Pardoux, É.
Markov Processes and Applications : Algorithms, Networks, Genome and Finance. - 1st ed. - 1 online resource (323 pages) - Wiley Series in Probability and Statistics Ser. ; v.796 . - Wiley Series in Probability and Statistics Ser. .
Intro -- Markov Processes and Applications -- Contents -- Preface -- 1 Simulations and the Monte Carlo method -- 1.1 Description of the method -- 1.2 Convergence theorems -- 1.3 Simulation of random variables -- 1.4 Variance reduction techniques -- 1.5 Exercises -- 2 Markov chains -- 2.1 Definitions and elementary properties -- 2.2 Examples -- 2.2.1 Random walk in E = Zd -- 2.2.2 Bienaymé-Galton-Watson process -- 2.2.3 A discrete time queue -- 2.3 Strong Markov property -- 2.4 Recurrent and transient states -- 2.5 The irreducible and recurrent case -- 2.6 The aperiodic case -- 2.7 Reversible Markov chain -- 2.8 Rate of convergence to equilibrium -- 2.8.1 The reversible finite state case -- 2.8.2 The general case -- 2.9 Statistics of Markov chains -- 2.10 Exercises -- 3 Stochastic algorithms -- 3.1 Markov chain Monte Carlo -- 3.1.1 An application -- 3.1.2 The Ising model -- 3.1.3 Bayesian analysis of images -- 3.1.4 Heated chains -- 3.2 Simulation of the invariant probability -- 3.2.1 Perfect simulation -- 3.2.2 Coupling from the past -- 3.3 Rate of convergence towards the invariant probability -- 3.4 Simulated annealing -- 3.5 Exercises -- 4 Markov chains and the genome -- 4.1 Reading DNA -- 4.1.1 CpG islands -- 4.1.2 Detection of the genes in a prokaryotic genome -- 4.2 The i.i.d. model -- 4.3 The Markov model -- 4.3.1 Application to CpG islands -- 4.3.2 Search for genes in a prokaryotic genome -- 4.3.3 Statistics of Markov chains Mk -- 4.3.4 Phased Markov chains -- 4.3.5 Locally homogeneous Markov chains -- 4.4 Hidden Markov models -- 4.4.1 Computation of the likelihood -- 4.4.2 The Viterbi algorithm -- 4.4.3 Parameter estimation -- 4.5 Hidden semi-Markov model -- 4.5.1 Limitations of the hidden Markov model -- 4.5.2 What is a semi-Markov chain? -- 4.5.3 The hidden semi-Markov model -- 4.5.4 The semi-Markov Viterbi algorithm. 4.5.5 Search for genes in a prokaryotic genome -- 4.6 Alignment of two sequences -- 4.6.1 The Needleman-Wunsch algorithm -- 4.6.2 Hidden Markov model alignment algorithm -- 4.6.3 A posteriori probability distribution of the alignment -- 4.6.4 A posteriori probability of a given match -- 4.7 A multiple alignment algorithm -- 4.8 Exercises -- 5 Control and filtering of Markov chains -- 5.1 Deterministic optimal control -- 5.2 Control of Markov chains -- 5.3 Linear quadratic optimal control -- 5.4 Filtering of Markov chains -- 5.5 The Kalman-Bucy filter -- 5.5.1 Motivation -- 5.5.2 Solution of the filtering problem -- 5.6 Linear-quadratic control with partial observation -- 5.7 Exercises -- 6 The Poisson process -- 6.1 Point processes and counting processes -- 6.2 The Poisson process -- 6.3 The Markov property -- 6.4 Large time behaviour -- 6.5 Exercises -- 7 Jump Markov processes -- 7.1 General facts -- 7.2 Infinitesimal generator -- 7.3 The strong Markov property -- 7.4 Embedded Markov chain -- 7.5 Recurrent and transient states -- 7.6 The irreducible recurrent case -- 7.7 Reversibility -- 7.8 Markov models of evolution and phylogeny -- 7.8.1 Models of evolution -- 7.8.2 Likelihood methods in phylogeny -- 7.8.3 The Bayesian approach to phylogeny -- 7.9 Application to discretized partial differential equations -- 7.10 Simulated annealing -- 7.11 Exercises -- 8 Queues and networks -- 8.1 M/M/1 queue -- 8.2 M/M/1/K queue -- 8.3 M/M/s queue -- 8.4 M/M/s/s queue -- 8.5 Repair shop -- 8.6 Queues in series -- 8.7 M/G/∞ queue -- 8.8 M/G/1 queue -- 8.8.1 An embedded chain -- 8.8.2 The positive recurrent case -- 8.9 Open Jackson network -- 8.10 Closed Jackson network -- 8.11 Telephone network -- 8.12 Kelly networks -- 8.12.1 Single queue -- 8.12.2 Multi-class network -- 8.13 Exercises -- 9 Introduction to mathematical finance -- 9.1 Fundamental concepts. 