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Stochastic Processes
From Applications to Theory
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Book Description
Unlike traditional books presenting stochastic processes in an academic way, this book includes concrete applications that students will find interesting such as gambling, finance, physics, signal processing, statistics, fractals, and biology. Written with an important illustrated guide in the beginning, it contains many illustrations, photos and pictures, along with several website links. Computational tools such as simulation and Monte Carlo methods are included as well as complete toolboxes for both traditional and new computational techniques.
Table of Contents
An illustrated guide
Motivating examples
Lost in the Great Sloan Wall
Meeting Alice in Wonderland
The lucky MIT Blackjack team
The Kruskal's magic trap card
The magic fern from Daisetsuzan
The Kepler22b Eve
Poisson's typos
Exercises
Selected topics
Stabilizing populations
The traps of Reinforcement
Casino roulette
Surfing Google's waves
Pinging hackers
Exercises
Computational & theoretical aspects
From Monte Carlo to Los Alamos
Signal processing & Population dynamics
The lost equation
Towards a general theory
The theory of speculation
Exercises
Stochastic simulation
Simulation toolbox
Inversion technique
Change of variables
Rejection techniques
Sampling probabilities
Bayesian inference
Laplace's rule of successions
Fragmentation and coagulation
Conditional probabilities
Bayes' formula
The regression formula
Gaussian updates
Conjugate priors
Spatial Poisson point processes
Some preliminary results
Conditioning principles
PoissonGaussian clusters
Exercises
Monte Carlo integration
Law of large numbers
Importance sampling
Twisted distributions
Sequential Monte Carlo
Tails distributions
Exercises
Some illustrations
Stochastic processes
Markov chain models
Blackbox type models
BoltzmannGibbs measures
The Ising model
The SherringtonKirkpatrick model
The traveling salesman model
Filtering & Statistical learning
The Bayes formula
The Singer's radar model
Exercises
Discrete time processes
Markov chains
Description of the models
Elementary transitions
Markov integral operators
Equilibrium measures
Stochastic matrices
Random dynamical systems
Linear Markov chain model
Two states Markov models
Transition diagrams
The tree of outcomes
General state space models
Nonlinear Markov chains
Selfinteracting processes
Mean field particle models
McKeanVlasov diffusions
Interacting jump processes
Exercises
Analysis toolbox
Linear algebra
Diagonalisation type techniques
The Perron Frobenius theorem
Functional analysis
Spectral decompositions
Total variation norms
Contraction inequalities
The Poisson equation
Vnorms
Geometric drift conditions
V norm contractions
Stochastic analysis
Coupling techniques
The total variation distance
The Wasserstein metric
Stopping times and coupling
Strong stationary times
Some illustrations
Minorization condition and coupling
Markov chains on complete graphs
Kruskal random walk
Martingales
Some preliminaries
Applications to Markov chains
Martingales with Fixed terminal values
A DoeblinIto formula
Occupation measures
Optional stopping theorems
A gambling model
Fair games
Unfair games
Maximal inequalities
Limit theorems
Topological aspects
Irreducibility and aperiodicity
Recurrent and transient states
Continuous state spaces
Path space models
Exercises
Computational toolbox
A weak ergodic theorem
Some illustrations
Parameter estimation
A Gaussian subset shaker
Exploration of the unit disk
Markov Chain Monte Carlo methods
Introduction
Metropolis and Hastings models
GibbsGlauber dynamics
The Propp and Wilson sampler
Time inhomogeneous MCMC models
Simulated annealing algorithm
A perfect sampling algorithm
FeynmanKac path integration
Weighted Markov chains
Evolution equations
Particle absorption models
Doob hprocesses
Quasiinvariant measures
Cauchy problems with terminal conditions
DirichletPoisson problems
CauchyDirichletPoisson problems
FeynmanKac particle methodology
Mean field genetic type particle models
Path space models
Backward integration
A random particle matrix model
A conditional formula for ancestral trees
Particle Markov Chain Monte Carlo methods
Manybody FeynmanKac measures
A particle MetropolisHastings model
Duality formulae for manybody models
A couple of particle Gibbs samplers
Quenched and annealed measures
FeynmanKac models
Particle Gibbs models
Particle MetropolisHastings models
Some application domains
Interacting MCMC algorithms
Nonlinear Filtering models
Markov chain restrictions
Selfavoiding walks
Importance twisted measures
KalmanBucy Filters
Forward Filters
Backward Filters
Ensemble Kalman Filters
Interacting Kalman Filters
Exercises
Continuous time processes
