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Applied Stochastic Processes and Control for
Jump-Diffusions: Modeling, Analysis and
Computation
Floyd B. Hanson
2007 by the Society for Industrial and Applied Mathematics.

Contents
Preface xvii
1 Stochastic Jump and Diffusion Processes 1
1.1 Poisson and Wiener Processes Basics . . . . . . . . . . . . . . . 1
1.2 Wiener Process Basic Properties . . . . . . . . . . . . . . . . . . 3
1.3 More Wiener Process Moments . . . . . . . . . . . . . . . . . . 6
1.4 Wiener Process Non-Differentiability . . . . . . . . . . . . . . . 9
1.5 Wiener Process Expectations Conditioned on Past . . . . . . . . 10
1.6 Poisson Process Basic Properties . . . . . . . . . . . . . . . . . . 11
1.7 Poisson Process Moments . . . . . . . . . . . . . . . . . . . . . . 16
1.8 Poisson Poisson Zero-One Jump Law . . . . . . . . . . . . . . . 18
1.9 Temporal, Non-Stationary Poisson Process . . . . . . . . . . . . 21
1.10 Poisson Process Expectations Conditioned on Past . . . . . . . 24
1.11 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2 Stochastic Integration for Diffusions 31
2.1 Ordinary or Riemann Integration . . . . . . . . . . . . . . . . . 32
2.2 Stochastic Integration in W(t): The Foundations . . . . . . . . 35
2.3 Stratonovich and other Stochastic Integration Rules . . . . . . . 56
2.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
2.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3 Stochastic Integration for Jumps 65
3.1 Stochastic Integration in P(t): The Foundations . . . . . . . . 65
3.2 Stochastic Jump Integration Rules and Expectations: . . . . . . 77
3.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
3.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
4 Stochastic Calculus for Jump-Diffusions 83
4.1 Diffusion Process Calculus Rules . . . . . . . . . . . . . . . . . . 83
4.1.1 Functions of Diffusions Alone, G(W(t)) . . . . . . 84
4.1.2 Functions of Diffusions and Time . . . . . . . . . . . 87
4.1.3 Itˆo Stochastic Natural Exponential Construction . . 90
4.1.4 Transformations of Linear Diffusion SDEs: . . . . . 94
4.1.5 Functions of General Diffusion States and Time . . 100
4.2 Poisson Jump Process Calculus Rules . . . . . . . . . . . . . . . 101
4.2.1 Jump Calculus Rule for h(dP(t)) . . . . . . . . . . . 101
4.2.2 Jump Calculus Rule for H(P(t), t) . . . . . . . . . . 102
4.2.3 Jump Calculus Rule with General State . . . . . . . 105
4.2.4 Transformations of Linear Jump with Drift SDEs . . 106
4.3 Jump-Diffusion Rules and SDEs . . . . . . . . . . . . . . . . . . 108
4.3.1 Jump-Diffusion Conditional Infinitesimal Moments . 109
4.3.2 Stochastic Jump-Diffusion Chain Rule . . . . . . . . 109
4.3.3 Linear Jump-Diffusion SDEs . . . . . . . . . . . . . 111
4.3.4 SDE Models Exactly Transformable . . . . . . . . . 121
4.4 Poisson Noise is White Noise Too! . . . . . . . . . . . . . . . . . 123
4.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
5 Stochastic Calculus for General Markov SDEs 131
5.1 Space-Time Poisson Process . . . . . . . . . . . . . . . . . . . . 132
5.2 State-Dependent Generalizations . . . . . . . . . . . . . . . . . . 141
5.2.1 State-Dependent Poisson Processes . . . . . . . . . . 141
5.2.2 State-Dependent Jump-Diffusion SDEs . . . . . . . 143
5.2.3 Linear State-Dependent SDEs . . . . . . . . . . . . 144
5.3 Multi-Dimensional Markov SDE . . . . . . . . . . . . . . . . . . 162
5.3.1 Conditional Infinitesimal Moments . . . . . . . . . . 163
5.3.2 Stochastic Chain Rule in Multi-Dimensions . . . . . 165
5.4 Distributed Jump SDE Models Exactly Transformable . . . . . 166
5.4.1 Jump SDE Models Exactly Transformable . . . . . . 167
5.4.2 Vector Jump SDE Models Exactly Transformable . 167
5.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
6 Stochastic Dynamic Programming 171
6.1 Stochastic Optimal Control Problem . . . . . . . . . . . . . . . 171
6.2 Bellman’s Principle of Optimality . . . . . . . . . . . . . . . . . 174
6.3 HJB Equation of Stochastic Dynamic Programming . . . . . . . 178
6.4 Linear Quadratic Jump-Diffusion (LQJD) Problem . . . . . . . 182
6.4.1 LQJD in Control Only (LQJD/U) Problem . . . . . 182
6.4.2 LLJD/U or the Case C2 ≡ 0: . . . . . . . . . . . . . 185
6.4.3 Canonical LQJD Problem . . . . . . . . . . . . . . . 186
6.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
7 Kolmogorov Equations 195
7.1 Dynkin’s Formula and the Backward Operator . . . . . . . . . . 195
7.2 Backward Kolmogorov Equations . . . . . . . . . . . . . . . . . 198
7.3 Forward Kolmogorov Equations . . . . . . . . . . . . . . . . . . 201
7.4 Multi-dimensional Backward and Forward Equations . . . . . . 205
7.5 Chapman-Kolmogorov Equation for Markov Processes . . . . . 208
7.6 Jump-Diffusion Boundary Conditions . . . . . . . . . . . . . . . 208
7.6.1 Absorbing Boundary Condition . . . . . . . . . . . . 208

