6,064 research outputs found
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
Sufficient stochastic maximum principle for the optimal control of semi-Markov modulated jump-diffusion with application to Financial optimization
The finite state semi-Markov process is a generalization over the Markov
chain in which the sojourn time distribution is any general distribution. In
this article we provide a sufficient stochastic maximum principle for the
optimal control of a semi-Markov modulated jump-diffusion process in which the
drift, diffusion and the jump kernel of the jump-diffusion process is modulated
by a semi-Markov process. We also connect the sufficient stochastic maximum
principle with the dynamic programming equation. We apply our results to finite
horizon risk-sensitive control portfolio optimization problem and to a
quadratic loss minimization problem.Comment: Forthcoming in Stochastic Analysis and Application
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