/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*

An overlapping generations model of an exchange economy with two sources of uncertainty is considered. Individuals have a finite expected life span and uncertain annual income. Conditions concerning birth, death, inheritance and bequests are fully specified. Under such conditions, the existence of a stationary Markovian equilibrium is established in some generality, and several explicitly solvable examples are examined in detail.

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A Stochastic Overlapping Generations Economy with Inheritance

We construct a planar diffusion process whose infinitesimal generator depends only on the order of the components of the process. Speaking informally and a bit imprecisely for the moment, imagine you run two Brownian-like particles on the real line. At any given time, you assign positive drift g and diffusion {\sigma} to the laggard; and you assign negative drift -h and diffusion {\rho} to the leader. We compute the transition probabilities of this process, discuss its realization in terms of appropriate systems of stochastic differential equations, study its dynamics under a time reversal, and note that these involve singularly continuous components governed by local time. Crucial in our analysis are properties of Brownian and semimartingale local time; properties of the generalized perturbed Tanaka equation which we study here in detail; and those of a one-dimensional diffusion with bang-bang drift. We also show that our planar diffusion can be represented in terms of a process with bang-bang drift, its local time at the origin, and an independent standard Brownian motion, in a form which can be construed as a two-dimensional analogue of the stochastic equation satisfied by the so-called skew Brownian motion.

Planar diffusions with rank-based characteristics and perturbed Tanaka equations

Diffusions with reflection on an orthant and associated initial-boundary value problems

This paper solves a Bayes sequential impulse control problem for a diffusion, whose drift has an unobservable parameter with a change point. The partially observed problem is reformulated into one with full observations, via a change of probability measure which removes the drift. The optimal impulse controls can be expressed in terms of the solutions and the current values of a Markov process adapted to the observation filtration. We shall illustrate the application of our results using the Longstaff–Schwartz algorithm for multiple optimal stopping times in a geometric Brownian motion stock price model with drift uncertainty.

Impulse control of a diffusion with a change point

We revisit the [JKO98] variational characterization of diffusion as entropic gradient flux, and provide for it a probabilistic interpretation based on stochastic calculus. It was shown by Jordan, Kinderlehrer, and Otto in [JKO98] that, for diffusions of Langevin-Smoluchowski type, the Fokker-Planck probability density flow minimizes the rate of relative entropy dissipation, as measured by the distance traveled in terms of the quadratic Wasserstein metric. We obtain novel, stochastic-process versions of these features, valid along almost every trajectory of the diffusive motion in both the forward and, most transparently, the backward, directions of time, using a very direct perturbation analysis; the original results follow then simply by taking expectations. As a bonus, we derive the Cordero-Erausquin version of the so-called HWI inequality relating relative entropy, Fisher information and Wasserstein distance.

Ioannis Karatzas

Papers

A Stochastic Overlapping Generations Economy with Inheritance

Planar diffusions with rank-based characteristics and perturbed Tanaka equations

Diffusions with reflection on an orthant and associated initial-boundary value problems

Impulse control of a diffusion with a change point

Pathwise Otto calculus