Stochastic processes and their first passage times: lecture notes
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Kingston, Ontario
Queen's Univ.
1994
|
| Series: | Queen's papers in pure and applied mathematics
96 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | IX, 616 S. graph. Darst. |
| ISBN: | 0889116733 |
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MARC
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| 245 | 1 | 0 | |a Stochastic processes and their first passage times |b lecture notes |c by M. T. Wasan |
| 264 | 1 | |a Kingston, Ontario |b Queen's Univ. |c 1994 | |
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Record in the Search Index
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|---|---|
| adam_text | Contents
Page
Preface
Chapter 1: Introduction
1. Introduction 1
2. Mathematical Formulation 2
3. Techniques to Find the First Passage Time Distribution 2
4. Summary 4
Chapter 2: Bernoulli Process
1. Introduction 8
2. First Passage Time of the Bernoulli Process 11
3.1 Random Walk First Passage Time 15
3.2 Application 17
4. Randomized Random Walk 20
5. Exercises 23
Chapter 3: Introduction to Markov Chain with Discrete Index
Parameter
1 . Introduction 25
2. Definitions and Main Theorems 44
3. Solved Problems 62
4. Exercises 67
5. Limit Theorems on Markov Chains (Introduction) 68
6. Occupation Times and First Passage Times 77
6a. Long Run and Stationary Properties
7. Solved Problems 87
8. Exercises 106
ii
Chapter 4: Markov Processes
1. Introduction 111
2. Poisson Process 112
3. Birth and Death Processes (Random Walk) 119
4. m|m|1 Queueing System 141
5. First Passage Times for State Dependent M|M|s|k Queueing Model 152
6. Combinatorial Methods in the First Passage Time of the Busy 162
Period
7. Green Function Method in the First Passage Times 171
8. First Passage Time, g|g|i Queues and Priority Scheme *
9. Branching Processes • 211
10. Exercises 228
Chapter 5: Renewal Process
1. Introduction . 234
2. The Renewal Counting Process 236
3. Distribution of Number of Renewals N(t) 237
4. The Renewal Function Renewal Density 240
5. Renewal Equation 244
6. Limiting Behaviour of Renewal Process 245
7. Population Age: Residual Lifetime 248
8. The Elementary Renewal Theorem ~ 253
9. Other Renewal Processes 255
TO. A Continuation of Renewal Theory 260
11. Applications 270
12. A First Passage Time of Renewal Process 276
iii
13. Special Case of the First Passage Time 281
14. First Passage for A Constant Barrier with Diffusion 283
Approximation
15. First Passage Time for Crossing A Linear Barrier 285
16. First Passage for Reaching A Linear Barrier • 286
17. First Passage Time for Reaching the Exponential Barrier 288
18. Imbedding Approach to First Passage Problem for Bounded 289
Processes
19. Exercises 294
Chapter 6: Brownian Motion Process
1 . Introduction 300
2. Definition and Preliminaries 301
3. First Passage Time Processes of Brownian Motion
3.1 Introduction 307
3.2 Basic Properties of T.B.M.P. Distribution 308
3.3 Various Forms of the Density 309
3.4 Moments of Density of a T.B.M.P. 310
3.5 A Reciprocal of T.B.M.P. Variate and Its Moments 311
3.6 Infinite Divisability _.„ 312
¦ 3.7 Properties of T.B.M.P. 313
3.8 First Passage Time Process 315
3.9 Properties of the Process 317
3.10 Characterization 320
3.11 A Cannonical Representation 333
3.12 Riemann Stieltjes Stochastic Integral T.B.M.P. Process 335
3.13 Decomposition 337
iv
3.14 Stochastic Integral for Nonanticipating Functionals 339
3.15 Behaviour at t = °° 343
3.16 First Passage Time Process of a Standard Brownian Motion 345
3.17 The Bivariate Process 355
4. Inverse Gaussian Measure and Its Applications to Approximation 368
Methods
5. Exercises 382
Chapter 7: Applications
1. Introduction 386
2. Monetary Control
2.1 Introduction 386
2.2 Summary of the Federal Reserves Control Procedure 389
2.3 A Random Walk Model of the Money Stock 392
