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

Convex Analysisの二,三の進展について

https://polymer.bu.edu/hes/cox79py538.pdf

Option pricing: A simplified approach☆

Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.

Convergence of Probability Measures

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

This paper describes a general type of stochastic system model that involves three basic elements: activities, resources, and stocks of material. A system manager chooses activity levels dynamically based on state observations, consuming some materials as inputs and producing other materials as outputs, subject to resource capacity constraints. A generalized notion of heavy traffic is described, in which exogenous input and output rates are approximately balanced with nominal activity rates derived from a static planning problem. A Brownian network model is then proposed as a formal approximation in the heavy traffic parameter regime. The current formulation is novel, relative to models analyzed in previous work, in that its definition of heavy traffic takes explicit account of the system manager's economic objective.

/pdf/a-broader-view-of-brownian-networks-2r2qzehqxu.pdf

A broader view of Brownian networks

Let $Z = \{ Z(t),t > 0\} $ be a reflected Brownian motion on the K-dimensional nonnegative orthant, with the direction of reflection constant over each boundary surface. Such processes arise in heavy traffic theory for K-station networks of queues. This paper continues our study of the diffusion process Z,with attention restricted to the case $k = 2$ for simplicity. (Most of the results extend directly to higher dimensions; our notation and style of argument are designed to suggest appropriate generalizations for arbitrary K wherever possible.) The backward equation for the transition density function (with boundary and initial conditions), the corresponding forward equation and the equation for the steady-state distribution are all derived informally. Also presented are various calculations relating to steady-state distributions, including a moment formula and the derivation of a condition (involving the drifts and directions of reflection), that we conjecture to be necessary and sufficient for existence...

On the Distribution of Multidimensional Reflected Brownian Motion

Motivated by applications in telephone call centers, we consider a service system model with m customer classes and r server pools. The model is one with doubly stochastic arrivals, which means that the m-vector ? of instantaneous arrival rates is allowed to vary both temporally and stochastically. Two levels of dynamic control are considered: customers may be either blocked or accepted at the time of their arrival, and then accepted customers of each class must be routed, either immediately upon acceptance or after some period of waiting, to a server pool that is qualified to handle that class. Customers who are made to wait before commencement of their service are liable to defect. The objective is to minimize the expected sum of blocking costs, waiting costs and defection costs over a fixed and finite planning horizon. We consider an asymptotic parameter regime in which (i) the arrival rates, service rates and defection rates are uniformly accelerated by a large factor ?, then (ii) arrival rates are increased by an additional factor g(?), and the number of servers in each pool is increased by g(?) as well. This produces a separation of time scales, justifying a pointwise stationary stochastic fluid approximation for our original system model. In the stochastic fluid approximation, optimal admission control and routing decisions are determined by a simple linear program that uses the current arrival rate vector ? as data. We explain how to implement the fluid model's optimal control policy in our original service system context, and prove that the proposed implementation is asymptotically optimal in the first-order sense.

/pdf/dynamic-routing-and-admission-control-in-high-volume-service-1h40cxf46h.pdf

Dynamic Routing and Admission Control in High-Volume Service Systems: Asymptotic Analysis via Multi-Scale Fluid Limits

Consider a single-server queueing system with K job classes, each having its own renewal input process and its own general service time distribution Further suppose the queue is in heavy traffic, meaning that its traffic intensity parameter is near the critical value of one A system manager must decide whether or not to accept new jobs as they arrive, and also the order in which to serve jobs that are accepted The goal is to minimize penalties associated with rejected jobs, subject to upper bound constraints on the throughput times for accepted jobs; both the penalty for rejecting a job and the bound on the throughput time may depend on job class This problem formulation does not make sense in a conventional queueing model, because throughput times are random variables, but we show that the formulation is meaningful in an asymptotic sense, as one approaches the heavy traffic limit under diffusion scaling Moreover, using a method developed recently by Bramson and Williams, we prove that a relatively simple dynamic control policy is asymptotically optimal in this framework Our proposed policy rejects jobs from one particular class when the server's nominal workload is above a threshold value, accepting all other arrivals; and the sequencing rule for accepted jobs is one that maintains near equality of the relative backlogs for different classes, defined in a natural sense

A Multiclass Queue in Heavy Traffic with Throughput Time Constraints: Asymptotically Optimal Dynamic Controls

Direct and to the point, this book from one of the field's leaders covers Brownian motion and stochastic calculus at the graduate level, and illustrates the use of that theory in various application domains, emphasizing business and economics. The mathematical development is narrowly focused and briskly paced, with many concrete calculations and a minimum of abstract notation. The applications discussed include: the role of reflected Brownian motion as a storage model, queuing model, or inventory model; optimal stopping problems for Brownian motion, including the influential McDonald–Siegel investment model; optimal control of Brownian motion via barrier policies, including optimal control of Brownian storage systems; and Brownian models of dynamic inference, also called Brownian learning models or Brownian filtering models.

/pdf/brownian-models-of-performance-and-control-9jpej50yr3.pdf

J. Michael Harrison

Papers

A broader view of Brownian networks

On the Distribution of Multidimensional Reflected Brownian Motion

Dynamic Routing and Admission Control in High-Volume Service Systems: Asymptotic Analysis via Multi-Scale Fluid Limits

A Multiclass Queue in Heavy Traffic with Throughput Time Constraints: Asymptotically Optimal Dynamic Controls

Brownian Models of Performance and Control