Probability and Stochastic Processes

Ionut Florescu, PhD, is Research Associate Professor of Financial Engineering and Director of the Hanlon Financial Systems Lab at Stevens Institute of Technology. His areas of research interest include stochastic volatility, stochastic partial differential equations, Monte Carlo methods, and numerical methods for stochastic processes. He is also the coauthor of Handbook of Probability and coeditor of Handbook of Modeling High-Frequency Data in Finance, both published by Wiley.

• List of Figures xviiList of Tables xxPreface xxiAcknowledgments xxiiiIntroduction 1Part I Probability1 Elements of Probability Measure 91.1 Probability Spaces 101.1.1 Null element of ℱ. Almost sure (a.s.) statements. Indicator of a set 211.2 Conditional Probability 221.3 Independence 291.4 Monotone Convergence Properties of Probability 311.5 Lebesgue Measure on the Unit Interval (0,1] 37Problems 402 Random Variables 452.1 Discrete and Continuous Random Variables 482.2 Examples of Commonly Encountered Random Variables 522.3 Existence of Random Variables with Prescribed Distribution 652.4 Independence 682.5 Functions of Random Variables. Calculating Distributions 72Problems 823 Applied Chapter: Generating Random Variables 873.1 Generating One-Dimensional Random Variables by Inverting the cdf 883.2 Generating One-Dimensional Normal Random Variables 913.3 Generating Random Variables. Rejection Sampling Method 943.4 Generating Random Variables. Importance Sampling 109Problems 1194 Integration Theory 1234.1 Integral of Measurable Functions 1244.2 Expectations 1304.3 Moments of a Random Variable. Variance and the Correlation Coefficient 1434.4 Functions of Random Variables. The Transport Formula 1454.5 Applications. Exercises in Probability Reasoning 1484.6 A Basic Central Limit Theorem: The DeMoivre–LaplaceTheorem: 150Problems 1525 Conditional Distribution and Conditional Expectation 1575.1 Product Spaces 1585.2 Conditional Distribution and Expectation. Calculation in Simple Cases 1625.3 Conditional Expectation. General Definition 1655.4 Random Vectors. Moments and Distributions 168Problems 1776 Moment Generating Function. Characteristic Function 1816.1 Sums of Random Variables. Convolutions 1816.2 Generating Functions and Applications 1826.3 Moment Generating Function 1886.4 Characteristic Function 1926.5 Inversion and Continuity Theorems 1996.6 Stable Distributions. Lvy Distribution 2046.6.1 Truncated Lévy flight distribution 206Problems 2087 Limit Theorems 2137.1 Types of Convergence 2137.1.1 Traditional deterministic convergence types 2147.1.2 Convergence in Lp 2157.1.3 Almost sure (a.s.) convergence 2167.1.4 Convergence in probability. Convergence in distribution 2177.2 Relationships between Types of Convergence 2217.2.1 A.S. and Lp 2217.2.2 Probability, a.s., Lp convergence 2237.2.3 Uniform Integrability 2267.2.4 Weak convergence and all the others 2287.3 Continuous Mapping Theorem. Joint Convergence. Slutsky’s Theorem 2307.4 The Two Big Limit Theorems: LLN and CLT 2327.4.1 A note on statistics 2327.4.2 The order statistics 2347.4.3 Limit theorems for the mean statistics 2387.5 Extensions of CLT 2457.6 Exchanging the Order of Limits and Expectations 251Problems 2528 Statistical Inference 2598.1 The Classical Problems in Statistics 2598.2 Parameter Estimation Problem 2608.2.1 The case of the normal distribution, estimating mean when variance is unknown 2628.2.2 The case of the normal distribution, comparing variances 2648.3 Maximum Likelihood Estimation Method 2658.3.1 The bisection method 2678.4 The Method of Moments 2768.5 Testing, the Likelihood Ratio Test 2778.5.1 The likelihood ratio test 2808.6 Confidence Sets 284Problems 286Part II Stochastic Processes9 Introduction to Stochastic Processes 2939.1 General Characteristics of Stochastic Processes 2949.1.1 The index set I 2949.1.2 The state space S 2949.1.3 Adaptiveness, filtration, standard filtration 2949.1.4 Pathwise realizations 2969.1.5 The finite distribution of stochastic processes 2969.1.6 Independent components 2979.1.7 Stationary process 2989.1.8 Stationary and independent increments 2999.1.9 Other properties that characterize specific classes of stochastic processes 3009.2 A Simple Process – The Bernoulli Process 301Problems 30410 The Poisson Process 30710.1 Definitions 30710.2 Inter-Arrival and Waiting Time for a Poisson Process 31010.2.1 Proving that the inter-arrival times are independent 31110.2.2 Memoryless property of the exponential distribution 31510.2.3 Merging two independent Poisson processes 31610.2.4 Splitting the events of the Poisson process into types 31610.3 General Poisson