510 Mathematik
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The aim of this article is to establish a stochastic search algorithm for neural networks based on the fractional stochastic processes {๐ต๐ป๐ก,๐กโฅ0} with the Hurst parameter ๐ปโ(0,1). We define and discuss the properties of fractional stochastic processes, {๐ต๐ป๐ก,๐กโฅ0}, which generalize a standard Brownian motion. Fractional stochastic processes capture useful yet different properties in order to simulate real-world phenomena. This approach provides new insights to stochastic gradient descent (SGD) algorithms in machine learning. We exhibit convergence properties for fractional stochastic processes.
The aim of this work is to establish and generalize a relationship between fractional partial differential equations (fPDEs) and stochastic differential equations (SDEs) to a wider class of stochastic processes, including fractional Brownian motions and sub-fractional Brownian motions with Hurst parameter H โ (1/2,1). We start by establishing the connection between a fPDE and SDE via the Feynman-Kac Theorem, which provides a stochastic representation of a general Cauchy problem. In hindsight, we extend this connection by assuming SDEs with fractional and sub-fractional Brownian motions and prove the generalized Feynman-Kac formulas under a (sub-)fractional Brownian motion. An application of the theorem demonstrates, as a by-product, the solution of a fractional integral, which has relevance in probability theory.
This paper analyzes governance mechanisms for different group sizes. The European sovereign debt crisis has demonstrated the need of efficient governance for different group sizes. I find that self-governance only works for sufficiently homogenous and small neighbourhoods. Second, as long as the union expands, the effect of credible self-governance decreases. Third, spill-over effects amplify the size effect. Fourth, I show that sufficiently large monetary unions, are better off with costly but external governance or a free market mechanism. Finally, intermediate-size unions are most difficult to govern efficiently.
Applied mathematical theory for monetary-fiscal interaction in a supranational monetary union
(2014)
I utilize a differentiable dynamical system รก la Lotka-Voletrra and explain monetary and fiscal interaction in a supranational monetary union. The paper demonstrates an applied mathematical approach that provides useful insights about the interaction mechanisms in theoretical economics in general and a monetary union in particular. I find that a common central bank is necessary but not sufficient to tackle the new interaction problems in a supranational monetary union, such as the free-riding behaviour of fiscal policies. Moreover, I show that upranational institutions, rules or laws are essential to mitigate violations of decentralized fiscal policies.