Numerical methods for stochastic processes / Nicolas Bouleau, Dominique Lépingle

Bibliogr. p. 337-351. Index

In the preface the authors write that the main motivation for the book is to give greater rigor to numerical treatments of stochastic models. After the short preliminary first chapter, in Chapter 2 they consider problems of the computation of mathematical expectations. Along with the well-known Monte Carlo approach, which makes use of pseudo-random sequences to simulate random behavior, they examine at length the quasi-Monte Carlo approach based on equidistributed sequences and give their comparative analysis. At the end of Chapter 2 the important problem of conditional expectations is considered. In Chapter 3 the authors deal with simulation methods for computing quantities which are connected with stationary processes, Markov processes and processes with independent increments. Here (and in Chapter 5) they pay attention to the shift method which, just as the Monte Carlo method, exploits pseudo-random numbers. But while the Monte Carlo method is based on the law of large numbers the shift method is based on the pointwise ergodic theorem. They also touch upon a number of not widely known topics which are of great theoretical and applied interest. For instance, such a subject is the use of subordination in the Bochner sense (a special change of time in a Markov process) in simulation. The close connection between Markov processes and problems of mathematical physics is well known. Applying this connection it becomes possible to compute by deterministic methods some quantities related to Markov processes. Chapter 4 is devoted to deterministic resolution of some numerical problems involving Markov processes on the whole by means of methods from the potential theory. In the final chapter (Chapter 5) the authors consider a number of problems relating to the numerical integration of stochastic differential equations and to the computation of Wiener functionals. The book is remarkable for its concentrated exposition of the latest results (often in fairly concise form), many of which are available only in the periodical literature. (MathSciNet)