Bang–bang-type Nash equilibrium point for Markovian nonzero-sum stochastic differential game

Hamadène, Saïd; Mu, Rui

doi:10.1016/j.crma.2014.06.011

Cited by 17 publications

(16 citation statements)

References 27 publications

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“…To better understand the problem we start with an example where the feedback can be written in an explicit way. This problem was considered in [19] for bounded domains and in [13] in IR N . For the sake of brevity here in the following HJBI equation and also in the next section we denote by V y the derivative ∂V ∂y with respect to a generic variable y.…”

Section: Bounded Data and Discontinuous Hamiltonianmentioning

confidence: 99%

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Regularity of Nash payoffs of Markovian nonzero-sum stochastic differential games

Hamadène

Mannucci

2018

Stochastics

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In this paper we deal with the problem of existence of a smooth solution of the Hamilton-Jacobi-Bellman-Isaacs (HJBI for short) system of equations associated with nonzero-sum stochastic differential games. We consider the problem in unbounded domains either in the case of continuous generators or for discontinuous ones. In each case we show the existence of a smooth solution of the system. As a consequence, we show that the game has smooth Nash payoffs which are given by means of the solution of the HJBI system and the stochastic process which governs the dynamic of the controlled system. This article deals with a nonzero-sum stochastic differential game (NZSDG for short) which we describe hereafter. Let us consider a system, on which intervene two players π 1 and π 2 , whose dynamics is given by a solution of a stochastic differential equation of the following form:where: (i) B := (B t ) t≤T is a Brownian motion ;(ii) u 1 := (u 1t ) t≤T (resp. u 2 := (u 2t ) t≤T ) is a stochastic process with values in U 1 (resp. U 2 ) a compact metric space and adapted w.r.t (F t ) t≤T , the completed natural filtration of B. The process u 1 (resp. u 2 ) is the way by which the first (resp. second) player π 1 (resp. π 2 ) acts on the system ;(iii) f (·) and σ(·) are given functions.The system that one implies could be an asset in the financial market, an economic unit, a factor in the economic or financial spheres, etc. On the other hand, one can consider the differential game with more than two players and this does not rise a major issue, the treatment is the same.The conditional payoff of player π 1 (resp. π 2 ) from t to T , when she implements u 1 (resp. u 2 ), is denoted J 1 t (u 1 , u 2 ) (resp. J 2 t (u 1 , u 2 )) and given by: for i = 1, 2,The functions h 1 , h 2 (resp. g 1 , g 2 ) stand for the intantaneous (resp. terminal) payoffs of the players π 1 , π 2 , respectively. Then the problem of interest is to find a Nash equilibrium point for the game, i.e., a pair of controls of the players (u * 1 , u * 2 ) such that J 1 0 (

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Section: Bounded Data and Discontinuous Hamiltonianmentioning

confidence: 99%

“…In this section we want to study from the PDEs point of view the game studied in the paper [13] with probabilistic tools. We consider a stochastic game where the drift of the dynamics of the system is of type…”

Section: Bounded Data and Discontinuous Hamiltonianmentioning

confidence: 99%

Regularity of Nash payoffs of Markovian nonzero-sum stochastic differential games

Hamadène

Mannucci

2018

Stochastics

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“…To instead, we consider a drift f which is of linear growth on the state process x. This has already been considered in some classical game problems without recursive part by [9] and [10]. To our knowledge, this general recursive case has not been studied in literatures.…”

Section: Rui Mu and Zhen Wumentioning

confidence: 99%

Nash equilibrium points of recursive nonzero-sum stochastic differential games with unbounded coefficients and related multiple\\ dimensional BSDEs

Mu¹

2017

Mathematical Control &Amp; Related Fields

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This paper is concerned with recursive nonzero-sum stochastic differential game problem in Markovian framework when the drift of the state process is no longer bounded but only satisfies the linear growth condition. The costs of players are given by the initial values of related backward stochastic differential equations which, in our case, are multidimensional with continuous coefficients, whose generators are of linear growth on the volatility processes and stochastic monotonic on the value processes. We finally show the wellposedness of the costs and the existence of a Nash equilibrium point for the game under the generalized Isaacs assumption.

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“…There have been many works on the existence of Nash equilibriums, by using either PDE method or BSDE method, see e.g. Bensoussan & Frehse [3], Buckdahn, Cardaliaguet, & Rainer [4], Cardaliaguet & Plaskacz [5], El-Karoui & Hamadene [10], Friedman [14], Hamadene [15], Hamadene, Lepeltier, & Peng [16], Hamadene & Mannucci [17], Hamadene & Mu [18,19], Lin [22], Mannucci [25,26], Olsder [27], Rainer [30],…”

Section: Introductionmentioning

confidence: 99%

Dynamic Set Values for Nonzero Sum Games with Multiple Equilibriums

Feinstein

Rudloff

Zhang

2020

Preprint

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Nonzero sum games typically have multiple Nash equilibriums (or no equilibrium), and unlike the zero sum case, they may have different values at different equilibriums.Instead of focusing on the existence of individual equilibriums, we study the set of values over all equilibriums, which we call the set value of the game. The set value is unique by nature and always exists (with possible value ∅). Similar to the standard value function in control literature, it enjoys many nice properties such as regularity, stability, and more importantly the dynamic programming principle. There are two main features in order to obtain the dynamic programming principle: (i) we must use closed-loop controls (instead of open-loop controls); (ii) we must allow for path dependent controls, even if the problem is in a state dependent (Markovian) setting. We shall consider both discrete and continuous time models with finite time horizon. For the latter we will also provide a duality approach through certain standard PDE (or path dependent PDE), which is quite efficient for numerically computing the set value of the game.

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Bang–bang-type Nash equilibrium point for Markovian nonzero-sum stochastic differential game

Cited by 17 publications

References 27 publications

Regularity of Nash payoffs of Markovian nonzero-sum stochastic differential games

Regularity of Nash payoffs of Markovian nonzero-sum stochastic differential games

Nash equilibrium points of recursive nonzero-sum stochastic differential games with unbounded coefficients and related multiple\\ dimensional BSDEs

Dynamic Set Values for Nonzero Sum Games with Multiple Equilibriums

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