Equilibrium in Functional Stochastic Games with Mean-Field Interaction
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2023
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866914679425597440 |
|---|---|
| author | Jaber, Eduardo Abi Neuman, Eyal Voß, Moritz |
| author_facet | Jaber, Eduardo Abi Neuman, Eyal Voß, Moritz |
| contents | We consider a general class of finite-player stochastic games with mean-field interaction, in which the linear-quadratic cost functional includes linear operators acting on controls in $L^2$. We propose a novel approach for deriving the Nash equilibrium of the game semi-explicitly in terms of operator resolvents, by reducing the associated first order conditions to a system of stochastic Fredholm equations of the second kind and deriving their solution in semi-explicit form. Furthermore, by proving stability results for the system of stochastic Fredholm equations, we derive the convergence of the equilibrium of the $N$-player game to the corresponding mean-field equilibrium. As a by-product, we also derive an $\varepsilon$-Nash equilibrium for the mean-field game, which is valuable in this setting as we show that the conditions for existence of an equilibrium in the mean-field limit are less restrictive than in the finite-player game. Finally, we apply our general framework to solve various examples, such as stochastic Volterra linear-quadratic games, models of systemic risk and advertising with delay, and optimal liquidation games with transient price impact. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_05433 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Equilibrium in Functional Stochastic Games with Mean-Field Interaction Jaber, Eduardo Abi Neuman, Eyal Voß, Moritz Optimization and Control Probability Mathematical Finance 49N80, 49N90, 93E20, 91G80 We consider a general class of finite-player stochastic games with mean-field interaction, in which the linear-quadratic cost functional includes linear operators acting on controls in $L^2$. We propose a novel approach for deriving the Nash equilibrium of the game semi-explicitly in terms of operator resolvents, by reducing the associated first order conditions to a system of stochastic Fredholm equations of the second kind and deriving their solution in semi-explicit form. Furthermore, by proving stability results for the system of stochastic Fredholm equations, we derive the convergence of the equilibrium of the $N$-player game to the corresponding mean-field equilibrium. As a by-product, we also derive an $\varepsilon$-Nash equilibrium for the mean-field game, which is valuable in this setting as we show that the conditions for existence of an equilibrium in the mean-field limit are less restrictive than in the finite-player game. Finally, we apply our general framework to solve various examples, such as stochastic Volterra linear-quadratic games, models of systemic risk and advertising with delay, and optimal liquidation games with transient price impact. |
| title | Equilibrium in Functional Stochastic Games with Mean-Field Interaction |
| topic | Optimization and Control Probability Mathematical Finance 49N80, 49N90, 93E20, 91G80 |
| url | https://arxiv.org/abs/2306.05433 |