Poisson processes
A counting process
Memoryless property
Uniform random times
The DoeblinIto formula
The Bernoulli process
Time inhomogeneous models
Description of the models
Poisson thinning simulation
Geometric random clocks
Exercises
Markov chain embeddings
Homogeneous embeddings
Description of the models
Semigroup evolution equations
Some illustrations
A two states Markov process
Matrix valued equations
Discrete Laplacian
Spatially inhomogeneous models
Explosion phenomenon
Finite state space models
Time in homogenous models
Description of the models
Poisson thinning models
Exponential and geometric clocks
Exercises
Jump processes
A class of pure jump models
Semigroup evolution equations
Approximation schemes
Sum of generators
DoobMeyer decompositions
Discrete time models
Continuous time martingales
Optional stopping theorems
DoeblinItoTaylor formulae
Stability properties
Invariant measures
Dobrushin contraction properties
Exercises
Piecewise deterministic processes
Dynamical systems basics
Semigroup and flow maps
Time discretization schemes
Piecewise deterministic jump models
Excursion valued Markov chains
Evolution semigroups
Infinitesimal generators
The FokkerPlanck equation
A time discretization scheme
DoeblinItoTaylor formulae
Stability properties
Switching processes
Invariant measures
An application to Internet architectures
The Transmission Control Protocol
Regularity and stability properties
The limiting distribution
Exercises
Diffusion processes
Brownian motion
Discrete vs continuous time models
Evolution semigroups
The heat equation
A DoeblinItoTaylor formula
Stochastic differential equations
Diffusion processes
The DoeblinIto differential calculus
Evolution equations
The FokkerPlanck equation
Weak approximation processes
A backward stochastic differential equation
Multidimensional diffusions
Multidimensional stochastic differential equations
An integration by parts formula
Laplacian and Orthogonal transformations
The FokkerPlanck equation
Exercises
Jump diffusion processes
Piecewise diffusion processes
Evolution semigroups
The DoeblinIto formula
The FokkerPlanck equation
An abstract class of stochastic processes
Generators and carré du champ operators
Perturbation formulae
Jumpdiffusion processes with killing
FeynmanKac semigroups
Cauchy problems with terminal conditions
DirichletPoisson problems
CauchyDirichletPoisson problems
Some illustrations
1dimensional DirichletPoisson problems
A backward stochastic differential equation
Exercises
Nonlinear jump diffusion processes
Nonlinear Markov processes
Pure diffusion models
The Burgers equation
FeynmanKac jump type models
A jump type Langevin model
Mean field particle models
Some application domains
FouqueSun systemic risk model
Burgers equation
A LangevinMcKeanVlasov model
The Dyson equation
Exercises
Stochastic analysis toolbox
Time changes
Stability properties
Some illustrations
Gradient flow processes
1dimensional diffusions
FosterLyapunov techniques
Contraction inequalities
Minorization properties
Some applications
OrnsteinUhlenbeck processes
Stochastic gradient processes
Langevin diffusions
Spectral analysis
Hilbert spaces and Schauder bases
Spectral decompositions
Poincaré inequality
Exercises
Path space measures
Pure jump models
Likelihood functionals
Girsanov's transformations
Exponential martingales
Diffusion models
The Wiener measure
Path space diffusions
Girsanov transformations
Exponential change twisted measures
Diffusion processes
Pure jump processes
Some illustrations
Risk neutral Financial markets
Poisson markets
Diffusion markets
Elliptic diffusions
Nonlinear filtering
Diffusion observations
DuncanZakai equation
KushnerStratonovitch equation
KalmanBucy Filters
Nonlinear diffusion and Ensemble KalmanBucy Filters
Robust Filtering equations
Poisson observations
Exercises
Processes on manifolds
A review of differential geometry
Projection operators
Covariant derivatives of vector fields
First order derivatives
Second order derivatives
Divergence and mean curvature
Lie brackets and commutation formulae
Inner product derivation formulae
Second order derivatives and some trace formulae
The Laplacian operator
Ricci curvature
BochnerLichnerowicz formula
Exercises
Stochastic differential calculus on manifolds
Embedded manifolds
Brownian motion on manifolds
A diffusion model in the ambient space
The infinitesimal generator
Monte Carlo simulation
Stratonovitch differential calculus
Projected diffusions on manifolds
Brownian motion on orbifolds
Exercises
Parameterizations and charts
Differentiable manifolds and charts
Orthogonal projection operators
Riemannian structures
First order covariant derivatives
Pushed forward functions
Pushed forward vector fields
Directional derivatives
Second order covariant derivative
Tangent basis functions
