5,696 research outputs found
Solving generic nonarchimedean semidefinite programs using stochastic game algorithms
A general issue in computational optimization is to develop combinatorial
algorithms for semidefinite programming. We address this issue when the base
field is nonarchimedean. We provide a solution for a class of semidefinite
feasibility problems given by generic matrices. Our approach is based on
tropical geometry. It relies on tropical spectrahedra, which are defined as the
images by the valuation of nonarchimedean spectrahedra. We establish a
correspondence between generic tropical spectrahedra and zero-sum stochastic
games with perfect information. The latter have been well studied in
algorithmic game theory. This allows us to solve nonarchimedean semidefinite
feasibility problems using algorithms for stochastic games. These algorithms
are of a combinatorial nature and work for large instances.Comment: v1: 25 pages, 4 figures; v2: 27 pages, 4 figures, minor revisions +
benchmarks added; v3: 30 pages, 6 figures, generalization to non-Metzler sign
patterns + some results have been replaced by references to the companion
work arXiv:1610.0674
Modeling Financial System with Interbank Flows, Borrowing, and Investing
In our model, private actors with interbank cash flows similar to, but nore
general than (Carmona, Fouque, Sun, 2013) borrow from the outside economy at a
certain interest rate, controlled by the central bank, and invest in risky
assets. Each private actor aims to maximize its expected terminal logarithmic
utility. The central bank, in turn, aims to control the overall economy by
means of an exponential utility function. We solve all stochastic optimal
control problems explicitly. We are able to recreate occasions such as
liquidity trap. We study distribution of the number of defaults (net worth of a
private actor going below a certain threshold).Comment: 27 pages, 29 figures. Keywords: systemic risk, stochastic control,
principal-agent problem, stationary distribution, stochastic stability,
Lyapunov functio
A survey of random processes with reinforcement
The models surveyed include generalized P\'{o}lya urns, reinforced random
walks, interacting urn models, and continuous reinforced processes. Emphasis is
on methods and results, with sketches provided of some proofs. Applications are
discussed in statistics, biology, economics and a number of other areas.Comment: Published at http://dx.doi.org/10.1214/07-PS094 in the Probability
Surveys (http://www.i-journals.org/ps/) by the Institute of Mathematical
Statistics (http://www.imstat.org
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