scholarly journals Stochastic nonzero-sum games: a new connection between singular control and optimal stopping

2018 ◽  
Vol 50 (2) ◽  
pp. 347-372 ◽  
Author(s):  
Tiziano De Angelis ◽  
Giorgio Ferrari
Keyword(s):  
Nash Equilibrium ◽  
Optimal Stopping ◽  
Singular Control ◽  
Hitting Times ◽  
Value Functions

AbstractIn this paper we establish a new connection between a class of two-player nonzero-sum games of optimal stopping and certain two-player nonzero-sum games of singular control. We show that whenever a Nash equilibrium in the game of stopping is attained by hitting times at two separate boundaries, then such boundaries also trigger a Nash equilibrium in the game of singular control. Moreover, a differential link between the players' value functions holds across the two games.

Nash equilibrium in nonzero-sum games of optimal stopping for Brownian motion

2017 ◽  
Vol 49 (2) ◽  
pp. 430-445 ◽  
Author(s):  
Natalie Attard
Keyword(s):  
Brownian Motion ◽  
Nash Equilibrium ◽  
Free Boundary ◽  
Boundary Problem ◽  
Free Boundary Problem ◽  
Optimal Stopping ◽  
Value Functions ◽  
Pass Through ◽  
A Free Boundary Problem ◽  
Zero Sum

Abstract We present solutions to nonzero-sum games of optimal stopping for Brownian motion in [0, 1] absorbed at either 0 or 1. The approach used is based on the double partial superharmonic characterisation of the value functions derived in Attard (2015). In this setting the characterisation of the value functions has a transparent geometrical interpretation of 'pulling two ropes' above 'two obstacles' which must, however, be constrained to pass through certain regions. This is an extension of the analogous result derived by Peskir (2009), (2012) (semiharmonic characterisation) for the value function in zero-sum games of optimal stopping. To derive the value functions we transform the game into a free-boundary problem. The latter is then solved by making use of the double smooth fit principle which was also observed in Attard (2015). Martingale arguments based on the Itô–Tanaka formula will then be used to verify that the solution to the free-boundary problem coincides with the value functions of the game and this will establish the Nash equilibrium.

Optimal stopping, randomized stopping and singular control with general information flow

2021 ◽  
Vol 66 (4) ◽  
pp. 760-773
Author(s):  
Nacira Agram ◽  
Nacira Agram ◽  
Свен Хаадем ◽  
Sven Haadem ◽  
Бернт Оксендаль ◽  
...  
Keyword(s):  
Information Flow ◽  
Optimal Stopping ◽  
Singular Control ◽  
General Information

В этой статье мы преследуем двоякую цель. Во-первых, мы распространяем хорошо известное соотношение между оптимальной остановкой и рандомизированной остановкой заданного случайного процесса на ситуацию, когда доступный поток информации - это фильтрация, которая априори не предполагается как-либо связанной с фильтрацией случайного процесса. В этом случае мы говорим об оптимальной остановке и рандомизированной остановке при общем потоке информации. Во-вторых, следуя идее Н. В. Крылова (1977), мы вводим специальную задачу сингулярного стохастического управления с общей информацией и показываем, что она эквивалентна задачам оптимального управления и рандомизированного управления с частичной информацией. Далее мы показываем, что решение указанной задачи сингулярного управления может быть выражено в терминах вариационных неравенств для частичной информации.

scholarly journals A policy iteration algorithm for nonzero-sum stochastic impulse games

2019 ◽  
Vol 65 ◽  
pp. 27-45
Author(s):  
René Aïd ◽  
Francisco Bernal ◽  
Mohamed Mnif ◽  
Diego Zabaljauregui ◽  
Jorge P. Zubelli
Keyword(s):  
Nash Equilibrium ◽  
Numerical Methods ◽  
Variational Inequalities ◽  
Convergence Analysis ◽  
Policy Iteration ◽  
Iteration Algorithm ◽  
Value Functions ◽  
Numerical Tests ◽  
Wide Range ◽  
Policy Iteration Algorithm

This work presents a novel policy iteration algorithm to tackle nonzero-sum stochastic impulse games arising naturally in many applications. Despite the obvious impact of solving such problems, there are no suitable numerical methods available, to the best of our knowledge. Our method relies on the recently introduced characterisation of the value functions and Nash equilibrium via a system of quasi-variational inequalities. While our algorithm is heuristic and we do not provide a convergence analysis, numerical tests show that it performs convincingly in a wide range of situations, including the only analytically solvable example available in the literature at the time of writing.

