyals Portfolio optimization setting Suppose we have
More informationReal Business Cycles (Solution)
Real Business Cycles (Solution) Exercise: A two-period real business cycle model Consider a representative household of a closed economy. The household has a planning horizon of two periods and is endowed
More information6. Martingales. = Zn. Think of Z n+1 as being a gambler s earnings after n+1 games. If the game if fair, then E [ Z n+1 Z n
6. Martingales For casino gamblers, a martingale is a betting strategy where (at even odds) the stake doubled each time the player loses. Players follow this strategy because, since they will eventually
More informationConfidence Intervals Introduction
Confidence Intervals Introduction A point estimate provides no information about the precision and reliability of estimation. For example, the sample mean X is a point estimate of the population mean μ
More informationQuantitative Risk Management
Quantitative Risk Management Asset Allocation and Risk Management Martin B. Haugh Department of Industrial Engineering and Operations Research Columbia University Outline Review of Mean-Variance Analysis
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 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 informationWhat can we do with numerical optimization?
Optimization motivation and background Eddie Wadbro Introduction to PDE Constrained Optimization, 2016 February 15 16, 2016 Eddie Wadbro, Introduction to PDE Constrained Optimization, February 15 16, 2016
More informationMultistage grid investments incorporating uncertainty in Offshore wind deployment
Multistage grid investments incorporating uncertainty in Offshore wind deployment Presentation by: Harald G. Svendsen Joint work with: Martin Kristiansen, Magnus Korpås, and Stein-Erik Fleten Content Transmission
More informationOPTIMIZATION MODELING FOR TRADEOFF ANALYSIS OF HIGHWAY INVESTMENT ALTERNATIVES
IIT Networks and Optimization Seminar OPTIMIZATION MODEING FOR TRADEOFF ANAYSIS OF HIGHWAY INVESTMENT ATERNATIVES Dr. Zongzhi i, Assistant Professor Dept. of Civil, Architectural and Environmental Engineering
More informationIntroduction to modeling using stochastic programming. Andy Philpott The University of Auckland
Introduction to modeling using stochastic programming Andy Philpott The University of Auckland Tutorial presentation at SPX, Tuscon, October 9th, 2004 Summary Introduction to basic concepts Risk Multi-stage
More informationAdditional questions for chapter 3
Additional questions for chapter 3 1. Let ξ 1, ξ 2,... be independent and identically distributed with φθ) = IEexp{θξ 1 })
More informationMartingale Pricing Theory in Discrete-Time and Discrete-Space Models
IEOR E4707: Foundations of Financial Engineering c 206 by Martin Haugh Martingale Pricing Theory in Discrete-Time and Discrete-Space Models These notes develop the theory of martingale pricing in a discrete-time,
More informationCourse notes for EE394V Restructured Electricity Markets: Locational Marginal Pricing
Course notes for EE394V Restructured Electricity Markets: Locational Marginal Pricing Ross Baldick Copyright c 2018 Ross Baldick www.ece.utexas.edu/ baldick/classes/394v/ee394v.html Title Page 1 of 160
More informationEngineering Statistics ECIV 2305
Engineering Statistics ECIV 2305 Section 5.3 Approximating Distributions with the Normal Distribution Introduction A very useful property of the normal distribution is that it provides good approximations
More informationElementary Statistics Lecture 5
Elementary Statistics Lecture 5 Sampling Distributions Chong Ma Department of Statistics University of South Carolina Chong Ma (Statistics, USC) STAT 201 Elementary Statistics 1 / 24 Outline 1 Introduction
More informationUNIVERSITY OF OSLO DEPARTMENT OF ECONOMICS
UNIVERSITY OF OSLO DEPARTMENT OF ECONOMICS Postponed exam: ECON4310 Macroeconomic Theory Date of exam: Wednesday, January 11, 2017 Time for exam: 09:00 a.m. 12:00 noon The problem set covers 13 pages (incl.
More informationChapter 7: Portfolio Theory
Chapter 7: Portfolio Theory 1. Introduction 2. Portfolio Basics 3. The Feasible Set 4. Portfolio Selection Rules 5. The Efficient Frontier 6. Indifference Curves 7. The Two-Asset Portfolio 8. Unrestriceted
More informationMTH6154 Financial Mathematics I Stochastic Interest Rates
MTH6154 Financial Mathematics I Stochastic Interest Rates Contents 4 Stochastic Interest Rates 45 4.1 Fixed Interest Rate Model............................ 45 4.2 Varying Interest Rate Model...........................
