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

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Springer Science & Business Media, Jan 3, 2008 - Mathematics - 344 pages
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This book is a systematic, rigorous, and self-consistent introduction to the theory of continuous-time stochastic processes. But it is neither a tract nor a recipe book as such; rather, it is an account of fundamental concepts as they appear in relevant modern applications and literature. We make no pretense of it being complete. Indeed, we have omitted many results, which we feel are notdirectly relatedtothemain themeorthatare availablein easilyaccessible sources. (Thosereaderswhoareinterestedinthehistoricaldevelopmentofthe subject cannot ignore the volume edited by Wax (1954). ) Proofs are often omitted as technicalities might distract the reader from a conceptual approach. They are produced whenever they may serve as a guide to the introduction of new concepts and methods towards the app- cations; otherwise, explicit references to standard literature are provided. A mathematically oriented student may ?nd it interesting to consider proofs as exercises. The scope of the book is profoundly educational, related to modeling re- world problems with stochastic methods. The reader becomes critically aware oftheconceptsinvolvedincurrentappliedliterature,andismoreoverprovided with a ?rm foundation of the mathematical techniques. Intuition is always supported by mathematical rigor. Our book addresses three main groups: ?rst, mathematicians working in a di?erent ?eld; second, other scientists and professionals from a business or academic background; third, graduate or advanced undergraduate students of a quantitative subject related to stochastic theory and/or applications.
 

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Contents

Fundamentals of Probability
3
12 Random Variables and Distributions
8
13 Expectations
15
14 Independence
19
15 Conditional Expectations
26
16 Conditional and Joint Distributions
35
17 Convergence of Random Variables
41
18 Exercises and Additions
44
44 Kolmogorov Equations
185
45 Multidimensional Stochastic Differential Equations
194
46 Stability of Stochastic Differential Equations
196
47 Exercises and Additions
203
The Applications of Stochastic Processes
209
Applications to Finance and Insurance
210
51 ArbitrageFree Markets
212
52 The Standard BlackScholes Model
216

Stochastic Processes
51
22 Stopping Times
58
23 Canonical Form of a Process
59
24 Gaussian Processes
60
25 Processes with Independent Increments
61
26 Martingales
63
27 Markov Processes
72
28 Brownian Motion and the Wiener Process
90
29 Counting Poisson and LÚvy Processes
102
210 Marked Point Processes
111
211 Exercises and Additions
118
The It˘ Integral
127
32 Stochastic Integrals as Martingales
139
33 It˘ Integrals of Multidimensional Wiener Processes
143
34 The Stochastic Differential
146
35 It˘s Formula
149
36 Martingale Representation Theorem
150
37 Multidimensional Stochastic Differentials
152
38 Exercises and Additions
155
Stochastic Differential Equations
161
42 The Markov Property of Solutions
176
43 Girsanov Theorem
182
53 Models of Interest Rates
222
54 Contingent Claims under Alternative Stochastic Processes
227
55 Insurance Risk
230
56 Exercises and Additions
236
Applications to Biology and Medicine
239
Continuous Approximation of Jump Models
250
IndividualBased Models
253
64 Neurosciences
270
65 Exercises and Additions
275
Appendices
280
Measure and Integration A1 Rings and 𝛔Algebras
281
A2 Measurable Functions and Measure
284
A3 Lebesgue Integration
288
A4 LebesgueStieltjes Measure and Distributions
292
A5 Stochastic Stieltjes Integration
296
Convergence of Probability Measures on Metric Spaces
297
B2 Prohorovs Theorem
304
Maximum Principles of Elliptic and Parabolic Operators
313
C2 Maximum Principles of Parabolic Equations
315
Stability of Ordinary Differential Equations
321
References
324
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