Sequential stochastic optimization
Sequential Stochastic Optimization provides mathematicians and applied researchers with a well-developed framework in which stochastic optimization problems can be formulated and solved. Offering much material that is either new or has never before appeared in book form, it lucidly presents a unified theory of optimal stopping and optimal sequential control of stochastic processes. This book has been carefully organized so that little prior knowledge of the subject is assumed; its only prerequisites are a standard graduate course in probability theory and some familiarity with discrete-parameter martingales.
Major topics covered in Sequential Stochastic Optimization include:
* Fundamental notions, such as essential supremum, stopping points, accessibility, martingales and supermartingales indexed by INd
* Conditions which ensure the integrability of certain suprema of partial sums of arrays of independent random variables
* The general theory of optimal stopping for processes indexed by Ind
* Structural properties of information flows
* Sequential sampling and the theory of optimal sequential control
* Multi-armed bandits, Markov chains and optimal switching between random walks
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Sums of Independent Random Variables
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accessible stopping point assertion assume belongs bidimensional filtration Chapter Clearly clique concave conclude condition converges Dalang denote the set deterministic increasing path dimension E(Xr E(XT E(Zr elements equal esssup establish example Exercise exists Fatou's lemma follows function defined hypothesis implies index controls index set indexed by G induced subgraph induction inequality intersection graph Lemma lim sup linear Markov chain Markov property martingale maximal Moreover natural filtration null sets observe optimal accessible stopping optimal control optimal stopping points optimal strategy optional increasing path predictable increasing path probability space problem process indexed Proof Proposition prove random variables real-valued reward process right-hand side satisfies CQI Section 3.3 sequence set F sigma field Snell's envelope Statistical Stochastic stopping time relative subset superharmonic supermartingale supremum theorem of Section uniformly integrable variables with values vertices y e Gf