scholarly journals A quantization algorithm for solving multidimensional discrete-time optimal stopping problems

Bernoulli ◽  
2003 ◽  
Vol 9 (6) ◽  
pp. 1003-1049 ◽  
Author(s):  
Vlad Bally ◽  
Gilles Pagès
Keyword(s):  
Discrete Time ◽  
Optimal Stopping ◽  
Time Optimal ◽  
Optimal Stopping Problems

Optimal stopping with random intervention times

2002 ◽  
Vol 34 (01) ◽  
pp. 141-157 ◽  
Author(s):  
Paul Dupuis ◽  
Hui Wang
Keyword(s):  
Optimal Stopping ◽  
Continuous Time ◽  
Value Function ◽  
Value Functions ◽  
Optimal Exercise Boundary ◽  
Exercise Boundary ◽  
Time Optimal ◽  
Optimal Stopping Problems ◽  
Necessary And Sufficient ◽  
The Value Function

We consider a class of optimal stopping problems where the ability to stop depends on an exogenous Poisson signal process - we can only stop at the Poisson jump times. Even though the time variable in these problems has a discrete aspect, a variational inequality can be obtained by considering an underlying continuous-time structure. Depending on whether stopping is allowed at t = 0, the value function exhibits different properties across the optimal exercise boundary. Indeed, the value function is only 𝒞 0 across the optimal boundary when stopping is allowed at t = 0 and 𝒞 2 otherwise, both contradicting the usual 𝒞 1 smoothness that is necessary and sufficient for the application of the principle of smooth fit. Also discussed is an equivalent stochastic control formulation for these stopping problems. Finally, we derive the asymptotic behaviour of the value functions and optimal exercise boundaries as the intensity of the Poisson process goes to infinity or, roughly speaking, as the problems converge to the classical continuous-time optimal stopping problems.

Optimal stopping with random intervention times

2002 ◽  
Vol 34 (1) ◽  
pp. 141-157 ◽  
Author(s):  
Paul Dupuis ◽  
Hui Wang
Keyword(s):  
Optimal Stopping ◽  
Continuous Time ◽  
Value Function ◽  
Value Functions ◽  
Optimal Exercise Boundary ◽  
Exercise Boundary ◽  
Time Optimal ◽  
Optimal Stopping Problems ◽  
Necessary And Sufficient ◽  
The Value Function

We consider a class of optimal stopping problems where the ability to stop depends on an exogenous Poisson signal process - we can only stop at the Poisson jump times. Even though the time variable in these problems has a discrete aspect, a variational inequality can be obtained by considering an underlying continuous-time structure. Depending on whether stopping is allowed att= 0, the value function exhibits different properties across the optimal exercise boundary. Indeed, the value function is only𝒞0across the optimal boundary when stopping is allowed att= 0 and𝒞2otherwise, both contradicting the usual𝒞1smoothness that is necessary and sufficient for the application of the principle of smooth fit. Also discussed is an equivalent stochastic control formulation for these stopping problems. Finally, we derive the asymptotic behaviour of the value functions and optimal exercise boundaries as the intensity of the Poisson process goes to infinity or, roughly speaking, as the problems converge to the classical continuous-time optimal stopping problems.

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.

Optimal Stopping Problem with Controlled Recall

1998 ◽  
Vol 12 (1) ◽  
pp. 91-108 ◽  
Author(s):  
Tsuyoshi Saito
Keyword(s):  
Discrete Time ◽  
Optimal Stopping ◽  
Search Process ◽  
Optimal Stopping Problem ◽  
Search Cost ◽  
Net Profit ◽  
Time Optimal

This paper deals with the following discrete-time optimal stopping problem. For fixed search costs, a random offer, w ~ F(w), will be found for each time. This offer is either accepted, rejected, or “reserved” for recall later. The reserving cost for any offer depends on its value, regardless of how long the offer is reserved. The objective is to maximize the expected discounted net profit, provided that an offer must be accepted. The major finding is that no previously reserved offer should be accepted prior to the deadline of the search process.

scholarly journals On general optimal stopping problems using penalty method

2012 ◽  
Vol 45 (2) ◽  
Author(s):  
Ł. Stettner
Keyword(s):  
Stochastic Processes ◽  
Markov Processes ◽  
Discrete Time ◽  
Optimal Stopping ◽  
Penalty Method ◽  
Value Function ◽  
Polish Space ◽  
Finite Horizon ◽  
Optimal Stopping Problem ◽  
Optimal Stopping Problems

AbstractIn the paper we use penalty method to approximate a number of general stopping problems over finite horizon. We consider optimal stopping of discrete time or right continuous stochastic processes, and show that suitable version of Snell’s envelope can by approximated by solutions to penalty equations. Then we study optimal stopping problem for Markov processes on a general Polish space, and again show that the optimal stopping value function can be approximated by a solution to a Markov version of the penalty equation.

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.

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