scholarly journals Optimal Stopping for Processes with Independent Increments, and Applications

2009 ◽  
Vol 46 (4) ◽  
pp. 1130-1145 ◽  
Author(s):  
G. Deligiannidis ◽  
H. Le ◽  
S. Utev
Keyword(s):  
Optimal Stopping ◽  
Optimization Problems ◽  
Infinite Horizon ◽  
Unified Approach ◽  
Reward Function ◽  
Finance Industry ◽  
Independent Increments ◽  
Reward Functions ◽  
First Time ◽  
The Value Function

In this paper we present an explicit solution to the infinite-horizon optimal stopping problem for processes with stationary independent increments, where reward functions admit a certain representation in terms of the process at a random time. It is shown that it is optimal to stop at the first time the process crosses a level defined as the root of an equation obtained from the representation of the reward function. We obtain an explicit formula for the value function in terms of the infimum and supremum of the process, by making use of the Wiener–Hopf factorization. The main results are applied to several problems considered in the literature, to give a unified approach, and to new optimization problems from the finance industry.

scholarly journals Optimal Stopping for Processes with Independent Increments, and Applications

2009 ◽  
Vol 46 (04) ◽  
pp. 1130-1145 ◽  
Author(s):  
G. Deligiannidis ◽  
H. Le ◽  
S. Utev
Keyword(s):  
Optimal Stopping ◽  
Optimization Problems ◽  
Infinite Horizon ◽  
Unified Approach ◽  
Reward Function ◽  
Finance Industry ◽  
Independent Increments ◽  
Reward Functions ◽  
First Time ◽  
The Value Function

In this paper we present an explicit solution to the infinite-horizon optimal stopping problem for processes with stationary independent increments, where reward functions admit a certain representation in terms of the process at a random time. It is shown that it is optimal to stop at the first time the process crosses a level defined as the root of an equation obtained from the representation of the reward function. We obtain an explicit formula for the value function in terms of the infimum and supremum of the process, by making use of the Wiener–Hopf factorization. The main results are applied to several problems considered in the literature, to give a unified approach, and to new optimization problems from the finance industry.

scholarly journals Construction of the Value Function and Optimal Rules in Optimal Stopping of One-Dimensional Diffusions

2010 ◽  
Vol 42 (1) ◽  
pp. 158-182 ◽  
Author(s):  
Kurt Helmes ◽  
Richard H. Stockbridge
Keyword(s):  
Optimal Stopping ◽  
Value Function ◽  
Optimization Problems ◽  
Strong Duality ◽  
Stopping Rules ◽  
One Dimensional ◽  
Reward Function ◽  
Restricted Form ◽  
The Family ◽  
The Value Function

A new approach to the solution of optimal stopping problems for one-dimensional diffusions is developed. It arises by imbedding the stochastic problem in a linear programming problem over a space of measures. Optimizing over a smaller class of stopping rules provides a lower bound on the value of the original problem. Then the weak duality of a restricted form of the dual linear program provides an upper bound on the value. An explicit formula for the reward earned using a two-point hitting time stopping rule allows us to prove strong duality between these problems and, therefore, allows us to either optimize over these simpler stopping rules or to solve the restricted dual program. Each optimization problem is parameterized by the initial value of the diffusion and, thus, we are able to construct the value function by solving the family of optimization problems. This methodology requires little regularity of the terminal reward function. When the reward function is smooth, the optimal stopping locations are shown to satisfy the smooth pasting principle. The procedure is illustrated using two examples.

scholarly journals Construction of the Value Function and Optimal Rules in Optimal Stopping of One-Dimensional Diffusions

2010 ◽  
Vol 42 (01) ◽  
pp. 158-182 ◽  
Author(s):  
Kurt Helmes ◽  
Richard H. Stockbridge
Keyword(s):  
Optimal Stopping ◽  
Value Function ◽  
Optimization Problems ◽  
Strong Duality ◽  
Stopping Rules ◽  
One Dimensional ◽  
Reward Function ◽  
Restricted Form ◽  
The Family ◽  
The Value Function

