8,520 research outputs found
A spectral-based numerical method for Kolmogorov equations in Hilbert spaces
We propose a numerical solution for the solution of the
Fokker-Planck-Kolmogorov (FPK) equations associated with stochastic partial
differential equations in Hilbert spaces.
The method is based on the spectral decomposition of the Ornstein-Uhlenbeck
semigroup associated to the Kolmogorov equation. This allows us to write the
solution of the Kolmogorov equation as a deterministic version of the
Wiener-Chaos Expansion. By using this expansion we reformulate the Kolmogorov
equation as a infinite system of ordinary differential equations, and by
truncation it we set a linear finite system of differential equations. The
solution of such system allow us to build an approximation to the solution of
the Kolmogorov equations. We test the numerical method with the Kolmogorov
equations associated with a stochastic diffusion equation, a Fisher-KPP
stochastic equation and a stochastic Burgers Eq. in dimension 1.Comment: 28 pages, 10 figure
Stability of real parametric polynomial discrete dynamical systems
We extend and improve the existing characterization of the dynamics of
general quadratic real polynomial maps with coefficients that depend on a
single parameter , and generalize this characterization to cubic real
polynomial maps, in a consistent theory that is further generalized to real
-th degree real polynomial maps. In essence, we give conditions for the
stability of the fixed points of any real polynomial map with real fixed
points. In order to do this, we have introduced the concept of Canonical
Polynomial Maps which are topologically conjugate to any polynomial map of the
same degree with real fixed points. The stability of the fixed points of
canonical polynomial maps has been found to depend solely on a special function
termed Product Position Function for a given fixed point. The values of this
product position determine the stability of the fixed point in question, when
it bifurcates, and even when chaos arises, as it passes through what we have
termed stability bands. The exact boundary values of these stability bands are
yet to be calculated for regions of type greater than one for polynomials of
degree higher than three.Comment: 23 pages, 4 figures, now published in Discrete Dynamics in Nature and
Societ
Local induction and provably total computable functions
Let IŠâ
2 denote the fragment of Peano Arithmetic obtained by restricting the
induction scheme to parameter free Š2 formulas. Answering a question of R.
Kaye, L. Beklemishev showed that the provably total computable functions
of IŠâ
2 are, precisely, the primitive recursive ones. In this work we give a new
proof of this fact through an analysis of certain local variants of induction
principles closely related to IŠâ
2 . In this way, we obtain a more direct answer
to Kayeâs question, avoiding the metamathematical machinery (reflection
principles, provability logic,...) needed for Beklemishevâs original proof.
Our methods are modelâtheoretic and allow for a general study of IŠâ
n+1
for all n ž 0. In particular, we derive a new conservation result for these
theories, namely that IŠâ
n+1 is Šn+2âconservative over I§n for each n ž 1.Ministerio de Ciencia e InnovaciĂłn MTM2008â06435Ministerio de Ciencia e InnovaciĂłn MTM2011â2684
Local Induction and Provably Total Computable Functions: A Case Study
Let IÎ â2 denote the fragment of Peano Arithmetic obtained by restricting the induction scheme to parameter free Î 2 formulas. Answering a question of R. Kaye, L. Beklemishev showed that the provably total computable functions (p.t.c.f.) of IÎ â2 are, precisely, the primitive recursive ones. In this work we give a new proof of this fact through an analysis of the p.t.c.f. of certain local versions of induction principles closely related to IÎ â2 . This analysis is essentially based on the equivalence between local induction rules and restricted forms of iteration. In this way, we obtain a more direct answer to Kayeâs question, avoiding the metamathematical machinery (reflection principles, provability logic,...) needed for Beklemishevâs original proof.Ministerio de Ciencia e InnovaciĂłn MTM2008â0643
Market and Funding Liquidity Stress Testing of the Luxembourg Banking Sector
This paper performs market and funding liquidity stress testing of the Luxembourg banking sector using stochastic haircuts and run-off rates. It takes into account not only the shocks to the banking sector and banks? responses to them, but second-round effects due to the effects of banks? reactions on asset prices and reputation. In general, banks? business lines and, therefore their buffers? composition, determine the net effect of the shocks on banks? stochastic liquidity buffers. So, results differ across banks. Second-round effects exemplify the relevance of contagion effects that reduce the systemic benefits of diversification. While systemic liquidity risk is low following a shock to the interbank market, for Luxembourg, with its high number of subsidiaries of large foreign financial institutions, the results indicate the importance of monitoring the liquidity of parent groups to which Luxembourg institutions belong. In particular, shocks to related-party deposits are important. Finally, the results, including those of a run-on-deposits shock, show the relevance of system-wide measures to minimize the systemic effects of liquidity crises.stress test, liquidity risk, banks, stochastic, contagion, macro-prudential
Gravitational perturbations of the Higgs field
We study the possible effects of classical gravitational backgrounds on the
Higgs field through the modifications induced in the one-loop effective
potential and the vacuum expectation value of the energy-momentum tensor. We
concentrate our study on the Higgs self-interaction contribution in a perturbed
FRW metric. For weak and slowly varying gravitational fields, a complete set of
mode solutions for the Klein-Gordon equation is obtained to leading order in
the adiabatic approximation. Dimensional regularization has been used in the
integral evaluation and a detailed study of the integration of nonrational
functions in this formalism has been presented. As expected, the regularized
effective potential contains the same divergences as in flat spacetime, which
can be renormalized without the need of additional counterterms. We find that,
in contrast with other regularization methods, even though metric perturbations
affect the mode solutions, they do not contribute to the leading adiabatic
order of the potential. We also obtain explicit expressions of the complete
energy-momentum tensor for general nonminimal coupling in terms of the
perturbed modes. The corresponding leading adiabatic contributions are also
obtained.Comment: 15 pages. Version accepted for publication in PRD. Error corrected in
the angular integration in Appendix B. Conclusions changed. New section
include
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