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 Durrett, Richard, 1951
 Boca Raton : CRC Press, c1996.
 Description
 Book — vi, 341 p. ; 24 cm.
 Summary

 CHAPTER 1. BROWNIAN MOTION Definition and Construction Markov Property, Blumenthal's 01 Law Stopping Times, Strong Markov Property First Formulas
 CHAPTER 2. STOCHASTIC INTEGRATION Integrands: Predictable Processes Integrators: Continuous Local Martingales Variance and Covariance Processes Integration w.r.t. Bounded Martingales The KunitaWatanabe Inequality Integration w.r.t. Local Martingales Change of Variables, Ito's Formula Integration w.r.t. Semimartingales Associative Law Functions of Several Semimartingales Chapter Summary MeyerTanaka Formula, Local Time Girsanov's Formula
 CHAPTER 3. BROWNIAN MOTION, II Recurrence and Transience Occupation Times Exit Times Change of Time, Levy's Theorem Burkholder Davis Gundy Inequalities Martingales Adapted to Brownian Filtrations
 CHAPTER 4. PARTIAL DIFFERENTIAL EQUATIONS A. Parabolic Equations The Heat Equation The Inhomogeneous Equation The FeynmanKac Formula B. Elliptic Equations The Dirichlet Problem Poisson's Equation The Schrodinger Equation C. Applications to Brownian Motion Exit Distributions for the Ball Occupation Times for the Ball Laplace Transforms, Arcsine Law
 CHAPTER 5. STOCHASTIC DIFFERENTIAL EQUATIONS Examples Ito's Approach Extension Weak Solutions Change of Measure Change of Time
 CHAPTER 6. ONE DIMENSIONAL DIFFUSIONS Construction Feller's Test Recurrence and Transience Green's Functions Boundary Behavior Applications to Higher Dimensions
 CHAPTER 7. DIFFUSIONS AS MARKOV PROCESSES Semigroups and Generators Examples Transition Probabilities Harris Chains Convergence Theorems
 CHAPTER 8. WEAK CONVERGENCE In Metric Spaces Prokhorov's Theorems The Space C Skorohod's Existence Theorem for SDE Donsker's Theorem The Space D Convergence to Diffusions Examples Solutions to Exercises References Index.
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QA274.2 .D87 1996  Unknown 
2. Probability : theory and examples [2019]
 Durrett, Richard, 1951 author.
 Fifth edition  Cambridge ; New York, NY : Cambridge University Press, 2019
 Description
 Book — 1 online resource
 Summary

 1. Measure theory
 2. Laws of large numbers
 3. Central limit theorems
 4. Martingales
 5. Markov chains
 6. Ergodic theorems
 7. Brownian motion
 8. Applications to random walk
 9. Multidimensional Brownian motion Appendix. Measure theory details.
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MATH230B01, STATS310B01
 Course
 MATH230B01  Theory of Probability
 Instructor(s)
 Dembo, Amir
 Course
 STATS310B01  Theory of Probability II
 Instructor(s)
 Dembo, Amir
3. Essentials of stochastic processes [2016]
 Durrett, Richard, 1951 author.
 Third edition.  Cham, Switzerland : Springer, 2016.
 Description
 Book — 1 online resource (ix, 275 pages) : illustrations. Digital: text file; PDF.
 Summary

