Two-Interval Sturm–Liouville Theory for the Electrostatic Potential of a Point Charge near a Dielectric Cone

2010 ◽  
Vol 70 (7) ◽  
pp. 2329-2352 ◽  
Author(s):  
C. A. Liondas ◽  
D. P. Chrissoulidis
Keyword(s):  
Electrostatic Potential ◽  
Point Charge ◽  
Liouville Theory ◽  
Sturm Liouville

scholarly journals Spectral study of options based on CEV model with multidimensional volatility

2018 ◽  
Vol 15 (1) ◽  
pp. 18-25 ◽  
Author(s):  
Ivan Burtnyak ◽  
Anna Malytska
Keyword(s):  
Stochastic Volatility ◽  
Liouville Theory ◽  
Cev Model ◽  
Derivatives Pricing ◽  
Poisson Equations ◽  
The Cauchy Problem ◽  
Sturm Liouville ◽  
Explicit Formulae ◽  
Adjoint Operators ◽  
Local Variable

This article studies the derivatives pricing using a method of spectral analysis, a theory of singular and regular perturbations. Using a risk-neutral assessment, the authors obtain the Cauchy problem, which allows to calculate the approximate price of derivative assets and their volatility based on the diffusion equation with fast and slow variables of nonlocal volatility, and they obtain a model with multidimensional stochastic volatility. Applying a spectral theory of self-adjoint operators in Hilbert space and a theory of singular and regular perturbations, an analytic formula for approximate asset prices is established, which is described by the CEV model with stochastic volatility dependent on l-fast variables and r-slowly variables, l ≥ 1, r ≥ 1, l ∈ N, r ∈ N and a local variable. Applying the Sturm-Liouville theory, Fredholm’s alternatives, as well as the analysis of singular and regular perturbations at different time scales, the authors obtained explicit formulas for derivatives price approximations. To obtain explicit formulae, it is necessary to solve 2l Poisson equations.

Preliminaries

1996 ◽  
Author(s):  
Wolfgang Schmickler
Keyword(s):  
Charged Particle ◽  
Surface Science ◽  
Electrostatic Potential ◽  
Point Charge ◽  
Chemical Interaction ◽  
Dipole Moments ◽  
Great Success ◽  
Definition Of ◽  
Electrified Interfaces ◽  
Electrochemical Interfaces

In this chapter we introduce and discuss a number of concepts that are commonly used in the electrochemical literature and in the remainder of this book. In particular we will illuminate the relation of electrochemical concepts to those used in related disciplines. Electrochemistry has much in common with surface science, which is the study of solid surfaces in contact with a gas phase or, more commonly, with ultrahigh vacuum (uhv). A number of surface science techniques has been applied to electrochemical interfaces with great success. Conversely, surface scientists have become attracted to electrochemistry because the electrode charge (or equivalently the potential) is a useful variable which cannot be well controlled for surfaces in uhv. This has led to a laudable attempt to use similar terminologies for these two related sciences, and to introduce the concepts of the absolute scale of electrochemical potentials and the Fermi level of a redox reaction into electrochemistry. Unfortunately, there is some confusion of these terms in the literature, even though they are quite simple. Electrochemical interfaces are sometimes referred to as electrified interfaces, meaning that potential differences, charge densities, dipole moments, and electric currents occur. It is obviously important to have a precise definition of the electrostatic potential of a phase. There are two different concepts. The outer or Volta potential ψα of the phase a is the work required to bring a unit point charge from infinity to a point just outside the surface of the phase. By "just outside" we mean a position very close to the surface, but so far away that the image interaction with the phase can be ignored; in practice, that means a distance of about 10-5 — 10-3 cm from the surface. Obviously, the outer potential ψα is a measurable quantity. In contrast, the inner or Galvani potential ϕα is defined as the work required to bring a unit point charge from infinity to a point inside the phase α; so this is the electrostatic potential which is actually experienced by a charged particle inside the phase. Unfortunately, the inner potential cannot be measured: If one brings a real charged particle - as opposed to a point charge - into the phase, additional work is required due to the chemical interaction of this particle with other particles in the phase. For example, if one brings an electron into a metal, one has to do not only electrostatic work, but also work against the exchange and correlation energies.

scholarly journals Examining the Feasibility of the Sturm–Liouville Theory for Ross Recovery

Mathematics ◽  
2020 ◽  
Vol 8 (4) ◽  
pp. 550
Author(s):  
Shinmi Ahn ◽  
Hyungbin Park
Keyword(s):  
Markov Process ◽  
Limit Point ◽  
Integrability Condition ◽  
Liouville Theory ◽  
Physical Measure ◽  
State Variable ◽  
Risk Neutral ◽  
Market State ◽  
Positive Eigenfunction ◽  
Sturm Liouville

Recent studies have suggested that it is feasible to recover a physical measure from a risk-neutral measure. Given a market state variable modeled as a Markov process, the key concept is to extract a unique positive eigenfunction of the generator of the Markov process. In this work, the feasibility of this recovery theory is examined. We prove that, under a restrictive integrability condition, recovery is feasible if and only if both endpoints of the state variable are limit-point. Several examples with explicit positive eigenfunctions are considered. However, in general, a physical measure cannot be recovered from a risk-neutral measure. We provide a financial and mathematical rationale for such recovery failure.

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