More informationOptimizing Portfolios
Optimizing Portfolios An Undergraduate Introduction to Financial Mathematics J. Robert Buchanan 2010 Introduction Investors may wish to adjust the allocation of financial resources including a mixture
More informationChapter 3 - Lecture 4 Moments and Moment Generating Funct
Chapter 3 - Lecture 4 and s October 7th, 2009 Chapter 3 - Lecture 4 and Moment Generating Funct Central Skewness Chapter 3 - Lecture 4 and Moment Generating Funct Central Skewness The expected value of
More informationMORE REALISTIC FOR STOCKS, FOR EXAMPLE
MARTINGALES BASED ON IID: ADDITIVE MG Y 1,..., Y t,... : IID EY = 0 X t = Y 1 +... + Y t is MG MULTIPLICATIVE MG Y 1,..., Y t,... : IID EY = 1 X t = Y 1... Y t : X t+1 = X t Y t+1 E(X t+1 F t ) = E(X t
More informationStochastic Programming Modeling
IE 495 Lecture 3 Stochastic Programming Modeling Prof. Jeff Linderoth January 20, 2003 January 20, 2003 Stochastic Programming Lecture 3 Slide 1 Outline Review convexity Review Farmer Ted Expected Value
More informationPractice Problems 1: Moral Hazard
Practice Problems 1: Moral Hazard December 5, 2012 Question 1 (Comparative Performance Evaluation) Consider the same normal linear model as in Question 1 of Homework 1. This time the principal employs
More informationUQ, STAT2201, 2017, Lectures 3 and 4 Unit 3 Probability Distributions.
UQ, STAT2201, 2017, Lectures 3 and 4 Unit 3 Probability Distributions. Random Variables 2 A random variable X is a numerical (integer, real, complex, vector etc.) summary of the outcome of the random experiment.
More informationDRAFT. 1 exercise in state (S, t), π(s, t) = 0 do not exercise in state (S, t) Review of the Risk Neutral Stock Dynamics
Chapter 12 American Put Option Recall that the American option has strike K and maturity T and gives the holder the right to exercise at any time in [0, T ]. The American option is not straightforward
More informationCSCI 1951-G Optimization Methods in Finance Part 00: Course Logistics Introduction to Finance Optimization Problems
CSCI 1951-G Optimization Methods in Finance Part 00: Course Logistics Introduction to Finance Optimization Problems January 26, 2018 1 / 24 Basic information All information is available in the syllabus
More informationDASC: A DECOMPOSITION ALGORITHM FOR MULTISTAGE STOCHASTIC PROGRAMS WITH STRONGLY CONVEX COST FUNCTIONS
DASC: A DECOMPOSITION ALGORITHM FOR MULTISTAGE STOCHASTIC PROGRAMS WITH STRONGLY CONVEX COST FUNCTIONS Vincent Guigues School of Applied Mathematics, FGV Praia de Botafogo, Rio de Janeiro, Brazil vguigues@fgv.br
More informationFinancial Economics Field Exam January 2008
Financial Economics Field Exam January 2008 There are two questions on the exam, representing Asset Pricing (236D = 234A) and Corporate Finance (234C). Please answer both questions to the best of your
More informationAsymptotic methods in risk management. Advances in Financial Mathematics
Asymptotic methods in risk management Peter Tankov Based on joint work with A. Gulisashvili Advances in Financial Mathematics Paris, January 7 10, 2014 Peter Tankov (Université Paris Diderot) Asymptotic
More informationStatistics and Their Distributions
Statistics and Their Distributions Deriving Sampling Distributions Example A certain system consists of two identical components. The life time of each component is supposed to have an expentional distribution
More informationu (x) < 0. and if you believe in diminishing return of the wealth, then you would require
Chapter 8 Markowitz Portfolio Theory 8.7 Investor Utility Functions People are always asked the question: would more money make you happier? The answer is usually yes. The next question is how much more
More informationLecture 10: Performance measures
Lecture 10: Performance measures Prof. Dr. Svetlozar Rachev Institute for Statistics and Mathematical Economics University of Karlsruhe Portfolio and Asset Liability Management Summer Semester 2008 Prof.
