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Some Optimal PESKIR GORAN, SHIRYAEV ALBERT Optimal Stopping and FreeBoundary Problems Series: Lectures in Mathematics. ETH Zürich 2006, XXII + 500 pages ISBN-10: 0-8176-4434-2 ISBN-13: 978-0-8176-4434-5 Birkhäuser Verlag Boston. Basel. Berlin www.birkhauser.ch stopping theory has its basic definitions roots in classical calculus and properties of variations in some are reviewed problems formulated by about Lagrange, Bolza. Mayer Some and martingales and optimal related processes: stopping problems supermartingales, emerged from work by submartingales, Wald in relation to local martingales problems of sequential and Markov testing and represent a processes. Some method of inference statistical fundamental (sequential theorems probability ratio are test) recalled: the where the number of optimal sampling observations is not theorem, determined in advance of martingale the experiment. fundamental A convergence connection theorem and between optimal stopping maximal and free boundary inequalities. The problems was discovered stochastic by a number of integral and Ito’s researchers (among them formula is is Shiryaev) and this presented. equivalence deeply The penetrates solutions of some into the classical fundamentals of modern problems of the probability their theory connection analysis and potential with as theory Dirichlet through problem for the boundary value problems. Poisson equation, Studies of martingales lead to Markov various problems and Markov processes are chains and give a sequential central to optimal probabilistic stopping problems and method stopping and other times are for Markov processes and the Markov chains, by means of the smallest for theoretical fields as: the processes superharmonic and function. theory of probability for Levy processes. mathematical of The of statistics important finance equilibria mathematical as: for a Markov function process, with the the gain value V is equivalent to the A huge stimulus problem most to the development of finding for sequential statistics transformations and optimal characterized the solution stopping problem classical Brownian inequalities, and free-boundary formulated first chapter it problems from important motion, diffusion dominating sharp and Markov optima stopping problems was proved that anticipation) are key to reviewed: Some stopping Some Dynkin (non- processes real world applications. analysis, stopping of stochastic control. And problems. In the the concepts of filtration solving. (information) optimal in between optimal of the optimal stopping theory smallest was provided by option superharmonic of pricing theory, developed function V , stochastic time, change of in the late 1960s and the which dominates lead to change an space, change of 1970s. According to the the gain optimal stopping measure are modern theory of finance, function G . problem, which can be considered are pricing an American This reduced to a free- defined and these option in a complete leads to boundary problem. are used in order market is equivalent to differential or Therefore the principal to obtain solving an optimal integroproblems which arise are solutions in a stopping problem. differential to reformulate these closed form, The authors, A. equations which reductions and to find the solving optimal N. Shiryaev, the disciple the function V solution of the initial stopping of A. N. Kolmogorov, solves. problems. problems. one of the greatest Some In some previous In probabilists of all time, is methods of papers, Mikhalevich, Chapter III it is an eminent authority on solutions are considered free-boundary described the stochastic processes. He presented in the problems for solving connection is a Russian Chapter IV mathematician, for his known Reduction work probability statistics to any n and it is a stopping The problem is to time if , P-a.s. A determine which in free-boundary theory, problem, and financial superharmonic mathematics. at method of the optimal of the given two stopping problem is to values Goran characterization, Peskir is a well known the professor form compute the value V* sup EG of a true mean is. In the Bayesian the time change, the and to exhibit an optimal formulation it is of method of space stopping time , where assumed that the University Manchester. change, the E is the expectation. unknown of a A first method is has The first Chapter method mean given stopping: measure change. to use backward distribution and General facts offer a Chapter V is induction in order to in this chapter a survey of basic results devoted to some construct a sequence of solution of the Optimal concerning optimal stochastic stopping in the case of analysis aspects discrete time, considering which are closely a martingale and a related to sharp Markov approach. inequalities as; It is considered a those of Wald, sequence of random Bessel, variables, denoted Doob, by Hardy- Gn , n N defined filtered on a Littlewood. One probability space (, F , Fn , P), n N . of the objectives of the chapter VI is G n The variable is the study of the interpreted as the gain, if problem of the observation is stopped sequential testing at time t and Fn is the of two that problem for a solves in a stochastic Wiener process finite sense, but in the case of with horizon is made. finite horizon N , which A solution of the choose the optimal time Bayesian and among those which variational belong to the problem for a N class M n { | n N } Wiener process . This method led to the with finite general principle of horizon is also dynamic programming derived. The (the Bellman’s principle). same problems The method of are studied, essential supremum replacing the solves the problem in the Wiener processes case of infinite horizon N by Poisson random simple information available up hypotheses about . to the time n. A random the mean value variables processes. In the subsection Some variable is a Markov of an observed approach concrete Wiener process. Markovian time if { n} Fn , for some basic results of problems of based on studies optimal stopping when mathematical of the time is discrete and finance can be and on modern the process is Markovian. reformulated as and For of research of some the case continuous of problems time, a optimal stopping martingale approach is of given. stochastic processes Markov kind of process, authors relevant prestigious mathematicians. and A Markov time is a solved particular the by a reduction to freefor boundary The book addresses to those readers with a high level which the strong Markov problems of real of mathematical property information and remains analysis. These preserved. Therefore a kinds of brief review of stochastic situations processes is Chapter II. are made in studied in the understanding needs a good chapter VII and background VIII. probability Since the and stochastic the gain process in in processes. optimal The stopping problem origin depends on the concepts is in the future, study of financial an of the optimal engineering and prediction mathematical problem appears statistics and which is of a problems of particular interest general in can be solved in financial engineering. an abstract, but The monograph interest very is manner. general The book may be as an viewed ideal compendium for an interested reader who wishes to master stochastic calculus via fundamental examples. Areas of application where examples are worked out in full detail include financial mathematics, financial engineering, mathematical statistics, and stochastic analysis. Associated Professor Liliana Popa Department of Mathematics "Gheorghe Asachi" Technical University of Iasi