A Weak Convergence Approach to the Theory of Large Deviations

Dupuis, Paul; Ellis, Richard S.

doi:10.1002/9781118165904

Cited by 774 publications

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“…In fact, this equivalence is a special case of the duality between free energy and relative entropy, a well-known result in the theory of large deviations (Dupuis and Ellis 1997) that has also been used in the area of risk-sensitive control (Dai Pra et al 1996, Petersen et al 2000. For completeness and the benefit of the reader, we shall establish this equivalence through the associated optimality equations.…”

Section: Exponential Utility and Relative Entropymentioning

confidence: 84%

“…In particular, the value function of the robust problem can be interpreted as the certainty equivalent of the value function of the problem with an exponential utility, and the optimal pricing policies for both problems are shown to be the same. The equivalence is actually a special case of the duality between free energy and relative entropy from the theory of large deviations (Dupuis and Ellis 1997) that has been exploited in the area of risk-sensitive control (Dai Pra et al 1996, Petersen et al 2000. For the sake of completeness, we derive this correspondence via the associated optimality equations.…”

Section: Introductionmentioning

confidence: 99%

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Relative Entropy, Exponential Utility, and Robust Dynamic Pricing

Lim

Shanthikumar

2007

Operations Research

158

113

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In the area of dynamic revenue management, optimal pricing policies are typically computed on the basis of an underlying demand rate model. From the perspective of applications, this approach implicitly assumes that the model is an accurate representation of the real-world demand process and that the parameters characterizing this model can be accurately calibrated using data. In many situations, neither of these conditions are satisfied. Indeed, models are usually simplified for the purpose of tractability and may be difficult to calibrate because of a lack of data. Moreover, pricing policies that are computed under the assumption that the model is correct may perform badly when this is not the case. This paper presents an approach to single-product dynamic revenue management that accounts for errors in the underlying model at the optimization stage. Uncertainty in the demand rate model is represented using the notion of relative entropy, and a tractable reformulation of the "robust pricing problem" is obtained using results concerning the change of probability measure for point processes. The optimal pricing policy is obtained through a version of the so-called Isaacs' equation for stochastic differential games, and the structural properties of the optimal solution are obtained through an analysis of this equation. In particular, (i) closed-form solutions for the special case of an exponential nominal demand rate model, (ii) general conditions for the exchange of the "max" and the "min" in the differential game, and (iii) the equivalence between the robust pricing problem and that of single-product revenue management with an exponential utility function without model uncertainty, are established through the analysis of this equation.

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Section: Exponential Utility and Relative Entropymentioning

confidence: 84%

Section: Introductionmentioning

confidence: 99%

Relative Entropy, Exponential Utility, and Robust Dynamic Pricing

Lim

Shanthikumar

2007

Operations Research

158

113

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“…By Theorem 1.2.3 in [13] the fact that S n /n 1−γ satisfies the Laplace principle implies that S n /n 1−γ satisfies the LDP with the same speed n −v and the same rate function Γ; i.e., for any closed subset F in R lim sup…”

Section: Mdps Formentioning

confidence: 96%

Multiple Critical Behavior of Probabilistic Limit Theorems in the Neighborhood of a Tricritical Point

Costeniuc

Ellis²,

Otto

2007

J Stat Phys

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We derive probabilistic limit theorems that reveal the intricate structure of the phase transitions in a mean-field version of the Blume-Emery-Griffiths model [4]. These probabilistic limit theorems consist of scaling limits for the total spin and moderate deviation principles (MDPs) for the total spin. The model under study is defined by a probability distribution that depends on the parameters n, β, and K, which represent, respectively, the 1 Costeniuc, Ellis, and Otto: Critical Behavior of Probabilistic Limit Theorems 2 number of spins, the inverse temperature, and the interaction strength. The intricate structure of the phase transitions is revealed by the existence of 18 scaling limits and 18 MDPs for the total spin. These limit results are obtained as (β, K) converges along appropriate sequences (β n , K n ) to points belonging to various subsets of the phase diagram, which include a curve of second-order points and a tricritical point. The forms of the limiting densities in the scaling limits and of the rate functions in the MDPs reflect the influence of one or more sets that lie in neighborhoods of the critical points and the tricritical point. Of all the scaling limits, the structure of those near the tricritical point is by far the most complex, exhibiting new types of critical behavior when observed in a limit-theorem phase diagram in the space of the two parameters that parametrize the scaling limits.American Mathematical Society 2000 Subject Classifications. Primary 60F10, 60F05, Secondary 82B20

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“…The LDP result is that which would hold true if the solution were a continuous functional of the noise W . Our proof consists in transferring the LDP satisfied by the Hilbert-valued Brownian motion √ εW to that of a Polish-space valued measurable functional of √ εW as established in [3]; see also [4], [13] and [15]. This is related to the Laplace principle.…”

Section: Introductionmentioning

confidence: 99%

Stochastic 2D Hydrodynamical Type Systems: Well Posedness and Large Deviations

Čhuešhov¹,

2009

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Abstract. We deal with a class of abstract nonlinear stochastic models, which covers many 2D hydrodynamical models including 2D Navier-Stokes equations, 2D MHD models and 2D magnetic Bénard problem and also some shell models of turbulence. We first prove the existence and uniqueness theorem for the class considered. Our main result is a Wentzell-Freidlin type large deviation principle for small multiplicative noise which we prove by weak convergence method.

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A Weak Convergence Approach to the Theory of Large Deviations

Cited by 774 publications

References 0 publications

Relative Entropy, Exponential Utility, and Robust Dynamic Pricing

Relative Entropy, Exponential Utility, and Robust Dynamic Pricing

Multiple Critical Behavior of Probabilistic Limit Theorems in the Neighborhood of a Tricritical Point

Stochastic 2D Hydrodynamical Type Systems: Well Posedness and Large Deviations

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