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Numerical Analysis of a Free-Boundary Singular Control Problem in Financial Economics (Classic Reprint)

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9781332272198: Numerical Analysis of a Free-Boundary Singular Control Problem in Financial Economics (Classic Reprint)
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Excerpt from Numerical Analysis of a Free-Boundary Singular Control Problem in Financial Economics

We consider a frictionless securities market with two long lived and continuously traded securities: a stock and a bond. The stock is risky, pays no dividends, and sells for S (t) at time t. The bond is riskless, does not pay dividend, and sells for B(t) e at time t, where r is the constant riskless interest rate.

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Excerpt from Numerical Analysis of a Free-Boundary Singular Control Problem in Financial Economics

We analyze a numerical scheme for solving the problem of optimal life-time consumption and investment allocation studied in a companion paper by Hindy, Huang, and Zhu (1993), denoted HH&Z henceforth. HH&Z study a model that represents, among other things, habit forming preferences over the service flows from irreversible purchases of a durable good whose stock decays over time. We provide a brief statement of a special case of the control problem in section 2. HH&Z contains the complete formulation, the motivation of the study, the economic interpretations of the variables, and a detailed discussion of the numerical solution. In this paper, we only provide the theoretical framework that supports the analysis in HH&Z.

We use dynamic programming to solve the optimal stochastic control for the continuous time problem formulated in HH&Z. The important feature of this problem is that the admissible controls allow for cumulative consumption processes with possible jumps and singular sample paths. Furthermore, the controls do not appear directly in the objective functional. As a result, the associated Bellman's equation, formula (7) in the sequel, takes the form of a differential inequality involving a nonlinear second order partial differential equation with gradient constraint. The differential inequality reveals the "free-boundary" nature of the solution. In particular, consumption occurs only at some boundary surface in the domain of definition of the value function. This free-boundary is the area in the state space at which both the second order differential equation and the gradient constraint arc equal to zero. Furthermore, the optimal consumption is the amount required to keep the trajectory of the controlled process from crossing the free-boundary. This typically results in optimal consumption, and associated wealth, with singular sample paths; hence the term "singular control".

In section 3.1, we state the differential inequality that the value function solves. HH&Z contains the details of the derivation which is heuristic because the value function is assumed to be twice continuously differentiable.

About the Publisher

Forgotten Books publishes hundreds of thousands of rare and classic books. Find more at www.forgottenbooks.com

This book is a reproduction of an important historical work. Forgotten Books uses state-of-the-art technology to digitally reconstruct the work, preserving the original format whilst repairing imperfections present in the aged copy. In rare cases, an imperfection in the original, such as a blemish or missing page, may be replicated in our edition. We do, however, repair the vast majority of imperfections successfully; any imperfections that remain are intentionally left to preserve the state of such historical works.

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Buchbeschreibung Paperback. Zustand: New. Language: English. Brand new Book. Excerpt from Numerical Analysis of a Free-Boundary Singular Control Problem in Financial Economics We consider a frictionless securities market with two long lived and continuously traded securities: a stock and a bond. The stock is risky, pays no dividends, and sells for S (t) at time t. The bond is riskless, does not pay dividend, and sells for B(t) e at time t, where r is the constant riskless interest rate. About the Publisher Forgotten Books publishes hundreds of thousands of rare and classic books. Find more at This book is a reproduction of an important historical work. Forgotten Books uses state-of-the-art technology to digitally reconstruct the work, preserving the original format whilst repairing imperfections present in the aged copy. In rare cases, an imperfection in the original, such as a blemish or missing page, may be replicated in our edition. We do, however, repair the vast majority of imperfections successfully; any imperfections that remain are intentionally left to preserve the state of such historical works. ***** Print on Demand *****. Bestandsnummer des Verkäufers PWW9781332272198

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