 1) Markov Chains1.1 Definitions and Examples1.2 Multistep Transition Probabilities1.3 Classification of States 1.4 Stationary Distributions1.4.1 Doubly stochastic chains1.5 Detailed balance condition1.5.1 Reversibility 1.5.2 The MetropolisHastings algorithm1.5.3 Kolmogorow cycle condition 1.6 Limit Behavior 1.7 Returns to a fixed state 1.8 Proof of the convergence theorem*1.9 Exit Distributions 1.10 Exit Times1.11 Infinite State Spaces* 1.12 Chapter Summary1.13 Exercises 2) Poisson Processes 2.1 Exponential Distribution 2.2 Defining the Poisson Process2.2.1 Constructing the Poisson Process2.2.2 More realistic models2.3 Compound Poisson Processes 2.4 Transformations2.4.1 Thinning 2.4.2 Superposition2.4.3 Conditioning2.5 Chapter Summary2.6 Exercises 3) Renewal Processes3.1 Laws of Large Numbers3.2 Applications to Queueing Theory3.2.1 GI/G/1 queue3.2.2 Cost equations 3.2.3 M/G/1 queue3.3 Age and Residual Life*3.3.1 Discrete case3.3.2 General case 3.4 Chapter Summary 3.5 Exercises 4) Continuous Time Markov Chains 4.1 Definitions and Examples4.2 Computing the Transition Probability4.2.1 Branching Processes 4.3 Limiting Behavior 4.3.1 Detailed balance condition 4.4 Exit Distributions and Exit Times 4.5 Markovian Queues 4.5.1 Single server queues4.5.2 Multiple servers4.5.3 Departure Processes 4.6 Queueing Networks*4.7 Chapter Summary4.8 Exercises 5) Martingales 5.1 Conditional Expectation 5.2 Examples5.3 Gambling Strategies, Stopping Times 5.4 Applications 5.4.1 Exit distributions5.4.2 Exit times 5.4.3 Extinction and ruin probabilities5.4.4 Positive recurrence of the GI/G/1 queue*5.5 Exercises 6) Mathematical Finance6.1 Two Simple Examples6.2 Binomial Model 6.3 Concrete Examples 6.4 American Options6.5 BlackScholes formula6.6 Calls and Puts6.7 Exercises A) Review of Probability A.1 Probabilities, Independence A.2 Random Variables, Distributions A.3 Expected Value, MomentsA.4 Integration to the Limit.
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4. Branching process models of cancer [2015]
 Durrett, Richard, 1951 author.
 Cham : Springer, [2015]
 Description
 Book — 1 online resource (vii, 63 pages) : illustrations (some color).
 Summary

 Multistage Theory of Cancer. Mathematical Overview. Branching Process Results. Time for Z_0 to Reach Size M. Time Until the First Type 1. Mutation Before Detection?. Accumulation of Neutral Mutations. Properties of the Gamma Function. Growth of Z_1(t). Movements of Z_1(t). LuriaDelbruck Distributions. Number of Type 1's at Time T_M. Gwoth of Z_k(t). Transitions Between Waves. Time to the First Type \tau_k, k \ge 2. Application: Metastasis. Application: Ovarian Cancer. Application: Intratumor Heterogeneity.
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5. Essentials of stochastic processes [2012]
 Durrett, Richard, 1951
 2nd ed.  New York : Springer, ©2012.
 Description
 Book — 1 online resource (x, 265 pages) : illustrations (some color). Digital: text file; PDF.
 Summary

 Markov Chains
 Poisson Processes
 Renewal Processes
 Continuous Time Markov Chains
 Martingales
 Mathematical Finance.
Online 6. Probability : theory and examples [2010]
 Durrett, Richard, 1951
 4th ed.  Cambridge ; New York : Cambridge University Press, 2010.
 Description
 Book — x, 428 p. : ill. ; 27 cm.
 Summary

 1. Measure theory
 2. Laws of large numbers
 3. Central limit theorems
 4. Random walks
 5. Martingales
 6. Markov chains
 7. Ergodic theorems
 8. Brownian motion Appendix A. Measure theory details.
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 Also online at
Science Library (Li and Ma)
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QA273 .D865 2010  Unknown 
MATH230B01, STATS310B01
 Course
 MATH230B01  Theory of Probability
 Instructor(s)
 Dembo, Amir
 Course
 STATS310B01  Theory of Probability II
 Instructor(s)
 Dembo, Amir
7. Elementary probability for applications [2009]
 Durrett, Richard, 1951
 New York : Cambridge University Press, 2009.
 Description
 Book — ix, 243 p. : ill. ; 26 cm.
 Summary