More informationProbability and Stochastics for finance-ii Prof. Joydeep Dutta Department of Humanities and Social Sciences Indian Institute of Technology, Kanpur
Probability and Stochastics for finance-ii Prof. Joydeep Dutta Department of Humanities and Social Sciences Indian Institute of Technology, Kanpur Lecture - 07 Mean-Variance Portfolio Optimization (Part-II)
More informationSampling Distribution
MAT 2379 (Spring 2012) Sampling Distribution Definition : Let X 1,..., X n be a collection of random variables. We say that they are identically distributed if they have a common distribution. Definition
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 informationWorst-Case Value-at-Risk of Non-Linear Portfolios
Worst-Case Value-at-Risk of Non-Linear Portfolios Steve Zymler Daniel Kuhn Berç Rustem Department of Computing Imperial College London Portfolio Optimization Consider a market consisting of m assets. Optimal
More informationMultistage Stochastic Programming
Multistage Stochastic Programming John R. Birge University of Michigan Models - Long and short term - Risk inclusion Approximations - stages and scenarios Computation Slide Number 1 OUTLINE Motivation
More informationChapter 7: Estimation Sections
1 / 31 : Estimation Sections 7.1 Statistical Inference Bayesian Methods: 7.2 Prior and Posterior Distributions 7.3 Conjugate Prior Distributions 7.4 Bayes Estimators Frequentist Methods: 7.5 Maximum Likelihood
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 informationWorst-Case Value-at-Risk of Derivative Portfolios
Worst-Case Value-at-Risk of Derivative Portfolios Steve Zymler Berç Rustem Daniel Kuhn Department of Computing Imperial College London Thalesians Seminar Series, November 2009 Risk Management is a Hot
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 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 informationModes of Convergence
Moes of Convergence Electrical Engineering 126 (UC Berkeley Spring 2018 There is only one sense in which a sequence of real numbers (a n n N is sai to converge to a limit. Namely, a n a if for every ε
More informationStochastic Programming Modeling
Stochastic Programming Modeling IMA New Directions Short Course on Mathematical Optimization Jeff Linderoth Department of Industrial and Systems Engineering University of Wisconsin-Madison August 8, 2016
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 informationBuilding Consistent Risk Measures into Stochastic Optimization Models
Building Consistent Risk Measures into Stochastic Optimization Models John R. Birge The University of Chicago Graduate School of Business www.chicagogsb.edu/fac/john.birge JRBirge Fuqua School, Duke University
More informationVersion A. Problem 1. Let X be the continuous random variable defined by the following pdf: 1 x/2 when 0 x 2, f(x) = 0 otherwise.
Math 224 Q Exam 3A Fall 217 Tues Dec 12 Version A Problem 1. Let X be the continuous random variable defined by the following pdf: { 1 x/2 when x 2, f(x) otherwise. (a) Compute the mean µ E[X]. E[X] x
More informationRobust Dual Dynamic Programming
1 / 18 Robust Dual Dynamic Programming Angelos Georghiou, Angelos Tsoukalas, Wolfram Wiesemann American University of Beirut Olayan School of Business 31 May 217 2 / 18 Inspired by SDDP Stochastic optimization
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 informationProbability without Measure!
Probability without Measure! Mark Saroufim University of California San Diego msaroufi@cs.ucsd.edu February 18, 2014 Mark Saroufim (UCSD) It s only a Game! February 18, 2014 1 / 25 Overview 1 History of
More informationIn Discrete Time a Local Martingale is a Martingale under an Equivalent Probability Measure
In Discrete Time a Local Martingale is a Martingale under an Equivalent Probability Measure Yuri Kabanov 1,2 1 Laboratoire de Mathématiques, Université de Franche-Comté, 16 Route de Gray, 253 Besançon,
More informationApproximation of Continuous-State Scenario Processes in Multi-Stage Stochastic Optimization and its Applications
Approximation of Continuous-State Scenario Processes in Multi-Stage Stochastic Optimization and its Applications Anna Timonina University of Vienna, Abraham Wald PhD Program in Statistics and Operations
More informationSTAT Chapter 7: Central Limit Theorem
STAT 251 - Chapter 7: Central Limit Theorem In this chapter we will introduce the most important theorem in statistics; the central limit theorem. What have we seen so far? First, we saw that for an i.i.d
More informationOptimum Thresholding for Semimartingales with Lévy Jumps under the mean-square error
Optimum Thresholding for Semimartingales with Lévy Jumps under the mean-square error José E. Figueroa-López Department of Mathematics Washington University in St. Louis Spring Central Sectional Meeting