Processes 31710.3.1 Nonhomogenous Poisson process 31810.3.2 The compound Poisson process 31910.4 Simulation techniques. Constructing Poisson Processes 32310.4.1 One-dimensional simple Poisson process 323Problems 32611 Renewal Processes 33111.0.2 The renewal function 33311.1 Limit Theorems for the Renewal Process 33411.1.1 Auxiliary but very important results. Wald’s theorem. Discrete stopping time 33611.1.2 An alternative proof of the elementary renewal theorem 34011.2 Discrete Renewal Theory 34411.3 The Key Renewal Theorem 34911.4 Applications of the Renewal Theorems 35011.5 Special cases of renewal processes 35211.5.1 The alternating renewal process 35311.5.2 Renewal reward process 35811.6 The renewal Equation 35911.7 Age-Dependent Branching processes 363Problems 36612 Markov Chains 37112.1 Basic Concepts for Markov Chains 37112.1.1 Definition 37112.1.2 Examples of Markov chains 37212.1.3 The Chapman– Kolmogorov equation 37812.1.4 Communicating classes and class properties 37912.1.5 Periodicity 37912.1.6 Recurrence property 38012.1.7 Types of recurrence 38212.2 Simple Random Walk on Integers in d Dimensions 38312.3 Limit Theorems 38612.4 States in a MC. Stationary Distribution 38712.4.1 Examples. Calculating stationary distribution 39112.5 Other Issues: Graphs, First-Step Analysis 39412.5.1 First-step analysis 39412.5.2 Markov chains and graphs 39512.6 A general Treatment of the Markov Chains 39612.6.1 Time of absorption 39912.6.2 An example 400Problems 40613 Semi-Markov and Continuous-time Markov Processes 41113.1 Characterization Theorems for the General semi- Markov Process 41313.2 Continuous-Time Markov Processes 41713.3 The Kolmogorov Differential Equations 42013.4 Calculating Transition Probabilities for a Markov Process General Approach 42513.5 Limiting Probabilities for the Continuous-Time Markov Chain 42613.6 Reversible Markov Process 429Problems 43214 Martingales 43714.1 Definition and Examples 43814.1.1 Examples of martingales 43914.2 Martingales and Markov Chains 44014.2.1 Martingales induced by Markov chains 44014.3 Previsible Process. The Martingale Transform 44214.4 Stopping Time. Stopped Process 44414.4.1 Properties of stopping time 44614.5 Classical Examples of Martingale Reasoning 44914.5.1 The expected number of tosses until a binary pattern occurs 44914.5.2 Expected number of attempts until a general pattern occurs 45114.5.3 Gambler’s ruin probability – revisited 45214.6 Convergence Theorems. L1 Convergence. Bounded Martingales in L2 456Problems 45815 Brownian Motion 46515.1 History 46515.2 Definition 46715.2.1 Brownian motion as a Gaussian process 46915.3 Properties of Brownian Motion 47115.3.1 Hitting times. Reflection principle. Maximum value 47415.3.2 Quadratic variation 47615.4 Simulating Brownian Motions 48015.4.1 Generating a Brownian motion path 48015.4.2 Estimating parameters for a Brownian motion with drift 481Problems 48116 Stochastic Differential Equations 48516.1 The Construction of the Stochastic Integral 48716.1.1 Itȏ integral construction 49016.1.2 An illustrative example 49216.2 Properties of the Stochastic Integral 49416.3 Itȏ lemma 49516.4 Stochastic Differential Equations (SDEs) 49916.4.1 A discussion of the types of solution for an SDE 50116.5 Examples of SDEs 50216.5.1 An analysis of Cox– Ingersoll– Ross (CIR) type models 50716.5.2 Models similar to CIR 50716.5.3 Moments calculation for the CIR model 50916.5.4 Interpretation of the formulas for moments 51116.5.5 Parameter estimation for the CIR model 51116.6 Linear Systems of SDEs 51316.7 A Simple Relationship between SDEs and Partial Differential Equations (PDEs) 51516.8 Monte Carlo Simulations of SDEs 517Problems 522A Appendix: Linear Algebra and Solving Difference Equations and Systems of Differential Equations 527A.1 Solving difference equations with constant coefficients 528A.2 Generalized matrix inverse and pseudo-determinant 528A.3 Connection between systems of differential equations and matrices 529A.3.1 Writing a system of differential equations in matrix form 530A.4 Linear Algebra results 533A.4.1 Eigenvalues, eigenvectors of a square matrix 533A.4.2 Matrix Exponential Function 534A.4.3 Relationship between Exponential matrix and Eigenvectors 534A.5 Finding fundamental solution of the homogeneous system 535A.5.1 The case when all the eigenvalues are distinct and real 536A.5.2 The case when some of the eigenvalues are complex 536A.5.3 The case of repeated real eigenvalues 537A.6 The nonhomogeneous system 538A.6.1 The method of undetermined coefficients 538A.6.2 The method of variation of parameters 539A.7 Solving systems when P is non-constant 540Bibliography 541Index 547