9.1.1 Option -- 9.1.2 Arbitrage -- 9.1.3 Viable and complete markets -- 9.2 European options in the discrete model -- 9.2.1 The model -- 9.2.2 Admissible strategy -- 9.2.3 Martingales -- 9.2.4 Viable and complete market -- 9.2.5 Call and put pricing -- 9.2.6 The Black-Scholes formula -- 9.3 The Black-Scholes model and formula -- 9.3.1 Introduction to stochastic calculus -- 9.3.2 Stochastic differential equations -- 9.3.3 The Feynman-Kac formula -- 9.3.4 The Black-Scholes partial differential equation -- 9.3.5 The Black-Scholes formula (2) -- 9.3.6 Generalization of the Black-Scholes model -- 9.3.7 The Black-Scholes formula (3) -- 9.3.8 Girsanov's theorem -- 9.3.9 Markov property and partial differential equation -- 9.3.10 Contingent claim on several underlying stocks -- 9.3.11 Viability and completeness -- 9.3.12 Remarks on effective computation -- 9.3.13 Historical and implicit volatility -- 9.4 American options in the discrete model -- 9.4.1 Snell envelope -- 9.4.2 Doob's decomposition -- 9.4.3 Snell envelope and Markov chain -- 9.4.4 Back to American options -- 9.4.5 American and European options -- 9.4.6 American options and Markov model -- 9.5 American options in the Black-Scholes model -- 9.6 Interest rate and bonds -- 9.6.1 Future interest rate -- 9.6.2 Future interest rate and bonds -- 9.6.3 Option based on a bond -- 9.6.4 An interest rate model -- 9.7 Exercises -- 10 Solutions to selected exercises -- 10.1 Chapter 1 -- 10.2 Chapter 2 -- 10.3 Chapter 3 -- 10.4 Chapter 4 -- 10.5 Chapter 5 -- 10.6 Chapter 6 -- 10.7 Chapter 7 -- 10.8 Chapter 8 -- 10.9 Chapter 9 -- Reference -- Index.
"Well-written, this book is suitable as a textbook for teaching a postgraduate course on applied Markov processes." (Mathmatical Assoc of America, June 2009) "It does provide a good introduction to each of the five application areas." (Mathematical Reviews, July 2010).
9780470721865
Markov processes.
Electronic books.
QA274.7.P375 2008
519.2/33
Markov Processes and Applications : Algorithms, Networks, Genome and Finance. - 1st ed. - 1 online resource (323 pages) - Wiley Series in Probability and Statistics Ser. ; v.796 . - Wiley Series in Probability and Statistics Ser. .
Intro -- Markov Processes and Applications -- Contents -- Preface -- 1 Simulations and the Monte Carlo method -- 1.1 Description of the method -- 1.2 Convergence theorems -- 1.3 Simulation of random variables -- 1.4 Variance reduction techniques -- 1.5 Exercises -- 2 Markov chains -- 2.1 Definitions and elementary properties -- 2.2 Examples -- 2.2.1 Random walk in E = Zd -- 2.2.2 Bienaymé-Galton-Watson process -- 2.2.3 A discrete time queue -- 2.3 Strong Markov property -- 2.4 Recurrent and transient states -- 2.5 The irreducible and recurrent case -- 2.6 The aperiodic case -- 2.7 Reversible Markov chain -- 2.8 Rate of convergence to equilibrium -- 2.8.1 The reversible finite state case -- 2.8.2 The general case -- 2.9 Statistics of Markov chains -- 2.10 Exercises -- 3 Stochastic algorithms -- 3.1 Markov chain Monte Carlo -- 3.1.1 An application -- 3.1.2 The Ising model -- 3.1.3 Bayesian analysis of images -- 3.1.4 Heated chains -- 3.2 Simulation of the invariant probability -- 3.2.1 Perfect simulation -- 3.2.2 Coupling from the past -- 3.3 Rate of convergence towards the invariant probability -- 3.4 Simulated annealing -- 3.5 Exercises -- 4 Markov chains and the genome -- 4.1 Reading DNA -- 4.1.1 CpG islands -- 4.1.2 Detection of the genes in a prokaryotic genome -- 4.2 The i.i.d. model -- 4.3 The Markov model -- 4.3.1 Application to CpG islands -- 4.3.2 Search for genes in a prokaryotic genome -- 4.3.3 Statistics of Markov chains Mk -- 4.3.4 Phased Markov chains -- 4.3.5 Locally homogeneous Markov chains -- 4.4 Hidden Markov models -- 4.4.1 Computation of the likelihood -- 4.4.2 The Viterbi algorithm -- 4.4.3 Parameter estimation -- 4.5 Hidden semi-Markov model -- 4.5.1 Limitations of the hidden Markov model -- 4.5.2 What is a semi-Markov chain? -- 4.5.3 The hidden semi-Markov model -- 4.5.4 The semi-Markov Viterbi algorithm. 