Mean field games with controlled jump-diffusion dynamics: Existence results and an illiquid interbank market model

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.Comment: 37 pages, 6 figure

Poisson's equation for discrete-time quasi-birth-and-death processes

We consider Poisson's equation for quasi-birth-and-death processes (QBDs) and we exploit the special transition structure of QBDs to obtain its solutions in two different forms. One is based on a decomposition through first passage times to lower levels, the other is based on a recursive expression for the deviation matrix. We revisit the link between a solution of Poisson's equation and perturbation analysis and we show that it applies to QBDs. We conclude with the PH/M/1 queue as an illustrative example, and we measure the sensitivity of the expected queue size to the initial value

Rate control of a queue with quality-of-service constraint under bounded and unbounded action spaces

We consider a simple Markovian queue with Poisson arrivals and exponential service times for jobs. The controller can choose service rates from a specified action space depending on number of jobs in the queue. The queue has a finite buffer and when full, new jobs get rejected. The controller’s objective is to choose optimal (state-dependent) service rates that minimize a suitable long-run average cost, subject to an upper bound on the job rejection-rate (quality-of-service constraint). We solve this problem of finding and computing the optimal control under two cases: When the action space is unbounded (i.e. [0, ∞)) and when it is bounded (i.e. [0, μ ̄], for some μ ̄ \u3e 0). We also numerically compute and compare the solutions for different specific choices of the cost function

Risk-sensitive optimal control for Markov decision processes with monotone cost

The existence of an optimal feedback law is established for the risk-sensitive optimal control problem with denumerable state space. The main assumptions imposed are irreducibility and anear monotonicity condition on the one-step cost function. A solution can be found constructively using either value iteration or policy iteration under suitable conditions on initial feedback law