On the equality of solutions of max–min and min–max systems of variational inequalities with interconnected bilateral obstacles

Abstract

In this paper, we deal with the solutions of systems of PDEs with bilateral inter-connected obstacles of minmax and max-min types. These systems arise naturally in stochastic switching zero-sum game problems. We show that when the switching costs of one side are regular, the solutions of the min-max and max-min systems coincide. Then, this common viscosity solution is related to a multi-dimensional doubly reflected BSDE with bilateral interconnected obstacles. Finally, its relationship with the the values of a zero-sum switching game is studied.

A Penalty Scheme for Monotone Systems with Interconnected Obstacles: Convergence and Error Estimates

“…To the best of our knowledge, this is the first paper which proposes penalty approximations for QVIs in such a generality and presents rigorous error estimates for the penalization errors. Natural next steps would be to extend the penalty approach to interconnected obstacles with negative switching costs as in [21], to more general intervention operators as in [2], and to monotone systems with interconnected bilateral obstacles as in [10].…”

“…Following [10], we will refer to Mu as an interconnected obstacle, where M is a special form of the intervention operator in [2]. Similar to [8] in the continuous context, we shall also refer to a continuous function F = (F i ) i∈I with condition (1.2) as a monotone system.…”

A probabilistic verification theorem for the finite horizon two-player zero-sum optimal switching game in continuous time

“…Multipleperson optimal switching problems in a continuous-time stochastic setting, the topic under which the present work falls, have been studied less frequently in the literature (there is related work for deterministic systems such as [32] and [33]). In the two-player zero-sum game of optimal switching, which is the setting of the present work, there are works such as [12], [18], [21], and [30], and related studies for impulse control games [10], [29].…”

Viscosity Solutions of Systems of Variational Inequalities with Interconnected Bilateral Obstacles of Non-Local Type

“…In these latter articles, the Hamilton-Jacobi-Bellmans-Isaacs PDE, which is of min-max or max-min type, associated with the zero-sum switching stochastic game is studied from the point of view of viscosity solution theory. The probabilistic version of those works is considered in [10,23] where it is shown that the BSDE system associated with the zero-sum game has a solution. In [10], uniqueness of the solution, which is an involved question, is proved as well.…”

“…in the energy market, jumps of the prices due to sudden weather changes, etc. Therefore the main objective of this work is the extension to the model with jumps of the paper [10], where the authors have studied systems of variational inequalities with interconnected lower and upper obstacles, which arise as the Hamilton-Jacobi-Bellman-Isaacs equation in a multiple modes switching game between two players in the framework without jumps. Precisely we consider the following system of non-local variational inequalities or integral-partial differential equations (IPDEs for short): For every pair (i, j) in the finite set of modes A 1 × A 2 ,…”