Approximation via regularization of the local time of semimartingales and Brownian motion

Through a regularization procedure, few approximation schemes of the local time of a large class of one dimensional processes are given. We mainly consider the local time of continuous semimartingales and reversible diffusions, and the convergence holds in ucp sense. In the case of standard Brownian motion, we have been able to determine a rate of convergence in $L^2$, and a.s. convergence of some of our schemes.Comment: Accept\'e conditionnelement par Stochastic processes and their application

Quelques approximations du temps local brownien

We give some approximations of the local time process $(L_t^x)_{t\geqslant 0}$ at level $x$ of the real Brownian motion $(X_t)$. We prove that \frac{2}{\epsilon}\int_0^{t} X_{(u+\epsilon)\wedge t}^+ \indi_{\{X_u \leqslant 0\}} du + \frac{2}{\epsilon}\int_0^{t} X_{(u+\epsilon) \wedge t}^- \indi_{\{X_u>0\}} du and \frac{4}{\epsilon}\int_0^{t} X_u^- \indi_{\{X_{(u+\epsilon) \wedge t} > 0\}} du converge in the ucp sense to $L_t^0$, as $\epsilon \to 0$. We show that \frac{1}{\epsilon}\int_0^t (\indi_{\{x goes to $L_t^x$ in $L^2(\Omega)$ as $\epsilon \to 0$, and that the rate of convergence is of order $\epsilon^\alpha$, for any $\alpha < {1/4}$.Comment: Soumis dans les Comptes rendus - Math\'ematiqu

Mathematical model for resistance and optimal strategy

We propose a mathematical model for one pattern of charts studied in technical analysis: in a phase of consolidation, the price of a risky asset goes down $\xi$ times after hitting a resistance level. We construct a mathematical strategy and we calculate the expectation of the wealth for the logaritmic utility function. Via simulations, we compare the strategy with the standard one

Quelques approximations du temps local brownien

National audienceWe give some approximations of the local time process $(L_t^x)_{t\geqslant 0}$ at level $x$ of the real Brownian motion $(X_t)$. We prove that \frac{2}{\epsilon}\int_0^{t} X_{(u+\epsilon)\wedge t}^+ \indi_{ \{X_u \leqslant 0\} } du + \frac{2}{\epsilon}\int_0^{t} X_{(u+\epsilon) \wedge t}^- \indi_{ \{X_u>0\} } du and \frac{4}{\epsilon}\int_0^{t} X_u^- \indi_{ \{X_{(u+\epsilon) \wedge t} > 0\} } du converge in the ucp sense to $L_t^0$, as $\epsilon \to 0$. We show that $ \frac{1}{\epsilon}\int_0^t ( \indi_{\{

On Spin(7) holonomy metric based on SU(3)/U(1)

We investigate the $Spin(7)$ holonomy metric of cohomogeneity one with the principal orbit $SU(3)/U(1)$. A choice of U(1) in the two dimensional Cartan subalgebra is left as free and this allows manifest $\Sigma_3=W(SU(3))$ (= the Weyl group) symmetric formulation. We find asymptotically locally conical (ALC) metrics as octonionic gravitational instantons. These ALC metrics have orbifold singularities in general, but a particular choice of the U(1) subgroup gives a new regular metric of $Spin(7)$ holonomy. Complex projective space ${\bf CP}(2)$ that is a supersymmetric four-cycle appears as a singular orbit. A perturbative analysis of the solution near the singular orbit shows an evidence of a more general family of ALC solutions. The global topology of the manifold depends on a choice of the U(1) subgroup. We also obtain an $L^2$-normalisable harmonic 4-form in the background of the ALC metric.Comment: 21 pages, Latex, Introduction slightly expanded, an error in section 6 corrected and references added, (v3) minor correction

Brane Resolution Through Fibration

We consider p-branes with one or more circular directions fibered over the transverse space. The fibration, in conjunction with the transverse space having a blown-up cycle, enables these p-brane solutions to be completely regular. Some such circularly-wrapped D3-brane solutions describe flows from SU(N)^3 N=2 theory, F_0 theory, as well as an infinite family of superconformal quiver gauge theories, down to three-dimensional field theories. We discuss the operators that are turned on away from the UV fixed points. Similarly, there are wrapped M2-brane solutions which describe smooth flows from known three-dimensional supersymmetric Chern-Simons matter theories, such as ABJM theory. We also consider p-brane solutions on gravitational instantons, and discuss various ways in which U-duality can be applied to yield other non-singular solutions.Comment: 35 pages, additional referenc

Resolutions of Cones over Einstein-Sasaki Spaces

Recently an explicit resolution of the Calabi-Yau cone over the inhomogeneous five-dimensional Einstein-Sasaki space Y^{2,1} was obtained. It was constructed by specialising the parameters in the BPS limit of recently-discovered Kerr-NUT-AdS metrics in higher dimensions. We study the occurrence of such non-singular resolutions of Calabi-Yau cones in a more general context. Although no further six-dimensional examples arise as resolutions of cones over the L^{pqr} Einstein-Sasaki spaces, we find general classes of non-singular cohomogeneity-2 resolutions of higher-dimensional Einstein-Sasaki spaces. The topologies of the resolved spaces are of the form of an R^2 bundle over a base manifold that is itself an $S^2$ bundle over an Einstein-Kahler manifold.Comment: Latex, 23 page

Spectral geometry, homogeneous spaces, and differential forms with finite Fourier series

Let G be a compact Lie group acting transitively on Riemannian manifolds M and N. Let p be a G equivariant Riemannian submersion from M to N. We show that a smooth differential form on N has finite Fourier series if and only if the pull back has finite Fourier series on

A G_2 Unification of the Deformed and Resolved Conifolds

We find general first-order equations for G_2 metrics of cohomogeneity one with S^3\times S^3 principal orbits. These reduce in two special cases to previously-known systems of first-order equations that describe regular asymptotically locally conical (ALC) metrics \bB_7 and \bD_7, which have weak-coupling limits that are S^1 times the deformed conifold and the resolved conifold respectively. Our more general first-order equations provide a supersymmetric unification of the two Calabi-Yau manifolds, since the metrics \bB_7 and \bD_7 arise as solutions of the {\it same} system of first-order equations, with different values of certain integration constants. Additionally, we find a new class of ALC G_2 solutions to these first-order equations, which we denote by \wtd\bC_7, whose topology is an \R^2 bundle over T^{1,1}. There are two non-trivial parameters characterising the homogeneous squashing of the T^{1,1} bolt. Like the previous examples of the \bB_7 and \bD_7 ALC metrics, here too there is a U(1) isometry for which the circle has everywhere finite and non-zero length. The weak-coupling limit of the \wtd\bC_7 metrics gives S^1 times a family of Calabi-Yau metrics on a complex line bundle over S^2\times S^2, with an adjustable parameter characterising the relative sizes of the two S^2 factors.Comment: Latex, 14 pages, Major simplification of first-order equations; references amende

Pseudo-Riemannian manifolds with recurrent spinor fields

The existence of a recurrent spinor field on a pseudo-Riemannian spin manifold $(M,g)$ is closely related to the existence of a parallel 1-dimensional complex subbundle of the spinor bundle of $(M,g)$. We characterize the following simply connected pseudo-Riemannian manifolds admitting such subbundles in terms of their holonomy algebras: Riemannian manifolds; Lorentzian manifolds; pseudo-Riemannian manifolds with irreducible holonomy algebras; pseudo-Riemannian manifolds of neutral signature admitting two complementary parallel isotropic distributions.Comment: 13 pages, the final versio