/pdf/convex-analysisnoer-san-nojin-zhan-nituite-5dl2rbmoyr.pdf

Convex Analysisの二,三の進展について

* Measure Theory-Basic Notions * Measure Theory-Key Results * Processes, Distributions, and Independence * Random Sequences, Series, and Averages * Characteristic Functions and Classical Limit Theorems * Conditioning and Disintegration * Martingales and Optional Times * Markov Processes and Discrete-Time Chains * Random Walks and Renewal Theory * Stationary Processes and Ergodic Theory * Special Notions of Symmetry and Invariance * Poisson and Pure Jump-Type Markov Processes * Gaussian Processes and Brownian Motion * Skorohod Embedding and Invariance Principles * Independent Increments and Infinite Divisibility * Convergence of Random Processes, Measures, and Sets * Stochastic Integrals and Quadratic Variation * Continuous Martingales and Brownian Motion * Feller Processes and Semigroups * Ergodic Properties of Markov Processes * Stochastic Differential Equations and Martingale Problems * Local Time, Excursions, and Additive Functionals * One-Dimensional SDEs and Diffusions * Connections with PDEs and Potential Theory * Predictability, Compensation, and Excessive Functions * Semimartingales and General Stochastic Integration * Large Deviations * Appendix 1: Advanced Measure Theory * Appendix 2: Some Special Spaces * Historical and Bibliographical Notes * Bibliography * Indices

http://cyber.sibsutis.ru:82/Monarev/docs/nauka/PROBABILITY/Kallenberg%20O.%20Foundations%20of%20Modern%20Probability%20(Springer,1997)(ISBN%200387949577)(535s).pdf

Foundations of modern probability

This book is intended as an introduction to optimal stochastic control for continuous time Markov processes and to the theory of viscosity solutions. The authors approach stochastic control problems by the method of dynamic programming. The text provides an introduction to dynamic programming for deterministic optimal control problems, as well as to the corresponding theory of viscosity solutions. A new Chapter X gives an introduction to the role of stochastic optimal control in portfolio optimization and in pricing derivatives in incomplete markets. Chapter VI of the First Edition has been completely rewritten, to emphasize the relationships between logarithmic transformations and risk sensitivity. A new Chapter XI gives a concise introduction to two-controller, zero-sum differential games. Also covered are controlled Markov diffusions and viscosity solutions of Hamilton-Jacobi-Bellman equations. The authors have tried, through illustrative examples and selective material, to connect stochastic control theory with other mathematical areas (e.g. large deviations theory) and with applications to engineering, physics, management, and finance. In this Second Edition, new material on applications to mathematical finance has been added. Concise introductions to risk-sensitive control theory, nonlinear H-infinity control and differential games are also included.

/pdf/controlled-markov-processes-and-viscosity-solutions-4n8p6ywj5v.pdf

Controlled Markov processes and viscosity solutions

The author's preface gives an outline: "This book is about weakconvergence methods in metric spaces, with applications sufficient to show their power and utility. The Introduction motivates the definitions and indicates how the theory will yield solutions to problems arising outside it. Chapter 1 sets out the basic general theorems, which are then specialized in Chapter 2 to the space C[0, l ] of continuous functions on the unit interval and in Chapter 3 to the space D [0, 1 ] of functions with discontinuities of the first kind. The results of the first three chapters are used in Chapter 4 to derive a variety of limit theorems for dependent sequences of random variables. " The book develops and expands on Donsker's 1951 and 1952 papers on the invariance principle and empirical distributions. The basic random variables remain real-valued although, of course, measures on C[0, l ] and D[0, l ] are vitally used. Within this framework, there are various possibilities for a different and apparently better treatment of the material. More of the general theory of weak convergence of probabilities on separable metric spaces would be useful. Metrizability of the convergence is not brought up until late in the Appendix. The close relation of the Prokhorov metric and a metric for convergence in probability is (hence) not mentioned (see V. Strassen, Ann. Math. Statist. 36 (1965), 423-439; the reviewer, ibid. 39 (1968), 1563-1572). This relation would illuminate and organize such results as Theorems 4.1, 4.2 and 4.4 which give isolated, ad hoc connections between weak convergence of measures and nearness in probability. In the middle of p. 16, it should be noted that C*(S) consists of signed measures which need only be finitely additive if 5 is not compact. On p. 239, where the author twice speaks of separable subsets having nonmeasurable cardinal, he means "discrete" rather than "separable." Theorem 1.4 is Ulam's theorem that a Borel probability on a complete separable metric space is tight. Theorem 1 of Appendix 3 weakens completeness to topological completeness. After mentioning that probabilities on the rationals are tight, the author says it is an

