ε-Nash equilibrium in stochastic differential games with mean-field interaction and controlled jumps

Abstract

We consider a symmetric n-player nonzero-sum stochastic differential game with jump-diffusion dynamics and mean-field type interaction among the players. Under the assumption of existence of a regular Markovian solution for the corresponding limiting mean-field game, we construct an approximate Nash equilibrium for the n-player game for n large enough, and provide the rate of convergence. This extends to a class of games with jumps classical results in mean-field game literature. This paper complements our previous work [2] on the existence of solutions of mean-field games for jump-diffusions.

Deep Learning and Mean-Field Games: A Stochastic Optimal Control Perspective

“…Following [13] (see also [14] and the references therein), let {Ω, F , (F t ), P} be a filtered probability space, with filtration F t = {F t , t ∈ [0, T]}, T > 0, supporting: • a controlled state variable (X α t ) t∈[0,T] , where X t is an i.i.d. sequence of R n -valued F 0 -measurable random variables; • a sequence {W i t } i≥1 of independent and F t -adapted Brownian motions.…”

“…where x 0 is the input, e.g., an image, several time-series, etc., while x T is the final output, to be compared to some target y T by means of a given loss function. By moving from a discrete time formulation to a continuous one, the forward dynamics we are interested in will be described by a differential equation that takes the role of (14). The learning aim is to tune the trainable parameters θ 0 , .…”

“…Concerning possible numerical analysis of the DL algorithm based on the maximum principle, we refer to [4] (pp. [13][14][15], where a comparison can be found between usual gradient approaches to the discrete formulation of the Mean-Field PMP stated in Theorem 2, with a loss function based on Equation (24). The test and train losses of some variants of SGD algorithms are compared to the mean-field algorithm based on the discrete PMP.…”

On fully nonlinear parabolic mean field games with examples of nonlocal and local diffusions

“…Mean field game problems involving Lévy operators [21,25,29,44], or fractional time derivatives [15], have recently been studied. Control of the diffusion is a rare and novel subject, mostly addressed by stochastic methods [6,5,55], but some analytical results can be found in [68]. Lately, fully nonlinear problems of mean field game type have also been considered in [2], but the setting and techniques are different from ours.…”

Dynamic contagion in a banking system with births and defaults

“…In contrast, endogenizing N creates multiple scaling alternatives which is one of the main foci of our work. Finally, we should also mention [BCDP17a,BCDP17b] who analyzed mean-field games with jump-diffusions, however again they only consider interaction in the jump sizes.…”