Applied Stochastic Processes and Control for Jump-Diffusions

Transcription

1 Applied Stochastic Processes and Control for Jump-Diffusions Modeling, Analysis, and Computation Floyd B. Hanson University of Illinois at Chicago Chicago, Illinois siam.. Society for Industrial and Applied Mathematics Philadelphia

2 List of Figures List of Tables Preface xv xxi xxiii 1 Stochastic Jump and Diffusion Processes: Introduction Poisson and Wiener Processes Basics Wiener Process Basic Properties More Wiener Process Moments Wiener Process Nondifferentiability Wiener Process Expectations Conditioned on the Past Poisson Process Basic Properties Poisson Process Moments Poisson Zero-One Jump Law Temporal, Nonstationary Poisson Process Poisson Process Expectations Conditioned on the Past Exercises 25 2 Stochastic Integration for Diffusions Ordinary or Riemann Integration Stochastic Integration in W(t): The Foundations Stratonovich and Other Stochastic Integration Rules Conclusion Exercises 57 3 Stochastic Integration for Jumps Stochastic Integration in P(t): The Foundations Stochastic Jump Integration Rules and Expectations Conclusion Exercises 77 4 Stochastic Calculus for Jump-Diffusions: Elementary SDEs Diffusion Process Calculus Rules Functions of Diffusions Alone, G(W(t)) 82 vii

3 4.1.2 Functions of Diffusions and Time G(W(t)J) Itô Stochastic Natural Exponential Construction Transformations of Linear Diffusion SDEs Functions of General Diffusion States and Time F(X(t), t) Poisson Jump Process Calculus Rules Jump Calculus Rule for h{dp{t)) Jump Calculus Rule for H(P(t),t) Jump Calculus Rule with General State Y(t) = F(X(t),t) Transformations of Linear Jump with Drift SDEs Jump-Diffusion Rules and SDEs Jump-Diffusion Conditional Infinitesimal Moments Stochastic Jump-Diffusion Chain Rule Linear Jump-Diffusion SDEs SDE Models Exactly Transformable to Purely Time-Varying Coefficients Poisson Noise Is White Noise Too! Exercises 122 Stochastic Calculus for General Markov SDËs: Space-Time Poisson, State-Dependent Noise, and Multidimensions Space-Time Poisson Process State-Dependent Generalization of Jump-Diffusion SDEs State-Dependent Generalization for Space-Time Poisson Processes State-Dependent Jump-Diffusion SDEs Linear State-Dependent SDEs Multidimensional Markov SDE Conditional Infinitesimal Moments in Multidimensions Stochastic Chain Rule in Multidimensions Distributed Jump SDE Models Exactly Transformable Distributed Jump SDE Models Exactly Transformable Vector Distributed Jump SDE Models Exactly Transformable Exercises 165 Stochastic Optimal Control: Stochastic Dynamic Programming Stochastic Optimal Control Problem Bellman's Principle of Optimality Hamiton-Jacobi-Bellman (HJB) Equation of Stochastic Dynamic Programming (SDP) Linear Quadratic Jump-Diffusion (LQJD) Problem LQJD in Control Only (LQJD/U) Problem LLJD/U or the Case C 2 = Canonical LQJD Problem Exercises 188

4 ix 7 Kolmogorov Forward and Backward Equations and Their Applications Dynkin's Formula and the Backward Operator Backward Kolmogorov Equations Forward Kolmogorov Equations Multidimensional Backward and Forward Equations Chapman-Kolmogorov Equation for Markov Processes in Continuous Time Jump-Diffusion Boundary Conditions Absorbing Boundary Conditions Reflecting Boundary Conditions Stopping Times: Expected Exit and First Passage Times Expected Stochastic Exit Time Diffusion Approximation Basis Exercises Computational Stochastic Control Methods Finite Difference PDE Methods of SDP Linear Dynamics and Quadratic Control Costs Crank-Nicolson, Extrapolation-Predictor-Corrector Finite Difference Algorithm for SDP Upwinding Finite Differences If Not Diffusion-Dominated Multistate Systems and Bellman's Curse of Dimensionality Markov Chain Approximation for SDP MCA Formulation for Stochastic Diffusions MCA Local Diffusion Consistency Conditions MCA Numerical Finite Differences for State Derivatives and Construction of Transition Probabilities MCA Extensions to Include Jump Processes Stochastic Simulations SDE Simulation Methods Convergence and Stability for Stochastic Problems and Simulations Stochastic Diffusion Euler Simulations Milstein's Higher Order Diffusion Simulations Convergence and Stability of Jump-Diffusion Euler Simulations Jump-Diffusion Euler Simulation Procedures Monte Carlo Methods Basic Monte Carlo Simulations Inverse Method for Generating Nonuniform Variates Acceptance and Rejection Method of von Neumann Importance Sampling Stratified Sampling Antithetic Variates Control Variates 281

