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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*

We study the stochastic control problem of maximizing expected utility from terminal wealth and/or consumption, when the portfolio is constrained to take values in a given closed, convex subset of R^d The setting is that of a continuous-time, Ito process model for the underlying asset prices General existence results are established for optimal portfolio/consumption strategies, by suitably embedding the constrained problem in an appropriate family of unconstrained ones, and finding a member of this family for which the corresponding optimal policy obeys the constraints Equivalent conditions for optimality are obtained, and explicit solutions
leading to feedback formulae are derived for special utility functions and for deterministic coefficients Results on incomplete markets, on short-selling constraints and on different interest rates for borrowing and lending are covered as special cases The mathematical tools are those of continuous-time martingales, convex analysis and duality theory

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Convex Duality in Constrained Portfolio Optimization

The problem of valuation for contingent claims that can be exercised at any time before or at maturity, such as American options, is discussed in the manner of Bensoussan [1]. We offer an approach which both simplifies and extends the results of existing theory on this topic.

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On the pricing of American options

This paper solves a general consumption and investment decision problem in closed form. An investor seeks to maximize total expected discounted utility of consumption. There are N distinct risky investments, modelled by dependent geometric Brownian motion processes, and one riskless deterministic investment. The analysis allows for a general utility function and general rates of return. The model and analysis take into consideration the inherent nonnegativity of consumption and consider bankruptcy, so this paper generalizes many of the results of Lehoczky, Sethi, and Shreve Lehoczky, J., S. Sethi, S. Shreva. 1983. Optimal consumption and investment policies allowing consumption constraints and bankruptcy. Math. Oper. Res.8 613--636.. The value function is determined explicitly, as are the optimal consumption and investment policies. The analysis is extended to consider more general risky investments. Under certain conditions, the value functions derived for geometric Brownian motion are shown to provide upper and lower bounds on the value functions in the more general context.

Explicit Solution of a General Consumption/Investment Problem

We derive a formula for the minimal initial wealth needed to hedge an arbitrary contingent claim in a continuous-time model with proportional transaction costs; the expression obtained can be interpreted as the supremum of expected discounted values of the claim, over all (pairs of) probability measures under which the \wealth process" is a supermartingale. Next, we prove the existence of an optimal solution to the portfolio optimization problem of maximizing utility from terminal wealth in the same model; we also characterize this solution via a transformation to a hedging problem: the optimal portfolio is the one that hedges the inverse of marginal utility evaluated at the shadow state-price density solving the corresponding dual problem, if such exists. We can then use the optimal shadow state-price density for pricing contingent claims in this market. The mathematical tools are those of continuous-time martingales, convex analysis, functional analysis and duality theory.

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Hedging and portfolio optimization under transaction costs: a martingale approach ⁄

We establish existence and uniqueness results for adapted solutions of backward stochastic differential equations (BSDE's) with two reflecting barriers, generalizing the work of El Karoui, Kapoudjian, Pardoux, Peng and Quenez. Existence is proved first by solving a related pair of coupled optimal stopping problems, and then, under different conditions, via a penalization method. It is also shown that the solution coincides with the value of a certain Dynkin game, a stochastic game of optimal stopping. Moreover, the connection with the backward SDE enables us to provide a pathwise (deterministic) approach to the game.

/pdf/backward-stochastic-differential-equations-with-reflection-3es92mfw27.pdf

Ioannis Karatzas

Papers

Convex Duality in Constrained Portfolio Optimization

On the pricing of American options

Explicit Solution of a General Consumption/Investment Problem

Hedging and portfolio optimization under transaction costs: a martingale approach ⁄

Backward stochastic differential equations with reflection and Dynkin games