Convergence of probability measures

A voluminous literature has emerged for modeling the temporal dependencies in financial market volatility using ARCH and stochastic volatility models. While most of these studies have documented highly significant in-sample parameter estimates and pronounced intertemporal volatility persistence, traditional ex-post forecast evaluation criteria suggest that the models provide seemingly poor volatility forecasts. Contrary to this contention, we show that volatility models produce strikingly accurate interdaily forecasts for the latent volatility factor that would be of interest in most financial applications. New methods for improved ex-post interdaily volatility measurements based on high-frequency intradaily data are also discussed.

https://faculty.washington.edu/ezivot/econ589//////AndersenBollerslevIER1998.pdf

Answering the skeptics: yes, standard volatility models do provide accurate forecasts*

In the spirit of the famous Koml\'os (1967) theorem, we show that every sequence of nonnegative, measurable functions $ \{ f_n \}_{n \in \N} $ on a probability space, contains a subsequence which - along with all its subsequences - converges a.e. Ces\`aro mean to some measurable function $f \ge 0$. The sequence $ \{ f_n \}_{n \in \N} $ need not be bounded in $ \mathbb{ L}^1$; the limit $f$ may take the value $+\infty$; and the proof is quite elementary. The method sharpens a result of Delbaen \& Schachermayer, replacing general convex combinations by Ces\`aro means; as well as the original Koml\'os theorem.

A Strong Law of Large Numbers for Positive Random Variables

Modeling problems for a monetary economy are discussed and some examples are presented in the context of an infinite-horizon economy with one or two types of traders, who use fiat money to buy a single perishable consumption good. Three instances are considered, all with transactions in fiat money. The first model has no borrowing or lending. The second model permits both borrowing and lending, but all loans are secured. The third model has borrowing and unsecured lending, and takes into account the presence of debtors who are unable to honor their debts and go bankrupt. Borrowing and depositing take place through an outside bank, although in some circumstances a money market could be used instead. Conditions for different forms of lending are discussed. This is a survey of three technical papers, where the mathematical models are developed in detail and the proofs are supplied.

A Stochastic Infinite-Horizon Economy with Secured Lending, or Unsecured Lending and Bankruptcy

We describe conditions for the existence of a stationary Markovian equilibrium when total production or total endowment is a random variable. Apart from regularity assumptions, there are two crucial conditions: (i) low information—agents are ignorant of both total endowment and their own endowments when they make decisions in a given period, and (ii) proportional endowments—the endowment of each agent is in proportion, possibly random, to the total endowment. When these conditions hold, there is a stationary equilibrium. When they do not hold, such an equilibrium need not exist.

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Information and the Existence of Stationary Markovian Equilibrium

Every sequence f 1 , f 2 , · · · of random variables with lim M →∞ (cid:0) M sup k ∈ N P ( | f k | > M ) (cid:1) = 0 con-tains a subsequence f k 1 , f k 2 , · · · that satisﬁes, along with all its subsequences, the weak law of large numbers: lim N →∞ (cid:0) (1 /N ) P Nn =1 f k n − D N (cid:1) = 0 , in probability. Here D N is a “corrector” random variable with values in [ − N, N ], for each N ∈ N ; these correctors are all equal to zero if, in addition, lim inf k →∞ E (cid:0) f 2 k 1 {| f k |≤ M } (cid:1) = 0 holds for every M ∈ (0 , ∞ ) .

Ioannis Karatzas

Papers

A Strong Law of Large Numbers for Positive Random Variables

A Stochastic Infinite-Horizon Economy with Secured Lending, or Unsecured Lending and Bankruptcy

Information and the Existence of Stationary Markovian Equilibrium

A Weak Law of Large Numbers for Dependent Random Variables

An overview of stochastic filtering theory