Limit theorems for suprema, threshold-stopped random variables and last exits of i.i.d. random variables with costs and discounting, with applications to optimal stopping

1992 ◽  
Vol 24 (02) ◽  
pp. 241-266
Author(s):  
Douglas P. Kennedy ◽  
Robert P. Kertz
Keyword(s):  
Optimal Stopping ◽  
Limit Theorems ◽  
Continuous Mapping ◽  
Random Variables ◽  
Exit Times ◽  
Convergence Results ◽  
Process Convergence ◽  
Optimal Stopping Problems ◽  
The Cost ◽  
Linear Cost

For linear-cost-adjusted and geometric-discounted infinite sequences of i.i.d. random variables, point process convergence results are proved as the cost or discounting effect diminishes. These process convergence results are combined with continuous-mapping principles to obtain results on joint convergence of suprema and threshold-stopped random variables, and last-exit times and locations. Applications are made to several classical optimal stopping problems in these settings.

Limit theorems for suprema, threshold-stopped random variables and last exits of i.i.d. random variables with costs and discounting, with applications to optimal stopping

1992 ◽  
Vol 24 (2) ◽  
pp. 241-266 ◽  
Author(s):  
Douglas P. Kennedy ◽  
Robert P. Kertz
Keyword(s):  
Optimal Stopping ◽  
Limit Theorems ◽  
Continuous Mapping ◽  
Random Variables ◽  
Exit Times ◽  
Convergence Results ◽  
Process Convergence ◽  
Optimal Stopping Problems ◽  
The Cost ◽  
Linear Cost

For linear-cost-adjusted and geometric-discounted infinite sequences of i.i.d. random variables, point process convergence results are proved as the cost or discounting effect diminishes. These process convergence results are combined with continuous-mapping principles to obtain results on joint convergence of suprema and threshold-stopped random variables, and last-exit times and locations. Applications are made to several classical optimal stopping problems in these settings.

scholarly journals On approximative solutions of optimal stopping problems

2011 ◽  
Vol 43 (04) ◽  
pp. 1086-1108
Author(s):  
Andreas Faller ◽  
Ludger Rüschendorf
Keyword(s):  
Discrete Time ◽  
Extreme Value Theory ◽  
Optimal Stopping ◽  
Value Theory ◽  
Threshold Functions ◽  
Convergence Results ◽  
Process Convergence ◽  
Complete Discussion ◽  
Optimal Stopping Problems ◽  
Poisson Type

In this paper we establish an extension of the method of approximating optimal discrete-time stopping problems by related limiting stopping problems for Poisson-type processes. This extension allows us to apply this method to a larger class of examples, such as those arising, for example, from point process convergence results in extreme value theory. Furthermore, we develop new classes of solutions of the differential equations which characterize optimal threshold functions. As a particular application, we give a fairly complete discussion of the approximative optimal stopping behavior of independent and identically distributed sequences with discount and observation costs.

scholarly journals On approximative solutions of optimal stopping problems

2011 ◽  
Vol 43 (4) ◽  
pp. 1086-1108 ◽  
Author(s):  
Andreas Faller ◽  
Ludger Rüschendorf
Keyword(s):  
Discrete Time ◽  
Extreme Value Theory ◽  
Optimal Stopping ◽  
Value Theory ◽  
Threshold Functions ◽  
Convergence Results ◽  
Process Convergence ◽  
Complete Discussion ◽  
Optimal Stopping Problems ◽  
Poisson Type

In this paper we establish an extension of the method of approximating optimal discrete-time stopping problems by related limiting stopping problems for Poisson-type processes. This extension allows us to apply this method to a larger class of examples, such as those arising, for example, from point process convergence results in extreme value theory. Furthermore, we develop new classes of solutions of the differential equations which characterize optimal threshold functions. As a particular application, we give a fairly complete discussion of the approximative optimal stopping behavior of independent and identically distributed sequences with discount and observation costs.

Limit theorems for processes such as the Markov branching process

1975 ◽  
Vol 12 (02) ◽  
pp. 289-297
Author(s):  
Andrew D. Barbour
Keyword(s):  
Markov Process ◽  
Residence Time ◽  
Continuous Time ◽  
Limit Theorems ◽  
Branching Process ◽  
Continuous Mapping ◽  
Random Variables ◽  
Time Parameter ◽  
Image Position ◽  
Independent Identically Distributed

LetX(t) be a continuous time Markov process on the integers such that, ifσis a time at whichXmakes a jump,X(σ)– X(σ–) is distributed independently ofX(σ–), and has finite meanμand variance. Letq(j) denote the residence time parameter for the statej.Iftndenotes the time of thenth jump andXn≡X(tb), it is easy to deduce limit theorems forfrom those for sums of independent identically distributed random variables. In this paper, it is shown how, forμ> 0 and for suitableq(·), these theorems can be translated into limit theorems forX(t), by using the continuous mapping theorem.

