Dynamic Programming and a New Formalism in the Calculus of Variations

Bellman, Richard

doi:10.1073/pnas.40.4.231

Cited by 108 publications

(60 citation statements)

References 4 publications

(1 reference statement)

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“…all possible velocity. The two solutions, (22), (25), are obviously equivalent if |V | = c, but many other velocities are possible in the least action case. This means that Fermat's principle applies only to light rays in the geometrical optics limit and assumes that the velocity of light (or equivalently the refractive index relative to the vacuum) is known at all points in space-time.…”

Section: Fermat's Principle and Hamilton's Principle: Is There A Relamentioning

confidence: 99%

Fermat’s Principle and Hamilton’s Principle: Does a least action take a least time for happening?

Hadi¹

2019

Preprint

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We explore deeper and analyse in more detail Fermat's and Hamilton's principles. We try to address some questions: Is it possible to have δS negative? Is Hamilton's principle always valid for entire path of the system? Is there a relation between Fermat's principle and Hamilton's principle? We assume analogy with Hamilton's principle, is Fermat's principle always valid for entire path of the system? Does a least action take a least time for happening?

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Section: Fermat's Principle and Hamilton's Principle: Is There A Relamentioning

confidence: 99%

Fermat’s Principle and Hamilton’s Principle: Does a least action take a least time for happening?

Hadi¹

2019

Preprint

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“…This approach was introduced by Bellman [14] in 1954 for optimal deterministic control problems. The basic idea is to relate the optimal problem with a certain differential equation called the Hamilton-Jacobi-Bellman equation (HJB).…”

Section: Prefacementioning

confidence: 99%

Stochastic Optimization in Insurance

Azcue

Muler

2014

SpringerBriefs in Quantitative Finance

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“…At this point we should mention that the dynamic programming technique was firstly introduced by Richard Bellman in the 1950s to deal with calculus of variations and optimal control problems [2][3][4][5]. Further developments have been obtained since then by a number of scholars including Florentin [6,7] and Kushner [8], among others.…”

Section: Introductionmentioning

confidence: 95%

Dynamic programming for a Markov-switching jump–diffusion

Azevedo

Pinheiro

Weber³

2014

Journal of Computational and Applied Mathematics

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a b s t r a c tWe consider an optimal control problem with a deterministic finite horizon and state variable dynamics given by a Markov-switching jump-diffusion stochastic differential equation. Our main results extend the dynamic programming technique to this larger family of stochastic optimal control problems. More specifically, we provide a detailed proof of Bellman's optimality principle (or dynamic programming principle) and obtain the corresponding Hamilton-Jacobi-Belman equation, which turns out to be a partial integro-differential equation due to the extra terms arising from the Lévy process and the Markov process. As an application of our results, we study a finite horizon consumptioninvestment problem for a jump-diffusion financial market consisting of one risk-free asset and one risky asset whose coefficients are assumed to depend on the state of a continuous time finite state Markov process. We provide a detailed study of the optimal strategies for this problem, for the economically relevant families of power utilities and logarithmic utilities.

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Dynamic Programming and a New Formalism in the Calculus of Variations

Cited by 108 publications

References 4 publications

Fermat’s Principle and Hamilton’s Principle: Does a least action take a least time for happening?

Fermat’s Principle and Hamilton’s Principle: Does a least action take a least time for happening?

Stochastic Optimization in Insurance

Dynamic programming for a Markov-switching jump–diffusion

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