Mathematical Control Theory and Finance

ISBN-13: 9783642089084 / Angielski / Miękka / 2010 / 420 str.

Andrey Sarychev; Albert Shiryaev; Manuel Guerra

Opis

Control theory provides a large set of theoretical and computational tools with applications in a wide range of ?elds, running from pure branches of mathematics, like geometry, to more applied areas where the objective is to ?nd solutions to real life problems, as is the case in robotics, control of industrial processes or ?nance. The high tech character of modern business has increased the need for advanced methods. These rely heavily on mathematical techniques and seem indispensable for competitiveness of modern enterprises. It became essential for the ?nancial analyst to possess a high level of mathematical skills. C- versely, the complex challenges posed by the problems and models relevant to ?nance have, for a long time, been an important source of new research topics for mathematicians. The use of techniques from stochastic optimal control constitutes a well established and important branch of mathematical ?nance. Up to now, other branches of control theory have found comparatively less application in ?n- cial problems. To some extent, deterministic and stochastic control theories developed as di?erent branches of mathematics. However, there are many points of contact between them and in recent years the exchange of ideas between these ?elds has intensi?ed. Some concepts from stochastic calculus (e.g., rough paths) havedrawntheattentionofthedeterministiccontroltheorycommunity.Also, some ideas and tools usual in deterministic control (e.g., geometric, algebraic or functional-analytic methods) can be successfully applied to stochastic c- trol."

Spis treści

Extremals Flows and Infinite Horizon Optimization.- Laplace Transforms and the American Call Option.- Time Change, Volatility, and Turbulence.- External Dynamical Equivalence of Analytic Control Systems.- On Option-Valuation in Illiquid Markets: Invariant Solutions to a Nonlinear Model.- Predicting the Time of the Ultimate Maximum for Brownian Motion with Drift.- A Stochastic Demand Model for Optimal Pricing of Non-Life Insurance Policies.- Optimality of Deterministic Policies for Certain Stochastic Control Problems with Multiple Criteria and Constraints.- Higher-Order Calculus of Variations on Time Scales.- Finding Invariants of Group Actions oon Function Spaces, a General Methodology from Non-Abelian Harmonic Analysis.- Nonholonomic Interpolation for Kinematic Problems, Entropy and Complexity.- Instalment Options: A Closed-Form Solution and the Limiting Case.- Existence and Lipschitzian Regularity for Relaxed Minimizers.- Pricing of Defaultable Securities under Stochastic Interest.- Spline Cubatures for Expectations of Diffusion Processes and Optimal Stopping in Higher Dimensions (with Computational Finance in View).- An Approximate Solution for Optimal Portfolio in Incomplete Markets.- Carleman Linearization of Linearly Observable Polynomial Systems.- Observability of Nonlinear Control Systems on Time Scales - Sufficient Conditions.- Sufficient Optimality Conditions for a Bang-bang Trajectory in a Bolza Problem.- Modelling Energy Markets with Extreme Spikes.- Generalized Bayesian Nonlinear Quickest Detection Problems: On Markov Family of Sufficient Statistics.- Necessary Optimality Condition for a Discrete Dead Oil Isotherm Optimal Control Problem.- Managing Operational Risk: Methodology and Prospects.

Blurb

This book highlights recent developments in mathematical control theory and its applications to finance. It presents a collection of original contributions by distinguished scholars, addressing  a large spectrum of problems and techniques. Control theory provides a large set of theoretical and computational tools with applications in a wide range of fields, ranging from "pure" areas of mathematics up to applied sciences like finance. Stochastic optimal control is a well established and important tool of mathematical finance. Other branches of control theory have found comparatively less applications to financial problems, but the exchange of ideas and methods has intensified in recent years. This volume should contribute to establish bridges between these separate fields. The diversity of topics covered as well as the large array of techniques and ideas brought in to obtain the results make this volume a valuable resource for advanced students and researchers.