Stochastic Processes
This comprehensive guide to stochastic processes gives a complete overview of the theory and addresses the most important applications. Pitched at a level accessible to beginning graduate students and researchers from applied disciplines, it is both a course book and a rich resource for individual readers. Subjects covered include Brownian motion, stochastic calculus, stochastic differential equations, Markov processes, weak convergence of processes and semigroup theory. Applications include the Black–Scholes formula for the pricing of derivatives in financial mathematics, the Kalman–Bucy filter used in the US space program and also theoretical applications to partial differential equations and analysis. Short, readable chapters aim for clarity rather than full generality. More than 350 exercises are included to help readers put their new-found knowledge to the test and to prepare them for tackling the research literature.
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Contents
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1 Basic notions
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1 |
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2 Brownian motion
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6 |
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3 Martingales
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13 |
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4 Markov properties of Brownian motion
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25 |
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5 The Poisson process
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32 |
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6 Construction of Brownian motion
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36 |
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7 Path properties of Brownian motion
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43 |
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8 The continuity of paths
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49 |
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26 The RayKnight theorems
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209 |
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27 Brownian excursions
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214 |
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28 Financial mathematics
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218 |
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29 Filtering
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229 |
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30 Convergence of probability measures
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237 |
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31 Skorokhod representation
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244 |
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32 The space C01
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247 |
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33 Gaussian processes
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251 |
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9 Continuous semimartingales
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54 |
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10 Stochastic integrals
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64 |
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11 Itos formula
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71 |
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12 Some applications of Itos formula
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77 |
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13 The Girsanov theorem
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89 |
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14 Local times
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94 |
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15 Skorokhod embedding
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100 |
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16 The general theory of processes
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111 |
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17 Processes with jumps
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130 |
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18 Poisson point processes
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147 |
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19 Framework for Markov processes
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152 |
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20 Markov properties
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160 |
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21 Applications of the Markov properties
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167 |
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22 Transformations of Markov processes
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177 |
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23 Optimal stopping
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184 |
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24 Stochastic differential equations
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192 |
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25 Weak solutions of SDEs
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204 |
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34 The space D01
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259 |
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35 Applications of weak convergence
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269 |
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36 Semigroups
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279 |
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37 Infinitesimal generators
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286 |
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38 Dirichlet forms
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302 |
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39 Markov processes and SDEs
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312 |
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40 Solving partial differential equations
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319 |
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41 Onedimensional diffusions
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326 |
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42 Levy processes
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339 |
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Appendix A Basic probability
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348 |
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Appendix B Some results from analysis
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378 |
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Appendix C Regular conditional probabilities
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380 |
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Appendix D Kolmogorov extension theorem
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382 |
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Common terms and phrases
Bessel process Borel measurable Borel subset Brownian motion characteristic function compact support continuous function continuous paths continuous with left converges weakly Corollary d-dimensional Brownian motion define definition dominated convergence equal Exercise exists filtration finite first function f hence holds increasing process independent inf{t Itˆo’s formula jumps L´evy processes left limits Lemma Let f linear local martingale Markov process Markov property martingale with respect mean zero metric space monotone class non-negative normal random variable null set one-dimensional Brownian motion optional Poisson process predictable stopping probability measure Proof Let Proposition prove right continuous satisfies satisfying the usual semigroup semimartingale sequence solution square integrable square integrable martingale stochastic integral stochastic process strong Markov process strong Markov property submartingale taking values Theorem uniformly integrable uniqueness usual conditions variance Xt Px