*Patrick L. Anderson*

- Published in print:
- 2013
- Published Online:
- September 2013
- ISBN:
- 9780804758307
- eISBN:
- 9780804783224
- Item type:
- chapter

- Publisher:
- Stanford University Press
- DOI:
- 10.11126/stanford/9780804758307.003.0015
- Subject:
- Economics and Finance, Financial Economics

This chapter presents the theory behind the novel value functional method. This includes the importance of the definition of the firm introduced in this book, which includes separation, replicable ...
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This chapter presents the theory behind the novel value functional method. This includes the importance of the definition of the firm introduced in this book, which includes separation, replicable business practices, and an objective of the firm that is not restricted to profit maximization, the maximization of value, rather than profit, a whirlwind introduction to control theory, and the distinction between the familiar concept of a function and the obscure notion of a functional. The author then presents a functional equation (or Bellman equation) that relates the value of a firm to specific optimization by the manager or entrepreneur. This theory is the basis for the tenth approach to valuation described in this book: the “recursive” or “value functional” approach. The author concludes by proposing conditions for the existence of a solution to the value functional equation for actual firms, basing these in human transversality conditions that he outlines.Less

This chapter presents the theory behind the novel value functional method. This includes the importance of the definition of the firm introduced in this book, which includes separation, replicable business practices, and an objective of the firm that is not restricted to profit maximization, the maximization of value, rather than profit, a whirlwind introduction to control theory, and the distinction between the familiar concept of a function and the obscure notion of a functional. The author then presents a functional equation (or Bellman equation) that relates the value of a firm to specific optimization by the manager or entrepreneur. This theory is the basis for the tenth approach to valuation described in this book: the “recursive” or “value functional” approach. The author concludes by proposing conditions for the existence of a solution to the value functional equation for actual firms, basing these in human transversality conditions that he outlines.

*Emanuel Todorov*

- Published in print:
- 2006
- Published Online:
- August 2013
- ISBN:
- 9780262042383
- eISBN:
- 9780262294188
- Item type:
- chapter

- Publisher:
- The MIT Press
- DOI:
- 10.7551/mitpress/9780262042383.003.0012
- Subject:
- Neuroscience, Disorders of the Nervous System

Optimal control theory is a mathematical discipline for studying the neural control of movement. This chapter presents a mathematical introduction to optimal control theory and discusses the ...
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Optimal control theory is a mathematical discipline for studying the neural control of movement. This chapter presents a mathematical introduction to optimal control theory and discusses the following topics: Bellman equations, Hamilton-Jacobi-Bellman equations, Ricatti equations, and Kalman filter. It also examines the duality of optimal control and optimal estimation, and, finally, describes optimal control models and suggests future research directions.Less

Optimal control theory is a mathematical discipline for studying the neural control of movement. This chapter presents a mathematical introduction to optimal control theory and discusses the following topics: Bellman equations, Hamilton-Jacobi-Bellman equations, Ricatti equations, and Kalman filter. It also examines the duality of optimal control and optimal estimation, and, finally, describes optimal control models and suggests future research directions.

*Tomas Björk*

- Published in print:
- 1998
- Published Online:
- November 2003
- ISBN:
- 9780198775188
- eISBN:
- 9780191595981
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/0198775180.003.0014
- Subject:
- Economics and Finance, Financial Economics

This chapter gives a self‐contained introduction to optimal control of stochastic differential equations. We derive the Hamilton‐Jacobi‐Bellman equation as well as a verification theorem. The general ...
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This chapter gives a self‐contained introduction to optimal control of stochastic differential equations. We derive the Hamilton‐Jacobi‐Bellman equation as well as a verification theorem. The general theory is then applied to optimal consumption and investment problems.Less

This chapter gives a self‐contained introduction to optimal control of stochastic differential equations. We derive the Hamilton‐Jacobi‐Bellman equation as well as a verification theorem. The general theory is then applied to optimal consumption and investment problems.

*Kerry E. Back*

- Published in print:
- 2017
- Published Online:
- May 2017
- ISBN:
- 9780190241148
- eISBN:
- 9780190241179
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780190241148.003.0014
- Subject:
- Economics and Finance, Financial Economics

The Euler equation is defined. The static approach can be used to derive an optimal portfolio in a complete market and when the investment opportunity set is constant. In the latter case, the optimal ...
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The Euler equation is defined. The static approach can be used to derive an optimal portfolio in a complete market and when the investment opportunity set is constant. In the latter case, the optimal portfolio is proportional to the growth‐optimal portfolio and two‐fund separation holds. Dynamic programming and the Hamilton‐Jacobi‐Bellman equation are explained. An optimal portfolio consists of myopic and hedging demands. The envelope condition is explained. CRRA utility implies a CRRA value function. The CCAPM and ICAPM are derived.Less

The Euler equation is defined. The static approach can be used to derive an optimal portfolio in a complete market and when the investment opportunity set is constant. In the latter case, the optimal portfolio is proportional to the growth‐optimal portfolio and two‐fund separation holds. Dynamic programming and the Hamilton‐Jacobi‐Bellman equation are explained. An optimal portfolio consists of myopic and hedging demands. The envelope condition is explained. CRRA utility implies a CRRA value function. The CCAPM and ICAPM are derived.