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.
We study a family of mean field games with a state variable evolving as a multivariate jump diffusion process. The jump component is driven by a Poisson process with a time-dependent intensity function. All coefficients, i.e. drift, volatility and jump size, are controlled. Under fairly general conditions, we establish existence of a solution in a relaxed version of the mean field game and give conditions under which the optimal strategies are in fact Markovian, hence extending to a jump-diffusion setting previous results established in [30]. The proofs rely upon the notions of relaxed controls and martingale problems. Finally, to complement the abstract existence results, we study a simple illiquid inter-bank market model, where the banks can change their reserves only at the jump times of some exogenous Poisson processes with a common constant intensity, and provide some numerical results.
We show via the nonlinear semigroup theory in L 1 (R) that the 1-D dynamic programming equation associated with a stochastic optimal control problem with multiplicative noise has a unique mild solution ϕ ∈ C([0, T ]; W 1,∞ (R)) with ϕ xx ∈ C([0, T ]; L 1 (R)). The n-dimensional case is also investigated.
We prove the existence of an optimal feedback controller for a stochastic optimization problem constituted by a variation of the Heston model, where a stochastic input process is added in order to minimize a given performance criterion. The stochastic feedback controller is searched by solving a nonlinear backward parabolic equation for which one proves the existence of a martingale solution.
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