On the Coincidence of the Minimax Solution and the Value Function in a Time-Optimal Game with a Lifeline

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.

Download Full-text

ASYMPTOTICS OF VALUE FUNCTION IN MODELS OF ECONOMIC GROWTH

Tambov University Reports Series Natural and Technical Sciences ◽

10.20310/1810-0198-2018-23-124-605-616 ◽

2018 ◽

pp. 605-616

Author(s):

Alexander Leonidovich Bagno ◽

Alexander Mikhailovich Tarasyev

Keyword(s):

Economic Growth ◽

Optimal Control ◽

Asymptotic Behavior ◽

Value Function ◽

Growth Models ◽

Infinite Horizon ◽

Minimax Solution ◽

Growth Stability ◽

Economic Growth Models ◽

The Value Function

Asymptotic behavior of the value function is studied in an infinite horizon optimal control problem with an unlimited integrand index discounted in the objective functional. Optimal control problems of such type are related to analysis of trends of trajectories in models of economic growth. Stability properties of the value function are expressed in the infinitesimal form. Such representation implies that the value function coincides with the generalized minimax solution of the Hamilton–Jacobi equation. It is shown that that the boundary condition for the value function is substituted by the property of the sublinear asymptotic behavior. An example is given to illustrate construction of the value function as the generalized minimax solution in economic growth models.

Download Full-text

Optimal stopping with random intervention times

Advances in Applied Probability ◽

10.1239/aap/1019160954 ◽

2002 ◽

Vol 34 (1) ◽

pp. 141-157 ◽

Cited By ~ 35

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.

Download Full-text