5 10 Applications in Financial Engineering Classical Black-Scholes Option Pricing Model Merton 's Three Asset Option Pricing Model Version of Black-Scholes PDE of Option Pricing Final and Boundary Conditions for Option Pricing PDE Transforming PDE to Standard Diffusion PDE Jump-Diffusion Option Pricing Jump-Diffusions with Normal Jump-Amplitudes Risk-Neutral Option Pricing for Jump-Diffusions Optimal Portfolio and Consumption Models Log-Uniform Amplitude Jump-Diffusion for Log-Returns Log-Uniform Jump-Amplitude Model Optimal Portfolio and Consumption Policies Application CRRA Utility and Canonical Solution Reduction Important Financial Events Model: The Greenspan Process Stochastic Scheduled and Unscheduled Events Model with Stochastic Parameter Processes Further Properties of Quasi-Deterministic or Scheduled Event Processes: K(q\ A{t))dQ(t) Optimal Portfolio Utility, Stock Fraction, and Consumption Canonical CRRA Model Solution Exercises Applications in Mathematical Biology and Medicine Stochastic Bioeconomics: Optimal Harvesting Applications Optimal Harvesting of Logistically Growing Population Undergoing Random Jumps 340 ll. 1.2 Optimal Harvesting with Both Price and Population Random Dynamics Stochastic Biomedical Applications Diffusion Approximation of Tumor Growth and Tumor Doubling Time Application Optimal Drug Delivery to Brain PDE Model Applied Guide to Abstract Theory of Stochastic Processes Very Basic Probability Measure Background Mathematical Measure Theory Basics Change of Measure: Radon-Nikodym Theorem and Derivative Probability Measure Basics Stochastic Processes in Continuous Time on Filtered Probability Spaces Martingales in Continuous Time Jump-Diffusion Martingale Representation Change in Probability Measure: Radon-Nikodym Derivatives and Girsanov's Theorem 376

6 xi 2.1 Radon-Nikodym Theorem and Derivative for Change of Probability Measure Change in Measure for Stochastic Processes: Girsanov's Theorem Itô, Levy, and Jump-Diffusion Comparisons Itô Processes and Jump-Diffusion Processes Levy Processes and Jump-Diffusion Processes Exercise 401 Bibliography 403 Index 423 A Online Appendix: Deterministic Optimal Control Al A.I Hamilton's Equations: Hamiltonian and Lagrange Multiplier Formulation of Deterministic Optimal Control A2 A.1.1 Deterministic Computation and Computational Complexity.All A.2 Optimum Principles: The Basic Principles Approach A12 A.3 Linear Quadratic (LQ) Canonical Models A22 A.3.1 Scalar, Linear Dynamics, Quadratic Costs (LQ) A22 A.3.2 Matrix, Linear Dynamics, Quadratic Costs (LQ) A24 A.4 Deterministic Dynamic Programming (DDP) A28 A.4.1 Deterministic Principle of Optimality A29 A.4.2 Hamilton-Jacobi-Bellman (HJB) Equation of Deterministic Dynamic Programming A30 A.4.3 Computational Complexity for Deterministic Dynamic Programming A31 A.4.4 Linear Quadratic (LQ) Problem by Deterministic Dynamic Programming A32 A.5 Control of PDE Driven Dynamics: Distributed Parameter Systems (DPS) A34 A.5.1 DPS Optimal Control Problem A34 A.5.2 DPS Hamiltonian Extended Space Formulation A35 A.5.3 DPS Optimal State, Costate, and Control PDEs A37 A.6 Exercises A39 Β Online Appendix: Preliminaries in Probability and Analysis Bl B.l Distributions for Continuous Random Variables B2 Β. 1.1 Probability Distribution and Density Functions B2 Β. 1.2 Expectations and Higher Moments B4 Β. 1.3 Uniform Distribution B5 Β. 1.4 Normal Distribution and Gaussian Processes B8 Β. 1.5 Simple Gaussian Processes B9 Β. 1.6 Lognormal Distribution Β11 Β. 1.7 Exponential Distribution Β14 B.2 Distributions of Discrete Random Variables Β17 B.2.1 Poisson Distribution and Poisson Process Β18

