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作者 Graham, C. (Carl), author
書名 Stochastic simulation and Monte Carlo methods : mathematical foundations of stochastic simulation / Carl Graham, Denis Talay
出版項 Heidelberg : Springer, [2013]
國際標準書號 3642393624
9783642393624
book jacket
館藏地 索書號 處理狀態 OPAC 訊息 條碼
 數學所圖書室  QA274 G725 2013    在架上    30340200531956
說明 xvi, 260 pages : illustrations ; 24 cm
text rdacontent
unmediated rdamedia
volume rdacarrier
系列 Stochastic modelling and applied probability, 0172-4568 ; 68
Stochastic modelling and applied probability ; 68
附註 Includes bibliographical references (pages 253-255) and index
Also issued online
Principles of Monte Carlo methods -- Introduction -- Why use probabilistic models and simulations? -- What are the reasons for probabilistic models? -- What are the objectives of random simulations? -- Organization of the monograph -- Strong law of large numbers and Monte Carlo methods -- Strong law of large numbers, examples of Monte Carlo methods -- Strong law of large numbers, almost sure convergence -- Buffon's needle -- Neutron transport simulations -- Stochastic numerical methods for partial differential equations -- Simulation algorithms for simple probability distributions -- Uniform distributions -- Discrete distributions -- Gaussian distributions -- Cumulative distribution function inversion, exponential distributions -- Rejection method -- Discrete-time martingales, proof of the SLLN -- Reminders on conditional expectation -- Martingales and sub-martingales, backward martingales -- Proof of the strong law of large numbers -- Problems -- Non-asymptotic error estimates for Monte Carlo methods -- Convergence in law and characteristic functions -- Central limit theorem -- Asymptotic confidence intervals -- Berry-Esseen's theorem -- Bikelis' theorem -- Absolute confidence intervals -- Concentration inequalities -- Logarithmic Sobolev inequalities -- Concentration inequalities, absolute confidence intervals -- Elementary variance reduction techniques -- Control variate -- Importance sampling -- Problems -- Exact and approximate simulation of Markov processes -- Poisson processes as particular Markov processes -- Quick introduction to markov processes -- Some issues in Markovian modeling -- Rudiments on processes, sample paths, and laws -- Poisson processes: characterization, properties -- Point processes and poisson processes -- Simple and strong markov property -- Superposition and decomposition -- Simulation and approximation -- Simulation of inter-arrivals -- Simulation of independent poisson processes -- Long time or large intensity limit, applications -- Problems -- Discrete-space markov processes -- Characterization, specification, properties -- Measures, functions, and transition matrices -- Simple and strong Markov property -- Semigroup, infinitesimal generator, and evolution law -- Constructions, existence, simulation, equations -- Fundamental constructions -- Explosion or existence for a Markov process -- Fundamental simulation, fictitious jump method -- Kolmogorov equations, Feynman-Kac formula -- Generators and semigroups in bounded operator algebras -- A few case studies -- Problems -- Continuous-space Markov processes with jumps -- Preliminaries -- Measures, functions, and transition kernels -- Markov property, finite-dimensional marginals -- Semigroup, infinitesimal generator -- Markov processes evolving only by isolated jumps -- Semigroup, infinitesimal generator, and evolution law -- Construction, simulation, existence -- Kolmogorov equations, Feynman-Kac formula, bounded generator case -- Markov processes following an ordinary differential equation between jumps: PDMP -- Sample paths, evolution, integro-differential generator -- Construction, simulation, existence -- Kolmogorov equations, Feynman-Kac formula -- Application to kinetic equations -- Further extensions -- Problems -- Discretization of stochastic differential equations -- Reminders on Itô's stochastic calculus -- Stochastic integrals and Itô processes -- Itô's formula, existence and uniqueness of solutions of stochastic differential equations -- Markov properties, martingale problems and Fokker- Planck equations -- Euler and Milstein schemes -- Moments of the solution and of its approximations -- Convergence rates in Lp (... ) norm and almost surely -- Monte Carlo methods for parabolic partial differential equations -- The principle of the method -- Introduction of the error analysis -- Optimal convergence rate : the Talay-Tubaro expansion -- Romberg-Richardson extrapolation methods -- Probabilistic interpretation and estimates for parabolic partial differential equations -- Problems -- Variance reduction, Girsanov's theorem, and stochastic algorithms -- Variance reduction and stochastic differential equations -- Preliminary reminders on the Girsanov theorem -- Control variates method -- Variance reduction for sensitivity analysis -- Differentiable terminal conditions -- Non-differentiable terminal conditions -- Importance sampling method -- Statistical romberg method -- Problems -- Stochastic algorithms -- Introduction -- Study in an idealized framework -- Definitions -- The ordinary differential equation method, martingale increments -- Long-time behavior of the algorithm -- Variance reduction for Monte Carlo methods -- Searching for an importance sampling -- Variance reduction and stochastic algorithms -- Problems -- Appendix solutions to selected problems -- References -- Index
主題 Stochastic processes
Monte Carlo method
Alt Author Talay, D. (Denis), author
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