6,326 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
Open problems in Banach spaces and measure theory
We collect several open questions in Banach spaces, mostly related to measure
theoretic aspects of the theory. The problems are divided into five categories:
miscellaneous problems in Banach spaces (non-separable spaces,
compactness in Banach spaces, -null sequences in dual spaces),
measurability in Banach spaces (Baire and Borel -algebras, measurable
selectors), vector integration (Riemann, Pettis and McShane integrals), vector
measures (range and associated spaces) and Lebesgue-Bochner spaces
(topological and structural properties, scalar convergence)
Large Deviations for Multiscale Diffusions via Weak Convergence Methods
We study the large deviations principle for locally periodic stochastic
differential equations with small noise and fast oscillating coefficients.
There are three possible regimes depending on how fast the intensity of the
noise goes to zero relative to the homogenization parameter. We use weak
convergence methods which provide convenient representations for the action
functional for all three regimes. Along the way we study weak limits of related
controlled SDEs with fast oscillating coefficients and derive, in some cases, a
control that nearly achieves the large deviations lower bound at the prelimit
level. This control is useful for designing efficient importance sampling
schemes for multiscale diffusions driven by small noise
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