/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.

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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*

A process X is observed continuously in time; it behaves like Brownian motion with drift, which changes from zero to a known constant ϑ>0 at some time τ that is not directly observable. It is important to detect this change when it happens, and we attempt to do so by selecting a stopping rule T* that minimizes the “expected miss” E|T−τ| over all stopping rules T. Assuming that τ has an exponential distribution with known parameter λ>0 and is independent of the driving Brownian motion, we show that the optimal rule T* is to declare that the change has occurred, at the first time t for which

/pdf/a-note-on-bayesian-detection-of-change-points-with-an-fdonum28t7.pdf

A note on bayesian detection of change-points with an expected miss criterion *

We study three stochastic differential games. In each game, two players control a process X = {Xt , 0 ≤ t < ∞} which takes values in the interval I = (0, 1), is absorbed at the endpoints of I, and satisfies a stochastic differential equation dXt = μ ( Xt, α(Xt), β(Xt) ) dt + σ ( Xt, α(Xt), β(Xt) ) dWt , X0 = x ∈ I . The control functions α(·) and β(·) are chosen by players A and B, respectively. In the first of our games, which is zero-sum, player A has a continuous reward function u : [0, 1] → R . In addition to α(·) , player A chooses a stopping rule τ and seeks to maximize the expectation of u(Xτ ) ; whereas player B chooses β(·) and aims to minimize this expectation. In the second game, players A and B each have continuous reward functions u(·) and v(·) , choose stopping rules τ and ρ, and seek to maximize the expectations of u(Xτ ) and v(Xρ) , respectively. In the third game the two players again have continuous reward functions u(·) and v(·) , now assumed to be unimodal, and choose stopping rules τ and ρ. This game terminates at the minimum τ ∧ ρ of the stopping rules τ and ρ , and players A , B want to maximize the expectations of u(Xτ∧ρ) and v(Xτ∧ρ) , respectively. Under mild technical assumptions we show that the first game has a value, and find a saddle point of optimal strategies for the players. The other two games are not zero-sum, in general, and for them we construct Nash equilibria.

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Stochastic games of control and stopping for a linear diffusion

Nous etudions des systemes de particules browniennes sur l’axe reel qui interagissent de facon asymetrique en fonction de leurs temps locaux de collision. Nous prouvons l’existence et l’unicite au sens fort de tels processus et les identifions avec un systeme de particules constitue d’une famille de processus browniens ordonnes ou les coefficients de derive et de diffusion, ainsi que les temps locaux de collision entre les particules, dependent de leurs rangs. Ces systemes browniens peuvent etre compris comme des generalisations de processus stochastiques issus des marches boursiers et de la gestion de portefeuilles. Nous pouvons pallier certaines lacunes de ces modeles tout en conservant la possibilite d’effectuer des calculs explicites. Nous montrons aussi, qu’en plus de leur interet intrinseque, de tels systemes de particules browniennes apparaissent comme des limites universelles de processus de sauts sur un reseau avec des interactions locales. Une etape clef dans la preuve est l’analyse d’une generalisation des applications de Skorokhod qui inclut les temps locaux a l’intersection des faces de l’orthant positif. Le resultat generalise la convergence du processus d’exclusion simple totalement asymetrique (TASEP) vers sa version continue. Finalement, nous identifions parmi les systemes de particules browniennes ceux qui possedent une structure probabiliste determinantale.

/pdf/systems-of-brownian-particles-with-asymmetric-collisions-bo8xz0ss3l.pdf

Systems of Brownian particles with asymmetric collisions

We study the distribution of the time to explosion for one-dimensional diffusions. We relate this question to the computation of expectations of suitable nonnegative local martingales. Moreover, we characterize the distribution function of the time to explosion as the minimal solution to a certain Cauchy problem for an appropriate parabolic differential equation; this leads to alternative characterizations of Feller’s criterion for explosions. We discuss in detail several examples for which it is possible to obtain analytic expressions for the corresponding distribution of the time to explosion, using the methodologies developed in the paper.

Distribution of the time to explosion for one-dimensional diffusions

We argue that real uncertainty itself causes long-run nominal inflation. Consider an infinite horizon cash-in-advance economy with a representative agent and real uncertainty, modeled by independent, identically distributed endowments. Suppose the central bank fixes the nominal rate of interest. We show that the equilibrium long-run rate of inflation is strictly higher, on almost every path of endowment realizations, than it would be if the endowments were constant. Indeed, we present an explicit formula for the long-run rate of inflation, based on the famous Fisher equation. The Fisher equation says the short-run rate of inflation should equal the nominal rate of interest less the real rate of interest. The long-run Fisher equation for our stochastic economy is similar, but with the rate of inflation replaced by the harmonic mean of the growth rate of money.

/pdf/the-inflationary-bias-of-real-uncertainty-and-the-harmonic-54h66j6seq.pdf

Ioannis Karatzas

Papers

A note on bayesian detection of change-points with an expected miss criterion *

Stochastic games of control and stopping for a linear diffusion

Systems of Brownian particles with asymmetric collisions

Distribution of the time to explosion for one-dimensional diffusions

The inflationary bias of real uncertainty and the harmonic Fisher equation