. C35

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2010-7-11 23:14:48

7.6.2 Reflecting Boundary Conditions . . . . . . . . . . . 209
7.7 Stopping Times: Expected Exit and First Passage Times . . . . 210
7.7.1 Expected Stochastic Exit Time . . . . . . . . . . . . 211
7.8 Diffusion Approximation Basis . . . . . . . . . . . . . . . . . . . 216
7.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
8 Computational Stochastic Control Methods 223
8.1 Finite Difference PDE Methods of SDP . . . . . . . . . . . . . . 224
8.1.1 Linear Dynamics and Quadratic Control Costs . . . 225
8.1.2 Crank-Nicolson, Prediction-Correction for SDP . . . 226
8.1.3 Upwinding If Not Diffusion-Dominated . . . . . . . 232
8.1.4 Multi-state Systems andCurse of Dimensionality . . 233
8.2 Markov Chain Approximation for SDP . . . . . . . . . . . . . . 235
8.2.1 The MCA Formulation for Stochastic Diffusions . . 236
8.2.2 MCA Local Diffusion Consistency Conditions . . . . 237
8.2.3 MCA Numerical Finite Differenced Derivatives . . . 238
8.2.4 MCA Extensions to Include Jump Processes . . . . 241
9 Stochastic Simulations 247
9.1 SDE Simulation Methods . . . . . . . . . . . . . . . . . . . . . . 247
9.1.1 Convergence and Stability for Stochastic Simulations 248
9.1.2 Stochastic Diffusion Euler Simulations . . . . . . . . 250
9.1.3 Milstein’s Higher Order Diffusion Simulations . . . . 255
9.1.4 Convergence of Jump-Diffusion Simulations . . . . . 256
9.1.5 Jump-Diffusion Simulation Procedures . . . . . . . . 262
9.2 Monte Carlo Methods . . . . . . . . . . . . . . . . . . . . . . . . 265
9.2.1 Basic Monte Carlo Simulations . . . . . . . . . . . . 267
9.2.2 Inverse Generation for Non-Uniform Variates . . . . 275
9.2.3 Acceptance and Rejection Method of von Neumann 278
9.2.4 Importance Sampling . . . . . . . . . . . . . . . . . 282
9.2.5 Stratified Sampling . . . . . . . . . . . . . . . . . . 284
9.2.6 Antithetic Variates . . . . . . . . . . . . . . . . . . . 287
9.2.7 Control Variates . . . . . . . . . . . . . . . . . . . . 289
10 Applications in Financial Engineering 295
10.1 Classical Black-Scholes Option Pricing Model . . . . . . . . . . 296
10.2 Merton’s Three Asset Option Pricing Model . . . . . . . . . . . 300
10.2.1 PDE of Option Pricing . . . . . . . . . . . . . . . . 307
10.2.2 Final and Boundary Conditions for Option Pricing . 309
10.2.3 Transforming PDE to Standard Diffusion PDE . . . 312
10.3 Jump-Diffusion Option Pricing . . . . . . . . . . . . . . . . . . . 317
10.3.1 Jump-Diffusions with Normal Jump-Amplitudes . . 319
10.3.2 Risk-Neutral Option Pricing for Jump-Diffusions . . 320
10.4 Optimal Portfolio and Consumption Models . . . . . . . . . . . 326
10.4.1 Log-Uniform Jump-Diffusion for Log-Return . . . . 326
10.4.2 Log-Uniform Jump-Amplitude Model . . . . . . . . 328
10.4.3 Optimal Portfolio and Consumption Policies . . . . 330
10.4.4 CRRA Utility and Canonical Solution Reduction: . 334
10.5 Important Financial Events Model: The Greenspan Process . . 337
10.5.1 Scheduled and Unscheduled Events Model . . . . . . 338
10.5.2 Properties of Scheduled Event Processes . . . . . . . 339
10.5.3 Optimal Utility, Stock Fraction and Consumption . 340
10.5.4 Canonical CRRA Model Solution . . . . . . . . . . . 343
10.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345
11 Applications in Mathematical Biology and Medicine 349
11.1 Stochastic Bioeconomics: Optimal Harvesting Applications . . . 349
11.1.1 Optimal Harvesting of Jump-Logistic Population . . 350
11.1.2 Optimal Harvesting with Random Price Dynamics . 354
11.2 Stochastic Biomedical Applications . . . . . . . . . . . . . . . . 357
11.2.1 Tumor Doubling Time Diffusion Approximation . . 358
11.2.2 Optimal Drug Delivery to Brain PDE Model . . . . 363
12 Applied Guide to Abstract Stochastic Processes 373
12.1 Very Basic Probability Measure Background . . . . . . . . . . . 374
12.1.1 Mathematical Measure Theory Basics . . . . . . . . 374
12.1.2 Change of Measure: Radon-Nikod´ym Derivative: . . 380
12.1.3 Probability Measure Basics . . . . . . . . . . . . . . 381
12.1.4 Stochastic Processes on Filtered Probability Spaces 383
12.1.5 Martingales in Continuous Time . . . . . . . . . . . 385
12.1.6 Marked-Jump-Diffusion Martingale Representation . 388
12.2 Change in Probability Measure: Radon-Nikod´ym and Girsanov’s 390
12.2.1 Radon-Nikod´ym Change of Probability Measure . . 390
12.2.2 Girsanov Change in Probability Measure . . . . . . 395
12.3 Itˆo, L´evy and Jump-Diffusion Comparisons . . . . . . . . . . . . 403
12.3.1 Itˆo Processes and Jump-Diffusion Processes . . . . . 403
12.3.2 L´evy Processes and Jump-Diffusion Processes . . . . 404
12.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415
Bibliography 417
Index 438

