arXiv subjects

Davide Francesco Redaelli

Publications and source records attributed to Davide Francesco Redaelli.

6 recordsLinked to original sources

arXiv2026

On the unimportance of distant players in sparse stochastic differential network games

We study stochastic differential games with $N$ players, where interactions are determined by sequences of graphs in which the number of neighbours of each node remains bounded as $N$ grows, such as chain graphs or lattices. Our main goal is to quantify the phenomenon of the "unimportance of distant players" in such a large population, sparse regime: we show that, in order to determine the optimal trajectory in open-loop strategies of a given player with an arbitrarily small error, it suffices to consider a reduced game involving only the players at a certain distance in the graph, assigning arbitrary trajectories to the farther ones. Our main result provides an explicit non-asymptotic estimate in terms of the graph distance, valid independently of the time horizon $T$, under suitable convexity and monotonicity assumptions on the costs. Similar results are obtained for games in distributed strategies.

math.AP↗

arXiv2025

Short-time existence and uniqueness for some infinite-dimensional Nash systems

We prove local (in time) existence and uniqueness for a class of infinite-dimensional Nash systems, namely systems of infinitely many Hamilton-Jacobi-Bellman equations set in an infinite-dimensional Euclidean space. Such systems have been recently showed (see arXiv:2401.06534) to arise in the theory of stochastic differential games with interactions governed by sparse graphs, under structural assumptions that inspired the hypotheses exploited in the present work. Contextually, we also prove a general linear result, providing a priori estimates, stable with respect to the dimension, for transport-diffusion equations whose drifts (and their derivatives) enjoy appropriate decay properties.

math.AP↗

arXiv2025

A priori estimates and large population limits for some nonsymmetric Nash systems with semimonotonicity

We address the problem of regularity of solutions $u^i(t, x^1, \dots, x^N)$ to a family of semilinear parabolic systems of $N$ equations, which describe closed-loop equilibria of some $N$-player differential games with Lagrangian having quadratic behaviour in the velocity variable, running costs $f^i(x)$ and final costs $g^i(x)$. By global (semi)monotonicity assumptions on the data $f=(f^i)_{1 \leq i \leq N}$ and $g=(g^i)_{1 \leq i \leq N}$, and assuming that derivatives of $f^i, g^i$ in directions $x^j$ are of order $1/N$ for $j \neq i$, we prove that derivatives of $u^i$ enjoy the same property. The estimates are uniform in the number of players $N$. Such a behaviour of the derivatives of $f^i, g^i$ arise in the theory of Mean Field Games, though here we do not make any symmetry assumption on the data. Then, by the estimates obtained we address the convergence problem $N \to \infty$ in a "heterogeneous'' Mean Field framework, where players all observe the empirical measure of the whole population, but may react differently from one another. We also discuss some results on the joint $N \to \infty$ and vanishing viscosity limit.

math.AP↗

arXiv2025

A non-asymptotic approach to stochastic differential games with many players under semi-monotonicity

We consider stochastic differential games with a large number of players, with the aim of quantifying the gap between closed-loop, open-loop and distributed equilibria. We show that, under two different semi-monotonicity conditions, the equilibrium trajectories are close when the interactions between the players are weak. Our approach is non-asymptotic in nature, in the sense that it does not make use of any a priori identification of a limiting model, like in mean field game (MFG) theory. The main technical step is to derive bounds on solutions to systems of PDE/FBSDE characterizing the equilibria that are independent of the number of players. When specialized to the mean field setting, our estimates yield quantitative convergence results for both open-loop and closed-loop equilibria without any use of the master equation. In fact, our main bounds hold for games in which interactions are much sparser than those of MFGs, and so we can also obtain some "universality" results for MFGs, in which we show that games governed by dense enough networks converge to the usual MFG limit. Finally, we use our estimates to study a joint vanishing viscosity and large population limit in the setting of displacement monotone games without idiosyncratic noise.

math.PR↗

arXiv2025

A primer on semiconvex functions in general potential theories

This work is dedicated to foundational aspects of general (nonlinear second order) potential theories and fully nonlinear elliptic PDEs. In particular, we systematically develop the fundamental role played by semiconvex functions as a bridge between the classical and viscosity theory of generalized subharmonics determined by a given subequation constraint set in the bundle of $2$-jets over open subsets of Euclidian spaces. The first part is dedicated to four fundamental and complementary aspects of semiconvex analysis for the viscosity theory of subharmonic functions in general potential theories: differentiability properties of first and second order for locally semiconvex functions; a detailed analysis of their upper contact jets and upper contact points; the deep analytical results which ensure the existence of sets of contact points with positive measure; the well-known device of semiconvex approximation of upper semicontinuous functions. The second part is dedicated to general potential theories. The fundamental roles of duality and monotonicity are discussed, along with the main tools in the viscosity theory of subharmonics and the essential role that semiconvex functions play in that theory. The validity of the comparison principle is examined in two distinct regimes of sufficient monotonicity and minimal monotonicity.

math.AP↗

arXiv2024

Some remarks on Linear-quadratic closed-loop games with many players

We identify structural assumptions which provide solvability of the Nash system arising from a linear-quadratic closed-loop game, with stable properties with respect to the number of players. In a setting of interactions governed by a sparse graph, both short-time and long-time existence of a classical solution for the Nash system set in infinitely many dimensions are addressed, as well as convergence to the solution to the respective ergodic problem as the time horizon goes to infinity; in addition, equilibria for the infinite-dimensional game are shown to provide $ε$-Nash closed-loop equilibria for the $N$-player game. In a setting of generalized mean-field type (where the number of interactions is large but not necessarily symmetric), directly from the $N$-player Nash system estimates on the value functions are deduced on an arbitrary large time horizon, which should pave the way for a convergence result as $N$ goes to infinity.

math.OC↗