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 informationMath489/889 Stochastic Processes and Advanced Mathematical Finance Homework 5
Math489/889 Stochastic Processes and Advanced Mathematical Finance Homework 5 Steve Dunbar Due Fri, October 9, 7. Calculate the m.g.f. of the random variable with uniform distribution on [, ] and then
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 informationStochastic Optimal Control
Stochastic Optimal Control Lecturer: Eilyan Bitar, Cornell ECE Scribe: Kevin Kircher, Cornell MAE These notes summarize some of the material from ECE 5555 (Stochastic Systems) at Cornell in the fall of
More informationThe Normal Distribution
The Normal Distribution The normal distribution plays a central role in probability theory and in statistics. It is often used as a model for the distribution of continuous random variables. Like all models,
More informationProblem 1: Random variables, common distributions and the monopoly price
Problem 1: Random variables, common distributions and the monopoly price In this problem, we will revise some basic concepts in probability, and use these to better understand the monopoly price (alternatively
More informationChapter 18 Student Lecture Notes 18-1
Chapter 18 Student Lecture Notes 18-1 Business Statistics: A Decision-Making Approach 6 th Edition Chapter 18 Introduction to Decision Analysis 5 Prentice-Hall, Inc. Chap 18-1 Chapter Goals After completing
More informationThe Value of Stochastic Modeling in Two-Stage Stochastic Programs
The Value of Stochastic Modeling in Two-Stage Stochastic Programs Erick Delage, HEC Montréal Sharon Arroyo, The Boeing Cie. Yinyu Ye, Stanford University Tuesday, October 8 th, 2013 1 Delage et al. Value
More informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.265/15.070J Fall 2013 Lecture 11 10/9/2013. Martingales and stopping times II
MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.65/15.070J Fall 013 Lecture 11 10/9/013 Martingales and stopping times II Content. 1. Second stopping theorem.. Doob-Kolmogorov inequality. 3. Applications of stopping
More informationPricing theory of financial derivatives
Pricing theory of financial derivatives One-period securities model S denotes the price process {S(t) : t = 0, 1}, where S(t) = (S 1 (t) S 2 (t) S M (t)). Here, M is the number of securities. At t = 1,
More informationFinancial Risk Forecasting Chapter 9 Extreme Value Theory
Financial Risk Forecasting Chapter 9 Extreme Value Theory Jon Danielsson 2017 London School of Economics To accompany Financial Risk Forecasting www.financialriskforecasting.com Published by Wiley 2011
More informationA Hybrid Monte Carlo Local Branching Algorithm for the Single Vehicle Routing Problem with Stochastic Demands
A Hybrid Monte Carlo Local Branching Algorithm for the Single Vehicle Routing Problem with Stochastic Demands Walter Rei Michel Gendreau Patrick Soriano July 2007 A Hybrid Monte Carlo Local Branching Algorithm
More informationConsistency of option prices under bid-ask spreads
Consistency of option prices under bid-ask spreads Stefan Gerhold TU Wien Joint work with I. Cetin Gülüm MFO, Feb 2017 (TU Wien) MFO, Feb 2017 1 / 32 Introduction The consistency problem Overview Consistency
More informationIEOR E4602: Quantitative Risk Management
IEOR E4602: Quantitative Risk Management Risk Measures Martin Haugh Department of Industrial Engineering and Operations Research Columbia University Email: martin.b.haugh@gmail.com Reference: Chapter 8
More information1 Consumption and saving under uncertainty
1 Consumption and saving under uncertainty 1.1 Modelling uncertainty As in the deterministic case, we keep assuming that agents live for two periods. The novelty here is that their earnings in the second
More informationc 2014 CHUAN XU ALL RIGHTS RESERVED
c 2014 CHUAN XU ALL RIGHTS RESERVED SIMULATION APPROACH TO TWO-STAGE BOND PORTFOLIO OPTIMIZATION PROBLEM BY CHUAN XU A thesis submitted to the Graduate School New Brunswick Rutgers, The State University
More informationLecture 23: April 10
CS271 Randomness & Computation Spring 2018 Instructor: Alistair Sinclair Lecture 23: April 10 Disclaimer: These notes have not been subjected to the usual scrutiny accorded to formal publications. They
More informationCHOICE THEORY, UTILITY FUNCTIONS AND RISK AVERSION
CHOICE THEORY, UTILITY FUNCTIONS AND RISK AVERSION Szabolcs Sebestyén szabolcs.sebestyen@iscte.pt Master in Finance INVESTMENTS Sebestyén (ISCTE-IUL) Choice Theory Investments 1 / 65 Outline 1 An Introduction
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 informationIEOR 3106: Introduction to OR: Stochastic Models. Fall 2013, Professor Whitt. Class Lecture Notes: Tuesday, September 10.