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2010-7-11 23:15:11

A Appendix: Deterministic Optimal Control A1
A.1 Hamilton’s Equations . . . . . . . . . . . . . . . . . . . . . . . . A2
A.1.1 Deterministic Computational Complexity . . . . . . A11
A.2 Optimum Principles: The Basic Principles Approach . . . . . . A12
A.3 Linear Quadratic (LQ) Canonical Models . . . . . . . . . . . . . A23
A.3.1 Scalar, Linear Dynamics, Quadratic Costs (LQ) . . A23
A.3.2 Matrix, Linear Dynamics, Quadratic Costs (LQ) . . A25
A.4 Deterministic Dynamic Programming (DDP) . . . . . . . . . . . A29
A.4.1 Deterministic Principle of Optimality . . . . . . . . A30
A.4.2 Hamilton-Jacobi-Bellman (HJB) Equation of DDP . A31
A.4.3 Computational Complexity for DDP . . . . . . . . . A32
A.4.4 Linear Quadratic (LQ) Problem by DDP . . . . . . A33
A.5 Control of PDE Driven Dynamics (DPS) . . . . . . . . . . . . . A35
A.5.1 DPS Optimal Control Problem . . . . . . . . . . . . A35
A.5.2 DPS Hamiltonian Extended Space Formulation . . . A36
A.5.3 DPS Optimal State, Co-State and Control PDEs . . A38
A.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A40
B Appendix Online: Preliminaries in Probability and Analysis B1
B.1 Distributions for Continuous Random Variables . . . . . . . . . B2
B.1.1 Probability Distribution and Density Functions . . . B2
B.1.2 Expectations and Higher Moments . . . . . . . . . . B4
B.1.3 Uniform Distribution . . . . . . . . . . . . . . . . . B5
B.1.4 Normal Distribution and Gaussian Processes . . . . B8
B.1.5 Simple Gaussian Processes . . . . . . . . . . . . . . B10
B.1.6 Lognormal Distribution . . . . . . . . . . . . . . . . B11
B.1.7 Exponential Distribution . . . . . . . . . . . . . . . B15
B.2 Distributions of Discrete Random Variables . . . . . . . . . . . B18
B.2.1 Poisson Distribution and Poisson Process . . . . . . B19
B.3 Joint and Conditional Distribution Definitions . . . . . . . . . . B21
B.3.1 Conditional Distributions and Expectations . . . . . B26
B.3.2 Law of Total Probability . . . . . . . . . . . . . . . B29
B.4 Probability Distribution of a Sum: Convolutions . . . . . . . . . B31
B.5 Characteristic Functions . . . . . . . . . . . . . . . . . . . . . . B34
B.6 Sample Mean and Variance: Sums of IID Random Variables . . B37
B.7 Law of Large Numbers . . . . . . . . . . . . . . . . . . . . . . . B39
B.7.1 Weak Law of Large Numbers (WLLN) . . . . . . . . B39
B.7.2 Strong Law of Large Numbers (SLLN) . . . . . . . . B40
B.8 Central Limit Theorem . . . . . . . . . . . . . . . . . . . . . . . B40
B.9 Matrix Algebra and Analysis . . . . . . . . . . . . . . . . . . . . B40
B.10 Some Multivariate Distributions . . . . . . . . . . . . . . . . . . B46
B.10.1 Multivariate Normal Distribution . . . . . . . . . . . B46
