Jean–François Le Gall, Professor at Université de Paris–Orsay, France.
Markov processes is the class of stochastic processes whose past and future are conditionally independent, given their present state. They constitute important models in many applied fields. After an introduction to the Monte Carlo method, this book describes discrete time Markov chains, the Poisson process and continuous time Markov chains. It also presents numerous applications including Markov Chain Monte Carlo, Simulated Annealing, Hidden Markov Models, Annotation and Alignment of Genomic sequences, Control and Filtering, Phylogenetic tree reconstruction and Queuing networks. The last chapter is an introduction to stochastic calculus and mathematical finance.
- The Monte Carlo method, discrete time Markov chains, the Poisson process and continuous time jump Markov processes.
- An introduction to diffusion processes, mathematical finance and stochastic calculus.
- Applications of Markov processes to various fields, ranging from mathematical biology, to financial engineering and computer science.
- Numerous exercises and problems with solutions to most of them.
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.
2. Markov chains.
2.1 Definitions and elementary properties.
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.
2.9 Statistics of Markov chains.
3. Stochastic algorithms.
3.1 Markov chain Monte Carlo.
3.2 Simulation of the invariant probability.
3.3 Rate of convergence towards the invariant probability.
3.4 Simulated annealing.
4. Markov chains and the genome.
4.1 Reading DNA.
4.2 The i.i.d. model.
4.3 The Markov model.
4.4 Hidden Markov models.
4.5 Hidden semi–Markov model.
4.6 Alignment of two sequences.
4.7 A multiple alignment algorithm.
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.
5.6 Linear–quadratic control with partial observation.
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.
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.
7.8 Markov models of evolution and phylogeny.
7.9 Application to discretized partial differential equations.
7.10 Simulated annealing.
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.
8.9 Open Jackson network.
8.10 Closed Jackson network.
8.11 Telephone network.
8.12 Kelly networks.
9. Introduction to mathematical finance.
9.1 Fundamental concepts.
9.2 European options in the discrete model.
9.3 The Black–Scholes model and formula.
9.4 American options in the discrete model.
9.5 American options in the Black–Scholes model.
9.6 Interest rate and bonds.
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.