A new approach to the solution of optimal stopping problems for one-dimensional diffusions is developed. It arises by imbedding the stochastic problem in a linear programming problem over a space of measures. Optimizing over a smaller class of stopping rules provides a lower bound on the value of the original problem. Then the weak duality of a restricted form of the dual linear program provides an upper bound on the value. An explicit formula for the reward earned using a two-point hitting time stopping rule allows us to prove strong duality between these problems and, therefore, allows us to either optimize over these simpler stopping rules or to solve the restricted dual program. Each optimization problem is parameterized by the initial value of the diffusion and, thus, we are able to construct the value function by solving the family of optimization problems. This methodology requires little regularity of the terminal reward function. When the reward function is smooth, the optimal stopping locations are shown to satisfy the smooth pasting principle. The procedure is illustrated using two examples.

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.

scholarly journals Concurrent material and structure optimization of multiphase hierarchical systems within a continuum micromechanics framework

2021 ◽  
Author(s):  
Tarun Gangwar ◽  
Dominik Schillinger
Keyword(s):  
Optimization Problems ◽  
Computational Cost ◽  
Structure Optimization ◽  
Material Behavior ◽  
Hierarchical Systems ◽  
Material Optimization ◽  
Continuum Micromechanics ◽  
Benchmark Tests ◽  
Constraint Optimization Problems ◽  
First Time

AbstractWe present a concurrent material and structure optimization framework for multiphase hierarchical systems that relies on homogenization estimates based on continuum micromechanics to account for material behavior across many different length scales. We show that the analytical nature of these estimates enables material optimization via a series of inexpensive “discretization-free” constraint optimization problems whose computational cost is independent of the number of hierarchical scales involved. To illustrate the strength of this unique property, we define new benchmark tests with several material scales that for the first time become computationally feasible via our framework. We also outline its potential in engineering applications by reproducing self-optimizing mechanisms in the natural hierarchical system of bamboo culm tissue.

A Unified Approach to Robust Farkas-Type Results with Applications to Robust Optimization Problems

2017 ◽  
Vol 27 (2) ◽  
pp. 1075-1101 ◽  
Author(s):  
N. Dinh ◽  
T. H. Mo ◽  
G. Vallet ◽  
M. Volle
Keyword(s):  
Robust Optimization ◽  
Optimization Problems ◽  
Unified Approach

Rule Fitness and Pathology in Learning Classifier Systems

2004 ◽  
Vol 12 (1) ◽  
pp. 99-135 ◽  
Author(s):  
Tim Kovacs
Keyword(s):  
Value Functions ◽  
Classifier Systems ◽  
Reward Function ◽  
Learning Classifier ◽  
Learning Tasks ◽  
General Rules ◽  
Strength Based ◽  
The Difference ◽  
The Value Function ◽  
Do So

It has long been known that in some relatively simple reinforcement learning tasks traditional strength-based classifier systems will adapt poorly and show poor generalisation. In contrast, the more recent accuracy-based XCS, appears both to adapt and generalise well. In this work, we attribute the difference to what we call strong over general and fit over general rules. We begin by developing a taxonomy of rule types and considering the conditions under which they may occur. In order to do so an extreme simplification of the classifier system is made, which forces us toward qualitative rather than quantitative analysis. We begin with the basics, considering definitions for correct and incorrect actions, and then correct, incorrect, and overgeneral rules for both strength and accuracy-based fitness. The concept of strong overgeneral rules, which we claim are the Achilles' heel of strength-based classifier systems, are then analysed. It is shown that strong overgenerals depend on what we call biases in the reward function (or, in sequential tasks, the value function). We distinguish between strong and fit overgeneral rules, and show that although strong overgenerals are fit in a strength-based system called SB-XCS, they are not in XCS. Next we show how to design fit overgeneral rules for XCS (but not SB-XCS), by introducing biases in the variance of the reward function, and thus that each system has its own weakness. Finally, we give some consideration to the prevalence of reward and variance function bias, and note that non-trivial sequential tasks have highly biased value functions.

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