Introduction to Continuous-Time Stochastic Processes : Theory, Models, and Applications to Biology, Finance, and Engineering

by Capasso, V.; Bakstein, David

• ISBN13: 9780817632342
• ISBN10: 0817632344
• Format: Hardcover
• Copyright: 2004-10-30
• Publisher: Springer Verlag

Summary

This concisely written book is a rigorous and self-contained introduction to the theory of continuous-time stochastic processes. A balance of theory and applications, the work features concrete examples of modeling real-world problems from biology, medicine, industrial applications, finance, and insurance using stochastic methods. No previous knowledge of stochastic processes is required.Key topics covered include:* Interacting particles and agent-based models: from polymers to ants* Population dynamics: from birth and death processes to epidemics* Financial market models: the non-arbitrage principle* Contingent claim valuation models: the risk-neutral valuation theory* Risk analysis in insuranceAn Introduction to Continuous-Time Stochastic Processes will be of interest to a broad audience of students, pure and applied mathematicians, and researchers or practitioners in mathematical finance, biomathematics, biotechnology, and engineering. Suitable as a textbook for graduate or advanced undergraduate courses, the work may also be used for self-study or as a reference. Prerequisites include knowledge of calculus and some analysis; exposure to probability would be helpful but not required since the necessary fundamentals of measure and integration are provided.

Table of Contents

Preface	v
Part I The Theory of Stochastic Processes
1 Fundamentals of Probability	3	(48)
1.1 Probability and Conditional Probability	3	(5)
1.2 Random Variables and Distributions	8	(7)
1.3 Expectations	15	(4)
1.4 Independence	19	(7)
1.5 Conditional Expectations	26	(9)
1.6 Conditional and Joint Distributions	35	(6)
1.7 Convergence of Random Variables	41	(3)
1.8 Exercises and Additions	44	(7)
2 Stochastic Processes	51	(76)
2.1 Definition	51	(7)
2.2 Stopping Times	58	(1)
2.3 Canonical Form of a Process	59	(1)
2.4 Gaussian Processes	60	(1)
2.5 Processes with Independent Increments	61	(2)
2.6 Martingales	63	(9)
2.7 Markov Processes	72	(18)
2.8 Brownian Motion and the Wiener Process	90	(12)
2.9 Counting, Poisson, and Lévy Processes	102	(9)
2.10 Marked Point Processes	111	(7)
2.11 Exercises and Additions	118	(9)
3 The Itô Integral	127	(34)
3.1 Definition and Properties	127	(12)
3.2 Stochastic Integrals as Martingales	139	(4)
3.3 Itô Integrals of Multidimensional Wiener Processes	143	(3)
3.4 The Stochastic Differential	146	(3)
3.5 Ito's Formula	149	(1)
3.6 Martingale Representation Theorem	150	(2)
3.7 Multidimensional Stochastic Differentials	152	(3)
3.8 Exercises and Additions	155	(6)
4 Stochastic Differential Equations	161	(50)
4.1 Existence and Uniqueness of Solutions	161	(15)
4.2 The Markov Property of Solutions	176	(6)
4.3 Girsanov Theorem	182	(3)
4.4 Kolmogorov Equations	185	(9)
4.5 Multidimensional Stochastic Differential Equations	194	(2)
4.6 Stability of Stochastic Differential Equations	196	(7)
4.7 Exercises and Additions	203	(8)
Part II The Applications of Stochastic Processes
5 Applications to Finance and Insurance	211	(28)
5.1 Arbitrage-Free Markets	212	(4)
5.2 The Standard Black-Scholes Model	216	(6)
5.3 Models of Interest Rates	222	(5)
5.4 Contingent Claims under Alternative Stochastic Processes	227	(3)
5.5 Insurance Risk	230	(6)
5.6 Exercises and Additions	236	(3)
6 Applications to Biology and Medicine	239	(44)
6.1 Population Dynamics: Discrete-in-Space-Continuous-in-Time Models	239	(11)
6.2 Population Dynamics: Continuous Approximation of Jump Models	250	(3)
6.3 Population Dynamics: Individual-Based Models	253	(17)
6.4 Neurosciences	270	(5)
6.5 Exercises and Additions	275	(8)
Part III Appendices
A Measure and Integration	283	(14)
A.1 Rings and σ-Algebras	283	(1)
A.2 Measurable Functions and Measure	284	(4)
A.3 Lebesgue Integration	288	(4)
A.4 Lebesgue-Stieltjes Measure and Distributions	292	(4)
A.5 Stochastic Stieltjes Integration	296	(1)
B Convergence of Probability Measures on Metric Spaces	297	(16)
B.1 Metric Spaces	297	(7)
B.2 Prohorov's Theorem	304	(1)
B.3 Donsker's Theorem	304	(9)
C Maximum Principles of Elliptic and Parabolic Operators	313	(8)
C.1 Maximum Principles of Elliptic Equations	313	(2)
C.2 Maximum Principles of Parabolic Equations	315	(6)
D Stability of Ordinary Differential Equations	321	(4)
References	325