An Introduction to Continuous-Time Stochastic Processes Theory, Models, and ...

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An Introduction to Continuous-Time Stochastic Processes Theory, Models, and Applications to Finance.pdf
by Vincenzo Capasso, David Bakstein

Birkhäuser Boston
2004-12-07
ISBN: 0817632344
343 pages

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

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