B.10.2 Multinomial Distribution . . . . . . . . . . . . . . . B48
B.11 Basic Asymptotic Notation and Results . . . . . . . . . . . . . . B51
B.12 Generalized Functions: Combined Continuous and Discrete . . . B53
B.13 Fundamental Properties of Stochastic and Markov Processes . . B61
B.13.1 Basic Classification of Stochastic Processes . . . . . B61
B.13.2 Markov Processes and Markov Chains . . . . . . . . B61
B.13.3 Stationary Markov Processes and Markov Chains . . B62
B.14 Continuity, Jump Discontinuity and Non-Smoothness . . . . . . B63
B.14.1 Beyond Continuity Properties . . . . . . . . . . . . . B63
B.14.2 Taylor Approximations of Composite Functions . . . B65
B.15 Extremal Principles . . . . . . . . . . . . . . . . . . . . . . . . . B69
B.16 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B71
C Appendix Online: MATLAB Programs C1
C.1 Program: Uniform Distribution Simulation Histograms . . . . . C1
C.2 Program: Normal Distribution Simulation Histograms . . . . . . C2
C.3 Program: Lognormal Distribution Simulation Histograms . . . . C4
C.4 Program: Exponential Distribution Simulation Histograms . . . C5
C.5 Program: Poisson Distribution versus Jump Counter k . . . . . C6
C.6 Program: Binomial Distribution versus Binomial Frequency f1 . C7
C.7 Program: Simulated Diffusion W(t) Sample Paths . . . . . . . . C8
C.8 Program: Diffusion Sample Paths Time Step Variation . . . . . C9
C.9 Program: Simulated Simple Poisson P(t) Sample Paths . . . . . C11
C.10 Program: Simulated Incremental Poisson P(t) Sample Paths . C12
C.11 Program: Simulated Diffusion Integrals
R
!(dW)2 . . . . . . . . . C14
C.12 Program: Simulated Diffusion Integrals
R
g(W, t)dW . . . . . . . C15
C.13 Program: Simulated Diffusion Integrals
R
g(W, t)dW: Chain Rule C16
C.14 Program: Simulated Linear Jump-Diffusion Sample Paths . . . C18
C.15 Program: Simulated Linear Mark-Jump-Diffusion Sample Paths C21
C.16 Program: Euler-Maruyama Simulations for Linear Diffusion SDE C25
C.17 Program: Milstein Simulations for Linear Diffusion SDE . . . . C27
C.18 Program: Monte Carlo Simulation Comparing Uniform and Normal
Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C29
C.19 Program: Monte Carlo Simulation Comparing Uniform and Normal
Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C31
C.20 Program: Monte Carlo Acceptance-Rejection Technique . . . . . C33
C.21 Program: Monte Carlo Multidimensional Integration . . . . . . C35

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zgryyl

2010-9-30 13:20:42

没必要那么贵吧

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moxiang

2011-1-2 08:56:48

google research 上免费有的

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mathmli

2012-3-16 10:31:29

google上有免费的

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