Abstract
We study a toy model of linear-quadratic mean field game with delay. We “lift” the delayed dynamic into an infinite dimensional space, and recast the mean field game system which is made of a forward Kolmogorov equation and a backward Hamilton-Jacobi-Bellman equation. We identify the corresponding master equation. A solution to this master equation is computed, and we show that it provides an approximation to a Nash equilibrium of the finite player game.
Highlights
A linear quadratic stochastic game model of inter-bank borrowing and lending was proposed in (Carmona et al 2015)
We study the mean field game (MFG) corresponding to the model proposed in (Carmona et al 2018) as the number of banks goes to infinity
We identify the mean field game system, which is a system of coupled partial differential equations (PDEs)
Summary
A linear quadratic stochastic game model of inter-bank borrowing and lending was proposed in (Carmona et al 2015). The finding is that, in equilibrium, the central bank acts as a clearing house providing liquidity, and stability is enhanced This model was extended in (Carmona et al 2018), where a delay in the controls was introduced. We study the mean field game (MFG) corresponding to the model proposed in (Carmona et al 2018) as the number of banks goes to infinity. The master equation for our delayed mean field game is derived, a solution is given explicitly, and we show that it is the limit of the closed-loop Nash equilibrium of the N-player game system as N → ∞.
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