4.5.5 Search for genes in a prokaryotic genome -- 4.6 Alignment of two sequences -- 4.6.1 The Needleman-Wunsch algorithm -- 4.6.2 Hidden Markov model alignment algorithm -- 4.6.3 A posteriori probability distribution of the alignment -- 4.6.4 A posteriori probability of a given match -- 4.7 A multiple alignment algorithm -- 4.8 Exercises -- 5 Control and filtering of Markov chains -- 5.1 Deterministic optimal control -- 5.2 Control of Markov chains -- 5.3 Linear quadratic optimal control -- 5.4 Filtering of Markov chains -- 5.5 The Kalman-Bucy filter -- 5.5.1 Motivation -- 5.5.2 Solution of the filtering problem -- 5.6 Linear-quadratic control with partial observation -- 5.7 Exercises -- 6 The Poisson process -- 6.1 Point processes and counting processes -- 6.2 The Poisson process -- 6.3 The Markov property -- 6.4 Large time behaviour -- 6.5 Exercises -- 7 Jump Markov processes -- 7.1 General facts -- 7.2 Infinitesimal generator -- 7.3 The strong Markov property -- 7.4 Embedded Markov chain -- 7.5 Recurrent and transient states -- 7.6 The irreducible recurrent case -- 7.7 Reversibility -- 7.8 Markov models of evolution and phylogeny -- 7.8.1 Models of evolution -- 7.8.2 Likelihood methods in phylogeny -- 7.8.3 The Bayesian approach to phylogeny -- 7.9 Application to discretized partial differential equations -- 7.10 Simulated annealing -- 7.11 Exercises -- 8 Queues and networks -- 8.1 M/M/1 queue -- 8.2 M/M/1/K queue -- 8.3 M/M/s queue -- 8.4 M/M/s/s queue -- 8.5 Repair shop -- 8.6 Queues in series -- 8.7 M/G/∞ queue -- 8.8 M/G/1 queue -- 8.8.1 An embedded chain -- 8.8.2 The positive recurrent case -- 8.9 Open Jackson network -- 8.10 Closed Jackson network -- 8.11 Telephone network -- 8.12 Kelly networks -- 8.12.1 Single queue -- 8.12.2 Multi-class network -- 8.13 Exercises -- 9 Introduction to mathematical finance -- 9.1 Fundamental concepts. 9.1.1 Option -- 9.1.2 Arbitrage -- 9.1.3 Viable and complete markets -- 9.2 European options in the discrete model -- 9.2.1 The model -- 9.2.2 Admissible strategy -- 9.2.3 Martingales -- 9.2.4 Viable and complete market -- 9.2.5 Call and put pricing -- 9.2.6 The Black-Scholes formula -- 9.3 The Black-Scholes model and formula -- 9.3.1 Introduction to stochastic calculus -- 9.3.2 Stochastic differential equations -- 9.3.3 The Feynman-Kac formula -- 9.3.4 The Black-Scholes partial differential equation -- 9.3.5 The Black-Scholes formula (2) -- 9.3.6 Generalization of the Black-Scholes model -- 9.3.7 The Black-Scholes formula (3) -- 9.3.8 Girsanov's theorem -- 9.3.9 Markov property and partial differential equation -- 9.3.10 Contingent claim on several underlying stocks -- 9.3.11 Viability and completeness -- 9.3.12 Remarks on effective computation -- 9.3.13 Historical and implicit volatility -- 9.4 American options in the discrete model -- 9.4.1 Snell envelope -- 9.4.2 Doob's decomposition -- 9.4.3 Snell envelope and Markov chain -- 9.4.4 Back to American options -- 9.4.5 American and European options -- 9.4.6 American options and Markov model -- 9.5 American options in the Black-Scholes model -- 9.6 Interest rate and bonds -- 9.6.1 Future interest rate -- 9.6.2 Future interest rate and bonds -- 9.6.3 Option based on a bond -- 9.6.4 An interest rate model -- 9.7 Exercises -- 10 Solutions to selected exercises -- 10.1 Chapter 1 -- 10.2 Chapter 2 -- 10.3 Chapter 3 -- 10.4 Chapter 4 -- 10.5 Chapter 5 -- 10.6 Chapter 6 -- 10.7 Chapter 7 -- 10.8 Chapter 8 -- 10.9 Chapter 9 -- Reference -- Index.
"Well-written, this book is suitable as a textbook for teaching a postgraduate course on applied Markov processes." (Mathmatical Assoc of America, June 2009) "It does provide a good introduction to each of the five application areas." (Mathematical Reviews, July 2010).
9780470721865
Markov processes.
Electronic books.
QA274.7.P375 2008
519.2/33