 1. Basic concepts
 2. Combinatorial probability
 3. Conditional probability
 4. Markov chains
 5. Continuous distributions
 6. Limit theorems
 7. Option pricing.
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QA273 .D8638 2009  Unknown 
QA273 .D8638 2009  Unknown 
 Durrett, Richard, 1951
 2nd ed.  New York : Springer, ©2008.
 Description
 Book — 1 online resource (xii, 431 pages) : illustrations. Digital: text file; PDF.
 Summary

 Basic models  Estimation and hypothesis testing  Recombination  Population complications  Stepping stone model  Natural selection  Diffusion process  Multidimensional diffusions  Genome rearrangement.
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Given a collection of DNA sequences, what underlying forces are responsible for the observed patterns of variability? This book introduces and analyzes a number of probability models: the WrightFisher model, the coalescent, the infinite alleles model, and the infinite sites model.
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APPPHYS23701
 Course
 APPPHYS23701  Quantitative Evolutionary Dynamics and Genomics
 Instructor(s)
 Good, Benjamin Harmar
 Durrett, Richard, 1951
 2nd ed.  New York : Springer, c2008.
 Description
 Book — xii, 431 p. : ill.
 Summary

 Basic models  Estimation and hypothesis testing  Recombination  Population complications  Stepping stone model  Natural selection  Diffusion process  Multidimensional diffusions  Genome rearrangement.
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Given a collection of DNA sequences, what underlying forces are responsible for the observed patterns of variability? This book introduces and analyzes a number of probability models: the WrightFisher model, the coalescent, the infinite alleles model, and the infinite sites model.
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APPPHYS23701
 Course
 APPPHYS23701  Quantitative Evolutionary Dynamics and Genomics
 Instructor(s)
 Good, Benjamin Harmar
10. Random graph dynamics [2007]
 Durrett, Richard, 1951
 Cambridge ; New York : Cambridge University Press, 2007.
 Description
 Book — ix, 212 p. : ill. ; 27 cm.
 Summary

 1. Overview
 2. ErdosRenyi random graphs
 3. Fixed degree distributions
 4. Power laws
 5. Small worlds
 6. Random walks
 7. CHKNS model.
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QA166.17 .D87 2007  Unknown 
11. Random graph dynamics [2007]
 Durrett, Richard, 1951
 Cambridge ; New York : Cambridge University Press, 2007.
 Description
 Book — 1 online resource (ix, 212 pages) : illustrations
 Summary

 1. Overview
 2. ErdosRenyi random graphs
 3. Fixed degree distributions
 4. Power laws
 5. Small worlds
 6. Random walks
 7. CHKNS model.
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12. Probability : theory and examples [2005]
 Durrett, Richard, 1951
 3rd ed.  Belmont, CA : Thomson Brooks/Cole, c2005.
 Description
 Book — xi, 497 p. ; 25 cm.
 Summary

 INTRODUCTORY LECTURE.
 1. LAWS OF LARGE NUMBERS. Basic Definitions. Random Variables. Expected Value. Independence. Weak Laws of Large Numbers. BorelCantelli Lemmas. Strong Law of Large Numbers. Convergence of Random Series. Large Deviations.
 2. CENTRAL LIMIT THEOREMS. The De MoivreLaplace Theorem. Weak Convergence. Characteristic Functions. Central Limit Theorems. Local Limit Theorems. Poisson Convergence. Stable Laws. Infinitely Divisible Distributions. Limit theorems in Rd.
 3. RANDOM WALKS. Stopping Times. Recurrence. Visits to 0, Arcsine Laws. Renewal Theory.
 4. MARTINGALES. Conditional Expectation. Martingales, Almost Sure Convergence. Examples. Doob's Inequality, LP Convergence. Uniform Integrability, Convergence in L1 / Backwards Martingales. Optional Stopping Theorems.
 5. MARKOV CHAINS. Definitions and Examples. Extensions of the Markov Property. Recurrence and Transience. Stationary Measures. Asymptotic Behavior. General State Space.
 6. ERGODIC THEOREMS. Definitions and Examples. Birkhoff's Ergodic Theorem. Recurrence. Mixing. Entropy. A Subadditive Ergodic Theorem. Applications.
 7. BROWNIAN MOTION. Definition and Construction. Markov Property, Blumenthal's 01 Law. Stopping Times, Strong Markov Property. Maxima and Zeros. Martingales. Donsker's Theorem. CLT's for Dependent Variables. Empirical Distributions, Brownian Bridge. Laws of the Iterated Logarithm. APPENDIX: MEASURE THEORY. REFERENCES. NOTATION. NORMAL TABLE. INDEX.
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QA273 .D865 2005  Unknown 
 Durrett, Richard, 1951
 Providence, RI : American Mathematical Society, 2002.
 Description
 Book — viii, 118 p. : ill. ; 26 cm.
 Summary

