scholarly journals The High Contact Principle with Reward Functions Involving Initial Points

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Dongmei Guo ◽  
Yi Hu ◽  
Jiankang Zhang ◽  
Chunli Chu
Keyword(s):  
Optimal Stopping ◽  
Initial Point ◽  
The Other ◽  
Optimal Stopping Problems ◽  
Points Of View ◽  
Reward Functions

This paper examines a class of general optimal stopping problems in which reward functions depend on initial points. Two points of view on the initial point are introduced: one is to view it as a constant, and the other is to view it as a constant process starting from the point. Based on the two different views, two versions of the generalized high contact principle are derived. Finally, we apply the generalized high contact principle to one example.

Selection of order of observation in optimal stopping problems

1985 ◽  
Vol 22 (1) ◽  
pp. 177-184 ◽  
Author(s):  
Theodore P. Hill ◽  
Arie Hordijk
Keyword(s):  
Optimal Stopping ◽  
Random Variables ◽  
Stopping Rule ◽  
The Other ◽  
Other Hand ◽  
General Rules ◽  
Optimal Stopping Problems ◽  
Optimal Ordering ◽  
Uniformly Bounded ◽  
Selection Of

In optimal stopping problems in which the player is free to choose the order of observation of the random variables as well as the stopping rule, it is shown that in general there is no function of all the moments of individual integrable random variables, nor any function of the first n moments of uniformly bounded random variables, which can determine the optimal ordering. On the other hand, several fairly general rules for identification of the optimal ordering based on individual distributions are given, and applications are made to several special classes of distributions.

scholarly journals One-sided solutions for optimal stopping problems with logconcave reward functions

2019 ◽  
Vol 51 (01) ◽  
pp. 87-115
Author(s):  
Yi-Shen Lin ◽  
Yi-Ching Yao
Keyword(s):  
Random Walks ◽  
Optimal Stopping ◽  
Lévy Processes ◽  
Levy Processes ◽  
Stopping Times ◽  
Reward Function ◽  
Continuous Reward ◽  
Optimal Stopping Problems ◽  
Reward Functions ◽  
Optimal Stopping Times

AbstractIn the literature on optimal stopping, the problem of maximizing the expected discounted reward over all stopping times has been explicitly solved for some special reward functions (including (x+)ν, (ex − K)+, (K − e− x)+, x ∈ ℝ, ν ∈ (0, ∞), and K > 0) under general random walks in discrete time and Lévy processes in continuous time (subject to mild integrability conditions). All such reward functions are continuous, increasing, and logconcave while the corresponding optimal stopping times are of threshold type (i.e. the solutions are one-sided). In this paper we show that all optimal stopping problems with increasing, logconcave, and right-continuous reward functions admit one-sided solutions for general random walks and Lévy processes, thereby generalizing the aforementioned results. We also investigate in detail the principle of smooth fit for Lévy processes when the reward function is increasing and logconcave.

Selection of order of observation in optimal stopping problems

1985 ◽  
Vol 22 (01) ◽  
pp. 177-184 ◽  
Author(s):  
Theodore P. Hill ◽  
Arie Hordijk
Keyword(s):  
Optimal Stopping ◽  
Random Variables ◽  
Stopping Rule ◽  
The Other ◽  
Other Hand ◽  
General Rules ◽  
Optimal Stopping Problems ◽  
Optimal Ordering ◽  
Uniformly Bounded ◽  
Selection Of

In optimal stopping problems in which the player is free to choose the order of observation of the random variables as well as the stopping rule, it is shown that in general there is no function of all the moments of individual integrable random variables, nor any function of the first n moments of uniformly bounded random variables, which can determine the optimal ordering. On the other hand, several fairly general rules for identification of the optimal ordering based on individual distributions are given, and applications are made to several special classes of distributions.

Sin is Dead, Long Live Sin!

2017 ◽  
Author(s):  
Kate Kirkpatrick
Keyword(s):  
Religious Commitment ◽  
The Other ◽  
Realist Approach ◽  
Points Of View ◽  
Cultural Amnesia

Chapter 9 offers two concluding ‘provocations’: one on wretchedness without God, the other on wretchedness with God. The first brings Sartre into dialogue with Marilyn McCord Adams’s work ‘God because of Evil’, arguing that Sartre’s account lends credence to her view that optimism is not warranted if one takes a robust realist approach to evil. Read as a phenomenologist of fallenness, Sartre may serve the apologetic purpose of making options ‘live’, in William James’s language; or, to use the phrase of Stephen Mulhall, to ‘hold open the possibility of taking religious points of view seriously’. The second provocation—on the question of wretchedness with God—suggests that Sartre can be read ‘for edification’ to help us see our failures in love. The book concludes that reading Sartre in this light can help redress ‘damaging cultural amnesia’ about religious commitment, offering an account of sin that cultivates humility, love, and mercy.

scholarly journals All adapted topologies are equal

2020 ◽  
Vol 178 (3-4) ◽  
pp. 1125-1172
Author(s):  
Julio Backhoff-Veraguas ◽  
Daniel Bartl ◽  
Mathias Beiglböck ◽  
Manu Eder
Keyword(s):  
Stochastic Processes ◽  
Weak Topology ◽  
Equilibrium Problems ◽  
Diffusion Processes ◽  
Temporal Structure ◽  
Wasserstein Distance ◽  
Optimal Stopping Problems ◽  
Reward Functions ◽  
Nested Distance ◽  
The Stability

Abstract A number of researchers have introduced topological structures on the set of laws of stochastic processes. A unifying goal of these authors is to strengthen the usual weak topology in order to adequately capture the temporal structure of stochastic processes. Aldous defines an extended weak topology based on the weak convergence of prediction processes. In the economic literature, Hellwig introduced the information topology to study the stability of equilibrium problems. Bion–Nadal and Talay introduce a version of the Wasserstein distance between the laws of diffusion processes. Pflug and Pichler consider the nested distance (and the weak nested topology) to obtain continuity of stochastic multistage programming problems. These distances can be seen as a symmetrization of Lassalle’s causal transport problem, but there are also further natural ways to derive a topology from causal transport. Our main result is that all of these seemingly independent approaches define the same topology in finite discrete time. Moreover we show that this ‘weak adapted topology’ is characterized as the coarsest topology that guarantees continuity of optimal stopping problems for continuous bounded reward functions.

On wald-type optimal stopping for Brownian motion

1997 ◽  
Vol 34 (1) ◽  
pp. 66-73 ◽  
Author(s):  
S. E. Graversen ◽  
G. Peškir
Keyword(s):  
Brownian Motion ◽  
Optimal Stopping ◽  
Stopping Time ◽  
Maximal Inequality ◽  
Stopping Times ◽  
Standard Brownian Motion ◽  
Square Root ◽  
Real Analysis ◽  
Optimal Stopping Problems ◽  
Wald's Identity

The solution is presented to all optimal stopping problems of the form supτE(G(|Β τ |) – cτ), where is standard Brownian motion and the supremum is taken over all stopping times τ for B with finite expectation, while the map G : ℝ+ → ℝ satisfies for some being given and fixed. The optimal stopping time is shown to be the hitting time by the reflecting Brownian motion of the set of all (approximate) maximum points of the map . The method of proof relies upon Wald's identity for Brownian motion and simple real analysis arguments. A simple proof of the Dubins–Jacka–Schwarz–Shepp–Shiryaev (square root of two) maximal inequality for randomly stopped Brownian motion is given as an application.

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