Limit theorems for threshold-stopped random variables with applications to optimal stopping

1990 ◽  
Vol 22 (2) ◽  
pp. 396-411 ◽  
Author(s):  
Douglas P. Kennedy ◽  
Robert P. Kertz
Keyword(s):  
Asymptotic Behaviour ◽  
Optimal Stopping ◽  
Limit Theorems ◽  
Random Variables ◽  
Limit Distributions ◽  
Joint Distributions ◽  
Normalizing Constants

The extremal types theorem identifies asymptotic behaviour for the maxima of sequences of i.i.d. random variables. A parallel theorem is given which identifies the asymptotic behaviour of sequences of threshold-stopped random variables. Three new types of limit distributions arise, but normalizing constants remain the same as in the maxima case. Limiting joint distributions are also given for maxima and threshold-stopped random variables. Applications to the optimal stopping of i.i.d. random variables are given.

Limit theorems for processes such as the Markov branching process

1975 ◽  
Vol 12 (2) ◽  
pp. 289-297 ◽  
Author(s):  
Andrew D. Barbour
Keyword(s):  
Markov Process ◽  
Residence Time ◽  
Continuous Time ◽  
Limit Theorems ◽  
Branching Process ◽  
Continuous Mapping ◽  
Random Variables ◽  
Time Parameter ◽  
Image Position ◽  
Independent Identically Distributed

Let X(t) be a continuous time Markov process on the integers such that, if σ is a time at which X makes a jump, X(σ)– X(σ–) is distributed independently of X(σ–), and has finite mean μ and variance. Let q(j) denote the residence time parameter for the state j. If tn denotes the time of the nth jump and Xn ≡ X(tb), it is easy to deduce limit theorems for from those for sums of independent identically distributed random variables. In this paper, it is shown how, for μ > 0 and for suitable q(·), these theorems can be translated into limit theorems for X(t), by using the continuous mapping theorem.

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.

On the best order of observation in optimal stopping problems

1987 ◽  
Vol 24 (3) ◽  
pp. 773-778 ◽  
Author(s):  
David Gilat
Keyword(s):  
Optimal Stopping ◽  
Random Variables ◽  
Stopping Rule ◽  
Similar Nature ◽  
Optimal Stopping Problems

For optimal stopping problems in which the player is allowed to choose the order of the random variables as well as the stopping rule, a notion of order equivalence is introduced. It is shown that different (non-degenerate) distributions cannot be order-equivalent.This result unifies and generalizes two theorems of a similar nature recently obtained by Hill and Hordijk (1985).

Limit theorems for threshold-stopped random variables with applications to optimal stopping

1990 ◽  
Vol 22 (02) ◽  
pp. 396-411 ◽  
Author(s):  
Douglas P. Kennedy ◽  
Robert P. Kertz
Keyword(s):  
Asymptotic Behaviour ◽  
Optimal Stopping ◽  
Limit Theorems ◽  
Random Variables ◽  
Limit Distributions ◽  
Joint Distributions ◽  
Normalizing Constants

The extremal types theorem identifies asymptotic behaviour for the maxima of sequences of i.i.d. random variables. A parallel theorem is given which identifies the asymptotic behaviour of sequences of threshold-stopped random variables. Three new types of limit distributions arise, but normalizing constants remain the same as in the maxima case. Limiting joint distributions are also given for maxima and threshold-stopped random variables. Applications to the optimal stopping of i.i.d. random variables are given.

Optimal stopping problems with generalized objective functions

1990 ◽  
Vol 27 (04) ◽  
pp. 828-838
Author(s):  
T. P. Hill ◽  
D. P. Kennedy
Keyword(s):  
Optimal Stopping ◽  
Stopping Time ◽  
Random Variables ◽  
Stopping Times ◽  
Objective Functions ◽  
Monotone Functions ◽  
Optimal Stopping Problems ◽  
Universal Bounds ◽  
Optimal Stopping Times ◽  
Entire Vector

Optimal stopping of a sequence of random variables is studied, with emphasis on generalized objectives which may be non-monotone functions ofEXt, wheretis a stopping time, or may even depend on the entire vector (E[X1I{t=l}], · ··,E[XnI{t=n}]),such as the minimax objective to maximize minj{E[XjI{t=j}]}.Convexity is used to establish a prophet inequality and universal bounds for the optimal return, and a method for constructing optimal stopping times for such objectives is given.

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