 Introduction Perturbations of onedimensional systems Twospecies examples Lower bounding lemmas for PDE Perturbations of higherdimensional systems Lyapunov functions for twospecies Lotka Volterra systems Three species linear competition models Three species predatorprey systems Some asymptotic results for our ODE and PDE A list of the invadability conditions References.
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Shelved by Series title NO.740  Unknown 
 Durrett, Richard, 1951
 New York : Springer, c2002.
 Description
 Book — viii, 240 p. : ill. ; 25 cm.
 Summary

 Basic Models. Neutral Complications. Natural Selection. Statistical Tests. Genome Rearrangement.
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 Online
SAL3 (offcampus storage)
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QH438.4 .S73 D87 2002  Available 
15. Essentials of stochastic processes [1999]
 Durrett, Richard, 1951
 New York : Springer, c1999.
 Description
 Book — vi, 281 p. : ill. ; 25 cm.
 Summary

 1. Markov Chains
 2. Martingales
 3. Poisson Processes
 4. Markov Chains
 5. Renewal Theory
 6. Brownian Motion.
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 Online
Engineering Library (Terman), Science Library (Li and Ma)
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QA274 .D87 1999  Unavailable Checked out  Overdue 
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QA274 .D87 1999  Unknown 
16. Probability : theory and examples [1996]
 Durrett, Richard, 1951
 2nd ed.  Belmont, Calif. : Duxbury Press, c1996.
 Description
 Book — xiii, 503 p. : ill. ; 25 cm.
 Online
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QA273 .D865 1996  Unavailable Checked out  Overdue 
17. The essentials of probability [1994]
 Durrett, Richard, 1951
 Belmont, Calif. : Duxbury Press, c1994.
 Description
 Book — vi, 269 p. ; 25 cm.
 Summary

 Coins, dice and cards Conditional probability Distributions Expected value Limit theorems. Answers to selected exercises. Appendix: Formulas for important distributions.
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QA273 .D864 1994  Unknown 
18. Probability : theory and examples [1991]
 Durrett, Richard, 1951
 Pacific Grove, Calif. : Wadsworth & Brooks/Cole Advanced Books & Software, c1991.
 Description
 Book — ix, 453 p. : ill. ; 25 cm.
 Summary

 Introductory lecture. Laws of large numbers. Central limit theorems. Random walks. Martingales. Markov chains. Ergodic theorems. Brownian motion. The essentials of measure theory.
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 Online
SAL3 (offcampus storage), Science Library (Li and Ma)
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QA273 .D865 1991  Available 
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QA273 .D865 1991  Unknown 
19. Brownian motion and martingales in analysis [1984]
 Durrett, Richard, 1951
 Belmont, Calif. : Wadsworth Advanced Books & Software, c1984.
 Description
 Book — xi, 328 p. : ill. ; 25 cm.
 Summary

 Brownian motion. Stochastic integration. Conditioned Brownian motions. Boundary limits of harmonic functions. Complex Brownian motion and analytic functions. Hardy spaces and related spaces of martingales. H1 and BMO, m1 and BMO. PDE's which can be solved by running a Brownian motion. Stochastic differential equations.
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QA274.75 .D87 1984  Unknown 
 Durrett, Richard, 1951
 1976.
 Description
 Book — 79 leaves.
 Online
SAL3 (offcampus storage), Special Collections
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3781 1976 D  Available 
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3781 1976 D  Inlibrary use 
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