Optimal stopping and dynamic allocation

1987 ◽  
Vol 19 (4) ◽  
pp. 829-853 ◽  
Author(s):  
Fu Chang ◽  
Tze Leung Lai
Keyword(s):  
Wiener Process ◽  
Asymptotic Expansions ◽  
Optimal Stopping ◽  
Exponential Family ◽  
Optimal Solutions ◽  
Bandit Problem ◽  
Dynamic Allocation ◽  
Asymptotically Optimal ◽  
Simple Index ◽  
Optimal Stopping Problems

A class of optimal stopping problems for the Wiener process is studied herein, and asymptotic expansions for the optimal stopping boundaries are derived. These results lead to a simple index-type class of asymptotically optimal solutions to the classical discounted multi-armed bandit problem: given a discount factor 0<β <1 and k populations with densities from an exponential family, how should x1, x2,… be sampled sequentially from these populations to maximize the expected value of Ʃ∞1 βi−1xi, in ignorance of the parameters of the densities?

Optimal stopping and dynamic allocation

1987 ◽  
Vol 19 (04) ◽  
pp. 829-853 ◽  
Author(s):  
Fu Chang ◽  
Tze Leung Lai
Keyword(s):  
Wiener Process ◽  
Asymptotic Expansions ◽  
Optimal Stopping ◽  
Exponential Family ◽  
Optimal Solutions ◽  
Bandit Problem ◽  
Dynamic Allocation ◽  
Asymptotically Optimal ◽  
Simple Index ◽  
Optimal Stopping Problems

A class of optimal stopping problems for the Wiener process is studied herein, and asymptotic expansions for the optimal stopping boundaries are derived. These results lead to a simple index-type class of asymptotically optimal solutions to the classical discounted multi-armed bandit problem: given a discount factor 0&lt;β &lt;1 and k populations with densities from an exponential family, how should x 1, x 2,… be sampled sequentially from these populations to maximize the expected value of Ʃ∞ 1 β i−1 x i , in ignorance of the parameters of the densities?

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

2014 ◽  
Vol 51 (02) ◽  
pp. 492-511
Author(s):  
Martin Klimmek
Keyword(s):  
Optimal Stopping ◽  
Infinite Horizon ◽  
Gittins Index ◽  
Dynamic Allocation ◽  
One Dimensional ◽  
Optimal Thresholds ◽  
The Family ◽  
Natural Generalisation ◽  
Optimal Stopping Problems ◽  
Parameter Dependent

Consider the classic infinite-horizon problem of stopping a one-dimensional diffusion to optimise between running and terminal rewards, and suppose that we are given a parametrised family of such problems. We provide a general theory of parameter dependence in infinite-horizon 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.

Generalizing the secretary problem

1979 ◽  
Vol 11 (2) ◽  
pp. 384-396 ◽  
Author(s):  
Thomas J. Lorenzen
Keyword(s):  
Differential Equation ◽  
Difference Equation ◽  
Loss Function ◽  
Optimal Stopping ◽  
Secretary Problem ◽  
Optimal Solutions ◽  
Optimal Stopping Problems ◽  
Realistic Formulation ◽  
Finite Problem ◽  
General Conditions

The secretary problem refers to a certain class of optimal stopping problems based on relative ranks. To allow a more realistic formulation of the problem, this paper considers an arbitrary loss function. A finite and an infinite problem are defined and the optimal solutions are obtained. The solution for the infinite problem is given by a differential equation while the finite problem is given by a difference equation. Under general conditions, the finite problem tends to the infinite problem. An example involving the secretary problem with interview cost is considered and illustrates the usefulness of the present paper.

Generalizing the secretary problem

1979 ◽  
Vol 11 (02) ◽  
pp. 384-396 ◽  
Author(s):  
Thomas J. Lorenzen
Keyword(s):  
Differential Equation ◽  
Difference Equation ◽  
Loss Function ◽  
Optimal Stopping ◽  
Secretary Problem ◽  
Optimal Solutions ◽  
Optimal Stopping Problems ◽  
Realistic Formulation ◽  
Finite Problem ◽  
General Conditions

The secretary problem refers to a certain class of optimal stopping problems based on relative ranks. To allow a more realistic formulation of the problem, this paper considers an arbitrary loss function. A finite and an infinite problem are defined and the optimal solutions are obtained. The solution for the infinite problem is given by a differential equation while the finite problem is given by a difference equation. Under general conditions, the finite problem tends to the infinite problem. An example involving the secretary problem with interview cost is considered and illustrates the usefulness of the present paper.

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

2014 ◽  
Vol 51 (2) ◽  
pp. 492-511
Author(s):  
Martin Klimmek
Keyword(s):  
Optimal Stopping ◽  
Infinite Horizon ◽  
Gittins Index ◽  
Dynamic Allocation ◽  
One Dimensional ◽  
Optimal Thresholds ◽  
The Family ◽  
Natural Generalisation ◽  
Optimal Stopping Problems ◽  
Parameter Dependent

Consider the classic infinite-horizon problem of stopping a one-dimensional diffusion to optimise between running and terminal rewards, and suppose that we are given a parametrised family of such problems. We provide a general theory of parameter dependence in infinite-horizon 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.

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.

Optimal Stopping of the Maximum Process

2014 ◽  
Vol 51 (03) ◽  
pp. 818-836 ◽  
Author(s):  
Luis H. R. Alvarez ◽  
Pekka Matomäki
Keyword(s):  
Brownian Motion ◽  
Free Boundary ◽  
Optimal Stopping ◽  
Original Problem ◽  
Alternative Route ◽  
Linear Diffusion ◽  
Maximum Process ◽  
Running Maximum ◽  
Parameterized Family ◽  
Optimal Stopping Problems

We consider a class of optimal stopping problems involving both the running maximum as well as the prevailing state of a linear diffusion. Instead of tackling the problem directly via the standard free boundary approach, we take an alternative route and present a parameterized family of standard stopping problems of the underlying diffusion. We apply this family to delineate circumstances under which the original problem admits a unique, well-defined solution. We then develop a discretized approach resulting in a numerical algorithm for solving the considered class of stopping problems. We illustrate the use of the algorithm in both a geometric Brownian motion and a mean reverting diffusion setting.

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