A class of singular stochastic control problems

1983 ◽  
Vol 15 (02) ◽  
pp. 225-254 ◽  
Author(s):  
Ioannis Karatzas
Keyword(s):  
Brownian Motion ◽  
Optimal Stopping ◽  
Moving Boundary ◽  
Upper And Lower Bounds ◽  
Control Problems ◽  
Explicit Solutions ◽  
Value Functions ◽  
Singular Stochastic Control ◽  
Optimal Stopping Problem ◽  
The Third

We consider the problem of tracking a Brownian motion by a process of bounded variation, in such a way as to minimize total expected cost of both ‘action' and ‘deviation from a target state 0'. The former is proportional to the amount of control exerted to date, while the latter is being measured by a function which can be viewed, for simplicity, as quadratic. We discuss the discounted, stationary and finite-horizon variants of the problem. The answer to all three questions takes the form of exerting control in asingularmanner, in order not to exit from a certain region. Explicit solutions are found for the first and second questions, while the third is reduced to an appropriate optimal stopping problem. This reduction yields properties, as well as global upper and lower bounds, for the associated moving boundary. The pertinent Abelian and ergodic relationships for the corresponding value functions are also derived.

scholarly journals Dynkin games with heterogeneous beliefs

2017 ◽  
Vol 54 (1) ◽  
pp. 236-251 ◽  
Author(s):  
Erik Ekström ◽  
Kristoffer Glover ◽  
Marta Leniec
Keyword(s):  
Optimal Stopping ◽  
Nash Equilibria ◽  
Heterogeneous Beliefs ◽  
Stopping Times ◽  
Value Functions ◽  
Natural Conditions ◽  
Dynkin Games ◽  
Zero Sum ◽  
Stopping Games ◽  
The Given

AbstractWe study zero-sum optimal stopping games (Dynkin games) between two players who disagree about the underlying model. In a Markovian setting, a verification result is established showing that if a pair of functions can be found that satisfies some natural conditions then a Nash equilibrium of stopping times is obtained, with the given functions as the corresponding value functions. In general, however, there is no uniqueness of Nash equilibria, and different equilibria give rise to different value functions. As an example, we provide a thorough study of the game version of the American call option under heterogeneous beliefs. Finally, we also study equilibria in randomized stopping times.

Monotone stopping games

1987 ◽  
Vol 24 (02) ◽  
pp. 386-401 ◽  
Author(s):  
John W. Mamer
Keyword(s):  
Nash Equilibrium ◽  
Optimal Stopping ◽  
Nash Equilibria ◽  
Stopping Times ◽  
Optimal Stopping Problem ◽  
Optimal Stopping Problems ◽  
Zero Sum ◽  
Stopping Games

We consider the extension of optimal stopping problems to non-zero-sum strategic settings called stopping games. By imposing a monotone structure on the pay-offs of the game we establish the existence of a Nash equilibrium in non-randomized stopping times. As a corollary, we identify a class of games for which there are Nash equilibria in myopic stopping times. These games satisfy the strategic equivalent of the classical ‘monotone case' assumptions of the optimal stopping problem.

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.

scholarly journals Optimal Time-Consistent Investment Strategy for a Random Household Expenditure with Default Risk under Relative Performance

Complexity ◽  
2021 ◽  
Vol 2021 ◽  
pp. 1-23
Author(s):  
Wenjin Guan ◽  
Wei Yuan ◽  
Sheng Li
Keyword(s):  
Nash Equilibrium ◽  
Default Risk ◽  
Investment Strategy ◽  
Relative Performance ◽  
Equilibrium Strategy ◽  
Optimal Time ◽  
Household Expenditure ◽  
Value Functions ◽  
Nash Equilibrium Strategy ◽  
The Mind

Considering the mind of rivalry between families, each family focuses not only on its own wealth but also on other families, especially neighbors. In this paper, we investigate the non-zero-sum mean-variance game between two families with a random household expenditure under the default risk and relative performance. Applying the stochastic control theory within the framework of the game theory, the extended Hamilton–Jacobi–Bellman equation equations are derived. By solving this equation, we obtain the Nash equilibrium strategies of the two families and the corresponding equilibrium value functions. We also provide a numerical example to analyze the effects of relevant parameters on Nash equilibrium strategy and on the utility loss due to the mind of rivalry.

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