IEOR 3106: Introduction to OR: Stochastic Models Fall 2013, Professor Whitt Class Lecture Notes: Tuesday, September 10. The Central Limit Theorem and Stock Prices 1. The Central Limit Theorem (CLT See
More informationChapter 5. Continuous Random Variables and Probability Distributions. 5.1 Continuous Random Variables
Chapter 5 Continuous Random Variables and Probability Distributions 5.1 Continuous Random Variables 1 2CHAPTER 5. CONTINUOUS RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS Probability Distributions Probability
More informationTwo Equivalent Conditions
Two Equivalent Conditions The traditional theory of present value puts forward two equivalent conditions for asset-market equilibrium: Rate of Return The expected rate of return on an asset equals the
More informationTutorial 11: Limit Theorems. Baoxiang Wang & Yihan Zhang bxwang, April 10, 2017
Tutorial 11: Limit Theorems Baoxiang Wang & Yihan Zhang bxwang, yhzhang@cse.cuhk.edu.hk April 10, 2017 1 Outline The Central Limit Theorem (CLT) Normal Approximation Based on CLT De Moivre-Laplace Approximation
More informationThe Central Limit Theorem. Sec. 8.2: The Random Variable. it s Distribution. it s Distribution
The Central Limit Theorem Sec. 8.1: The Random Variable it s Distribution Sec. 8.2: The Random Variable it s Distribution X p and and How Should You Think of a Random Variable? Imagine a bag with numbers
More informationSimulation Wrap-up, Statistics COS 323
Simulation Wrap-up, Statistics COS 323 Today Simulation Re-cap Statistics Variance and confidence intervals for simulations Simulation wrap-up FYI: No class or office hours Thursday Simulation wrap-up
More informationAsymmetric Information: Walrasian Equilibria, and Rational Expectations Equilibria
Asymmetric Information: Walrasian Equilibria and Rational Expectations Equilibria 1 Basic Setup Two periods: 0 and 1 One riskless asset with interest rate r One risky asset which pays a normally distributed
More informationChapter 8: Sampling distributions of estimators Sections
Chapter 8 continued Chapter 8: Sampling distributions of estimators Sections 8.1 Sampling distribution of a statistic 8.2 The Chi-square distributions 8.3 Joint Distribution of the sample mean and sample
More informationDepartment of Mathematics. Mathematics of Financial Derivatives
Department of Mathematics MA408 Mathematics of Financial Derivatives Thursday 15th January, 2009 2pm 4pm Duration: 2 hours Attempt THREE questions MA408 Page 1 of 5 1. (a) Suppose 0 < E 1 < E 3 and E 2
More informationNumerical Simulation of Stochastic Differential Equations: Lecture 1, Part 1. Overview of Lecture 1, Part 1: Background Mater.
Numerical Simulation of Stochastic Differential Equations: Lecture, Part Des Higham Department of Mathematics University of Strathclyde Course Aim: Give an accessible intro. to SDEs and their numerical
More informationHigh Dimensional Edgeworth Expansion. Applications to Bootstrap and Its Variants
With Applications to Bootstrap and Its Variants Department of Statistics, UC Berkeley Stanford-Berkeley Colloquium, 2016 Francis Ysidro Edgeworth (1845-1926) Peter Gavin Hall (1951-2016) Table of Contents
More informationPORTFOLIO THEORY. Master in Finance INVESTMENTS. Szabolcs Sebestyén
PORTFOLIO THEORY Szabolcs Sebestyén szabolcs.sebestyen@iscte.pt Master in Finance INVESTMENTS Sebestyén (ISCTE-IUL) Portfolio Theory Investments 1 / 60 Outline 1 Modern Portfolio Theory Introduction Mean-Variance
More informationChapter 7: Estimation Sections
Chapter 7: Estimation Sections 7.1 Statistical Inference Bayesian Methods: 7.2 Prior and Posterior Distributions 7.3 Conjugate Prior Distributions Frequentist Methods: 7.5 Maximum Likelihood Estimators
More informationClass Notes on Financial Mathematics. No-Arbitrage Pricing Model
Class Notes on No-Arbitrage Pricing Model April 18, 2016 Dr. Riyadh Al-Mosawi Department of Mathematics, College of Education for Pure Sciences, Thiqar University References: 1. Stochastic Calculus for
More information