Existence of Optimal Stochastic Control Laws

Beneš, V. E.

doi:10.1137/0309034

Cited by 226 publications

(113 citation statements)

References 13 publications

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“…Establishing uniform integrability of these RadonNikodym derivatives, one obtained their relative sequential compactness in the σ(L 1 , L ∞ ) topology by the Dunford-Pettis theorem. After establishing that every limit point thereof in this topology was also a legal Girsanov functional for some controlled diffusion, this was improved to compactness [5], [6], [37]. (See [47], [80] for some precursors which use more restrictive hypotheses.)…”

Section: Complete Observationsmentioning

confidence: 99%

Controlled diffusion processes

Borkar¹

2005

Probab. Surveys

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This article gives an overview of the developments in controlled diffusion processes, emphasizing key results regarding existence of optimal controls and their characterization via dynamic programming for a variety of cost criteria and structural assumptions. Stochastic maximum principle and control under partial observations (equivalently, control of nonlinear filters) are also discussed. Several other related topics are briefly sketched.

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Section: Complete Observationsmentioning

confidence: 99%

Controlled diffusion processes

Borkar¹

2005

Probab. Surveys

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“…Of theoretical interest is the "existence" problem, which means determining in terms of the above three defining characteristics a class of control problems for which there exist control laws achieving minimum cost.. Published results ([1 ], [2], [3], see especially the excellent survey article [4] of Fleming) differ from one another and are not usually comparable because either the models are different or the set of admissible control laws is different.…”

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confidence: 99%

“…u(t)dt + a(t, x(" ), B(. )) dB(t), Bene [3] considers stochastic differential equations of the form (1) (c) Let =.…”

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confidence: 99%

On the Solutions of a Stochastic Control System

Duncan¹,

Varaiya²

1971

SIAM Journal on Control

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Abstract. The control system considered in this paper is modeled by the stochastic differential equation dx(t, to) f(t, x(., o), u(t, to)) dt + dB(t, to), where B is n-dimensional Brownian motion, and the control u is a nonanticipative functional of x(., to) taking its values in a fixed set U. Under various conditions on f it is shown that for every admissible control a solution is defined whose law is absolutely continuous with respect to the Wiener measure #, and the corresponding set of densities on the space C forms a strongly closed, convex subset of L I(C, I). Applications of this result to optimal control and two-person, zero-sum differential games are noted. Finally, an example is given which shows that in the case where only some of the components of x are observed, the set of attainable densities is not weakly closed in LI(C, t).

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“…Secondly, the existence result (420) was originally proved in [2 ] and [30] just by using the compactness properties of the density sets. However they were obliged to assume convexity of the "velocity set" f(t,x,U) in order that the set D(U) = { 6 (fu) : ueU} be convex (and can then be shown to be weakly closed).…”

Section: Controlled Stochastic Differential Equations -Complete Obsermentioning

confidence: 99%

“…We will mention more specific results for special cases below; see also references [2], [3], [12], [13], [30], [36], · [43], [56], [60], 1277]. [ 64,III 102].…”

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Martingale methods in stochastic control

Davis

Lecture Notes in Control and Information Sciences

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The martingale treatment of stochastic control problems is based on the idea that the correct formulation of Bellman's "principle of optiraality" for stochastic minimization problems is in terms of a submartingale inequality: the "value function" of dynamic programming is always a submartingale and is a martingale under a particular control strategy if and only if that strategy is optimal. Local conditions for optimality in the form of a minimum principle can be obtained by applying Meyer's submartingale decomposition along with martingale representation theorems; conditions for existence of an optimal strategy can also be stated. This paper gives an introduction to these methods and a survey of the results that have been obtained so far, as well as an indication of some shortcomings in the theory and open problems. By way of introduction we treat systems of controlled stochastic differential equations, the case for which the most definitive results have been obtained so far. We then outline a general semimartingale formulation of controlled processes, state some optimality conditions and indicate their application to other specific cases such as that of controlled jump processes.The martingale approach to some related problems -optimal stopping, impulse control and stochastic differential games -will also be outlined.

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Existence of Optimal Stochastic Control Laws

Cited by 226 publications

References 13 publications

Controlled diffusion processes

Controlled diffusion processes

On the Solutions of a Stochastic Control System

Martingale methods in stochastic control

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