Application of the Fractional Sturm–Liouville Theory to a Fractional Sturm–Liouville Telegraph Equation

2021 ◽  
Vol 15 (5) ◽  
Author(s):  
M. Ferreira ◽  
M. M. Rodrigues ◽  
N. Vieira
Keyword(s):  
Telegraph Equation ◽  
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.

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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