2.4 First Passage Probabilities: One Boundary 396
2.5 First Passage Probabilities: Two Boundaries 400
2.6 Conclusion 406
2.7 Appendix 407
3. Currency Exchange Rate 413
4. Neuronal Spike Train 423
5. Ruin Problem 429
6. Problems
6.1 Dam 435
6.2 Inventory 436
Chapter 8: Diffusion Equations
1. Introduction 439
v
2. General Discussion of Properties 439
3. Boundary Conditions for Homogeneous Diffusion Process 456
4. The Uhlenbeck Ornstein Process 467
5. Particular Cases 473
6. Transformations of Standard Brownian Motion 480
7. Approximation of Certain Discrete Time Markov Processes by 483
Diffusion Processes
8. Exercises 489
Chapter 9: Martingale Approach
1. Introduction 492
2. Martingales 493
3. Applications to Problems of First Passage Time 506
4. Martingales and Eigenvectors of the Transition Matrix 516
5. Exercises 519
Chapter 10: Stochastic Differentials
10.0 Introduction 524
10.1 Berstein Differentials 525
.,10.2 Diffusion Approximations and Stochastic Differentials
¦ 10.2.0 Introduction 526
10.2.1 Bernstein Stochastic Differentials Diffusion 528
Equations
10.2.2 Diffusion Approximation of Random Walk 534
10.2.3 Approximation of A Compound Poisson Process 539
10.2.4 The Birth Death Process with Constant Parameters 542
10.2.5. Processes Approximated by the Ornstein Uhlenbec 546
Diffusion Process
vi
10.2.6 The Birth Death Process with Parameters 550
Proportional to the State
10.2.7 The Galton Watson Branching Process 552
10.2.8 Diffusion Approximation of Certain Population 557
Dependent Branching Processes
10.2.9 The Multiple Galton Watson Process 562
10.3 The First Passage Time Probability for a Boundary Consisting 567
of Two Intersecting Straight Lines
10.4 Exercises 591
Appendix 598
References 606
vii
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| physical | IX, 616 S. graph. Darst. |
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| series | Queen's papers in pure and applied mathematics |
| series2 | Queen's papers in pure and applied mathematics |
| spelling | Wasan, Madanlal T. Verfasser aut Stochastic processes and their first passage times lecture notes by M. T. Wasan Kingston, Ontario Queen's Univ. 1994 IX, 616 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Queen's papers in pure and applied mathematics 96 Processus stochastiques ram Stochastic processes Stochastischer Prozess (DE-588)4057630-9 gnd rswk-swf Stochastischer Prozess (DE-588)4057630-9 s DE-604 Queen's papers in pure and applied mathematics 96 (DE-604)BV001889470 96 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=006698223&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Wasan, Madanlal T. Stochastic processes and their first passage times lecture notes Queen's papers in pure and applied mathematics Processus stochastiques ram Stochastic processes Stochastischer Prozess (DE-588)4057630-9 gnd |
| subject_GND | (DE-588)4057630-9 |
| title | Stochastic processes and their first passage times lecture notes |
| title_auth | Stochastic processes and their first passage times lecture notes |
| title_exact_search | Stochastic processes and their first passage times lecture notes |
| title_full | Stochastic processes and their first passage times lecture notes by M. T. Wasan |
| title_fullStr | Stochastic processes and their first passage times lecture notes by M. T. Wasan |
| title_full_unstemmed | Stochastic processes and their first passage times lecture notes by M. T. Wasan |
| title_short | Stochastic processes and their first passage times |
| title_sort | stochastic processes and their first passage times lecture notes |
| title_sub | lecture notes |
| topic | Processus stochastiques ram Stochastic processes Stochastischer Prozess (DE-588)4057630-9 gnd |
| topic_facet | Processus stochastiques Stochastic processes Stochastischer Prozess |
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