Reward functionals, salvage values, and optimal stopping

Alvarez, Luis H. R.

doi:10.1007/s001860100161

Cited by 73 publications

(100 citation statements)

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“…A number of authors have contributed to understanding of the smooth-fit principle by various means. With no aim to review the full history of these developments we refer to [2]- [14] and [16]- [18] (see also [1] for Lévy processes). These studies contain further references which are useful to consult.…”

Section: Introductionmentioning

confidence: 99%

Principle of smooth fit and diffusions with angles

Peškir

2007

Stochastics

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We show that there exists a regular diffusion process X and a differentiable gain function G such that the value function V of the optimal stopping problemfails to satisfy the smooth fit condition V (b) = G (b) at the optimal stopping point b . On the other hand, if the scale function S of X is differentiable at b , then the smooth fit condition V (b) = G (b) holds (whenever X is regular and G is differentiable at b ). We give an example showing that the latter can happen even when d

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Section: Introductionmentioning

confidence: 99%

Principle of smooth fit and diffusions with angles

Peškir

2007

Stochastics

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“…Athey [3] and Jewitt [18] prove that f : R 2 → R + and h : R 2 → R + are log-supermodular if and only if s f (x, s)h(s, θ)ds is log-supermodular. The first statement now follows from the fact (e.g Alvarez [2] or Rogers [23], V.50) that R(x, θ) = I r(x, y)c(y, θ)m(dy), where r(x, y) is a product of two single-variate functions and hence log-supermodular.…”

Section: Monotonicity Of the Optimal Stopping Threshold In The Paramementioning

confidence: 96%

“…Θ * (x), the indifference map at x, is the set of parameters θ ∈ Θ for which it is optimal to stop immediately when X 0 = x. The indifference map Θ * is a natural generalisation of the allocation index common in the theory of multi-armed bandits: while we make few assumptions on the reward functions, the multi-armed bandit or dynamic allocation literature is restricted to the standard problem (c(x, θ) = c(x) and G(x, θ) = θ), see for instance Gittins and Glazebrook [14], Whittle [28] and for a diffusion setting closer to the setting of this article, Karatzas [19] and Alvarez [2].…”

Section: The Envelope Theoremmentioning

confidence: 99%

Parameter Dependent Optimal Thresholds, Indifference Levels and Inverse Optimal Stopping Problems

Klimmek

2014

J. Appl. Probab.

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Consider the classic infinite-horizon problem of stopping a one-dimensional diffusion to optimise between running and terminal rewards and suppose we are given a parametrised family of such problems. We provide a general theory of parameter dependence in infinitehorizon stopping problems for which threshold strategies are optimal. The crux of the approach is a supermodularity condition which guarantees that the family of problems is indexable by a set valued map which we call the indifference map. This map is a natural generalisation of the allocation (Gittins) index, a classical quantity in the theory of dynamic allocation. Importantly, the notion of indexability leads to a framework for inverse optimal stopping problems.

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“…7 See Darling and Siegert (1953) for a seminal paper, where these discount factors were originally developed in a general Markovian setting. For a more recent treatment within an optimal stopping framework, see Alvarez (2001) and Dayanik and Karatzas (2003).…”

Section: One-way Rotationmentioning

confidence: 99%