Composition formulae
Hessian operators
BochnerLichnerowicz formula
Exercises
Stochastic calculus in chart spaces
Brownian motion on Riemannian manifolds
Diffusions on chart spaces
Brownian motion on spheres
The unit circle S = S1 _ R2
The unit sphere S = S2 _ R3
Brownian motion on the Torus
Diffusions on the simplex
Exercises
Some analytical aspects
Geodesics and the exponential map
A Taylor expansion
Integration on manifolds
The volume measure on the manifold
Wedge product and volume forms
The divergence theorem
Gradient flow models
Steepest descent model
Euclidian state spaces
Drift changes and irreversible Langevin diffusions
Langevin diffusions on closed manifolds
Riemannian Langevin diffusions
Metropolisadjusted Langevin models
Stability and some functional inequalities
Exercises
Some illustrations
Prototype manifolds
The Circle
The 2Sphere
The Torus
Information theory
Nash embedding theorem
Distribution manifolds
Bayesian statistical manifolds
The CramerRao lower bound
Some illustrations
BoltzmannGibbs measures
Multivariate normal distributions
Some application areas
Simple random walks
Random walk on lattices
Description
Dimension 1
Dimension 2
Dimension d > 3
Random walks on graphs
Simple exclusion process
Random walks on the circle
Markov chain on cycles
Markov chain on the circle
Spectral decomposition
Random walk on hypercubes
Description
A macroscopic model
A lazy random walk
Urn processes
Ehrenfest model
Pólya urn model
Exercises
Iterated random functions
Description
A motivating example
Uniform selection
An ancestral type evolution model
An absorbed Markov chain
Shuffling cards
Introduction
The topinat random shuffle
The random transposition shuffle
The riffle shuffle
Fractal models
Exploration of Cantor's discontinuum
Some fractal images
Exercises
Computational & Statistical physics
Molecular dynamics simulation
Newton's second law of motion
Langevin diffusion processes
The Schrödinger equation
A physical derivation
A FeynmanKac formulation
Brakets and path integral formalism
Spectral decompositions
The harmonic oscillator
Diffusion Monte Carlo models
Interacting particle systems
Introduction
Contact process
Voter process
Exclusion process
Exercises
Dynamic population models
Discrete time birth and death models
Continuous time models
Birth and death generators
Logistic processes
Epidemic model with immunity
LotkaVolterra predatorprey stochastic model
The Moran genetic model
Genetic evolution models
Branching processes
Birth and death models with linear rates
Discrete time branching
Continuous time branching processes
Absorption  death process
Birth type branching process
Birth and death branching processes
KolmogorovPetrovskiiPiskunov equations
Exercises
Gambling, ranking and control
The Google page rank
Gambling betting systems
Martingale systems
St. Petersburg martingales
Conditional gains and losses
Conditional gains
Conditional losses
Bankroll managements
The Grand Martingale
The D'Alembert Martingale
The Whittacker Martingale
Stochastic optimal control
Bellman equations
Control dependent value functions
Continuous time models
Optimal stopping
Games with Fixed terminal condition
Snell envelope
Continuous time models
Exercises
Mathematical finance
Stock price models
Up and down martingales
CoxRossRubinstein model
BlackScholesMerton model
European option pricing
Call and Put options
Selffinancing portfolios
Binomial pricing technique
BlackScholesMerton pricing model
The BlackScholes partial differential equation
Replicating portfolios
Option price and hedging computations
A numerical illustration
Exercises
Bibliography
Index
Author(s)
Biography
Pierre Del Moral and Spiridon Penev are professors in the School of Mathematics and Statistics at the University of New South Wales.
Reviews
"The title itself suggests that the reader should expect something different, applications to theory and not theory to applications. The title is correct, and that is the main theme of the book. Start with some general applications, and then build the theory around them. The range of applications and the depth of the discussions are impressive." (Igor Cialenco, Illinois Institute of Technology)
"This is a great reference… It lays out a lot of calculations in simple and direct ways. If you go through this book as a firstyear grad student, you will understand lots of material and be prepared for many things." (Richard Sowers, University of Illinois at UrbanaChampaign)
"(This book makes) theoretical tools developed in the stochastic analysis/probability community available to a significant community of applied mathematicians. As such, it should be highly successful, as it is well written and clear." (John Fricks, The Pennsylvania State University)
Support Material
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