7 xji B.3 Joint and Conditional Distribution Definitions B20 B.3.1 Conditional Distributions and Expectations B25 B.3.2 Law of Total Probability B29 B.4 Probability Distribution of a Sum: Convolutions B30 B.5 Characteristic Functions B33 B.6 Sample Mean and Variance: Sums of Independent, Identically Distributed (HD) Random Variables B36 B.7 Law of Large Numbers B38 B.7.1 Weak Law of Large Numbers (WLLN) B38 B.7.2 Strong Law of Large Numbers (SLLN) B38 B.8 Central Limit Theorem B39 B.9 Matrix Algebra and Analysis B39 B.IO Some Multivariate Distributions B45 B.10.1 Multivariate Normal Distribution B45 B.10.2 Multinomial Distribution B46 B.ll Basic Asymptotic Notation and Results B49 BJ2 Generalized Functions: Combined Continuous and Discrete Processes B52 B.13 Fundamental Properties of Stochastic and Markov Processes B59 B.13.1 Basic Classification of Stochastic Processes B59 B.13.2 Markov Processes and Markov Chains B59 B.13.3 Stationary Markov Processes and Markov Chains B61 B.14 Continuity, Jump Discontinuity, and Nonsmoothness Approximations. B61 B.I4.1 Beyond Continuity Properties B61 B.14.2 Taylor Approximations of Composite Functions B63 B.15 Extremal Principles B67 B.16 Exercises B69 C Online Appendix: MATLAB Programs CI C.I Program: Uniform Distribution Simulation Histograms CI C.2 Program: Normal Distribution Simulation Histograms C2 C3 Program; Lognormal Distribution Simulation Histograms C3 C.4 Program: Exponential Distribution Simulation Histograms C4 C.5 Program: Poisson Distribution Versus Jump Counter k CS C.6 Program: Binomial Distribution Versus Binomial Frequency f {... Cβ C.7 Program: Simulated Diffusion W(t) Sample Paths CI C.8 Program: Simulated Diffusion W(t) Sample Paths Showing Variation with Time Step Size C8 C.9 Program: Simulated Simple Poisson P(t) Sample Paths C9 CIO Program: Simulated Simple Incremental Poisson AP(t) Sample Paths. CIO Cll Program: Simulated Diffusion Integrals f(dw) 2 (t) by Itô Partial Sums C12 C12 Program: Simulated Diffusion Integrals fg(wj)äw: Direct Case by Itô Partial Sums C13 C.13 Program: Simulated Diffusion Integrals fg(x,t)dw: Chain Rule.... C14 C.14 Program: Simulated Linear Jump-Diffusion Sample Paths C16 C.I5 Program: Simulated Linear Mark-Jump-Diffusion Sample Paths... C18

8 xiii C.16 Program: Curse of Dimensionality C21 C.I7 Program: Euler-Maruyama Simulations for Linear Diffusion SDE.. C23 C.I8 Program: Milstein Simulations for Linear Diffusion SDE C25 C.19 Program: Monte Carlo Simulation Comparing Uniform and Normal Errors C27 C.20 Program: Monte Carlo Simulation Testing Uniform Distribution... C29 C.21 Program: Monte Carlo Acceptance-Rejection Technique C30 C.22 Program: Monte Carlo Multidimensional Integration C32 C.23 Program: Regular and Bang Control Examples C34 C.24 Program: Simple Optimal Control Example C37 C.25 Program: Bang-Bang Control with Control Switching Example... C38 C.26 Program: Singular Control Examples C40