The Lagrange and the vanishing discount techniques to controlled diffusions with cost constraints

Constrained stochastic differential games with additive structure: Average and discount payoffs

“…Recently, in [35] nonzero-sum constrained stochastic games was studied under the average payoff criterion with denumerable state space and Borel action spaces. In [23], as well as, in [35], under suitable conditions, it proves the existence of stationary optimal policies and constrained stationary Nash equilibrium via the 110 ESCOBEDO-TRUJILLO, ALAFFITA-HERNÁNDEZ AND LÓPEZ-MARTÍNEZ vanishing discount approach, respectively. In this work we use the vanishing discount approach to prove that every sequence of constrained Nash equilibria with discounted payoff converges, in some appropriately defined sense, to a constrained Nash equilibrium with average payoff, [23,35].…”

“…Remark 2. (a) The Assumption 2 (a Lyapunov-like condition) guarantees, in particular, the positive recurrence of the diffusion x π 1 ,π 2 (•), as well as, the existence of a unique invariant probability measure which is key to prove the existence of constrained Nash equilibrium with average payoff, see, [5,9,10,14,16,17,19,23,24,28]. (b) Under Assumption 2, for each π 1 , π 2 ∈ Π 1 ×Π 2 , the Markov process…”

“…Although we restrict ourselves to the case where only two players can modify the system and two restrictions (one for each player), it's relatively easy to extend our results to a more general context where there are N players with this ability, or two players with N restrictions. Our approach in this paper is mainly based on [23].…”

“…The Lagrange technique studied in [23,27] characterized the discounted problem with constraints by means of an equivalent unconstrained problem using the Lagrange multipliers method so that optimality for the unconstrained yield optimality for the original one when a suitable multiplier is selected.…”

“…The stochastic process u k (•) represents the control actions of player k = 1, 2 at each time t ≥ 0. We impose the following standard assumptions to guarantee the existence and uniqueness of solutions to the stochastic differential equation (1), [5,9,10,14,16,17,23,24]. (e) ( Uniform ellipticity.)…”