scholarly journals Positive Solutions of the Fractional SDEs with Non-Lipschitz Diffusion Coefficient

Mathematics ◽  
2020 ◽  
Vol 9 (1) ◽  
pp. 18
Author(s):  
Kęstutis Kubilius ◽  
Aidas Medžiūnas
Keyword(s):  
Diffusion Coefficient ◽  
Positive Solutions ◽  
Almost Sure Convergence ◽  
Euler Approximation ◽  
Asymptotically Normal ◽  
Backward Euler ◽  
Linear Growth Condition ◽  
Fractional Stochastic Differential Equations ◽  
Positivity Of Solutions ◽  
Asymptotically Normal Estimator

We study a class of fractional stochastic differential equations (FSDEs) with coefficients that may not satisfy the linear growth condition and non-Lipschitz diffusion coefficient. Using the Lamperti transform, we obtain conditions for positivity of solutions of such equations. We show that the trajectories of the fractional CKLS model with β>1 are not necessarily positive. We obtain the almost sure convergence rate of the backward Euler approximation scheme for solutions of the considered SDEs. We also obtain a strongly consistent and asymptotically normal estimator of the Hurst index H>1/2 for positive solutions of FSDEs.

scholarly journals A Finite Element Splitting Method for a Convection-Diffusion Problem

2020 ◽  
Vol 20 (4) ◽  
pp. 717-725 ◽  
Author(s):  
Vidar Thomée
Keyword(s):  
Finite Element ◽  
Splitting Method ◽  
Diffusion Problem ◽  
Euler Method ◽  
Euler Approximation ◽  
Convection Diffusion ◽  
Time Stepping ◽  
Backward Euler ◽  
Spatially Periodic ◽  
Forward Euler

AbstractFor a spatially periodic convection-diffusion problem, we analyze a time stepping method based on Lie splitting of a spatially semidiscrete finite element solution on time steps of length k, using the backward Euler method for the diffusion part and a stabilized explicit forward Euler approximation on {m\geq 1} intervals of length {k/m} for the convection part. This complements earlier work on time splitting of the problem in a finite difference context.

scholarly journals Estimation of the Hurst index of the solutions of fractional SDE with locally Lipschitz drift

2020 ◽  
Vol 25 (6) ◽  
pp. 1059-1078
Author(s):  
Kęstutis Kubilius
Keyword(s):  
Differential Equations ◽  
Positive Solution ◽  
Euler Scheme ◽  
Hurst Index ◽  
Unique Positive Solution ◽  
Backward Euler Scheme ◽  
Asymptotically Normal ◽  
Backward Euler ◽  
Locally Lipschitz ◽  
Strongly Consistent

Strongly consistent and asymptotically normal estimates of the Hurst index H are obtained for stochastic differential equations (SDEs) that have a unique positive solution. A strongly convergent approximation of the considered SDE solution is constructed using the backward Euler scheme. Moreover, it is proved that the Hurst estimator preserves its properties, if we replace the solution with its approximation.

scholarly journals On the Backward Euler Approximation of the Stochastic Allen-Cahn Equation

2015 ◽  
Vol 52 (02) ◽  
pp. 323-338 ◽  
Author(s):  
Mihály Kovács ◽  
Stig Larsson ◽  
Fredrik Lindgren
Keyword(s):  
Gaussian Noise ◽  
Smooth Boundary ◽  
Spatial Domain ◽  
Euler Method ◽  
Euler Approximation ◽  
Implicit Euler Method ◽  
Cahn Equation ◽  
Backward Euler

We consider the stochastic Allen-Cahn equation perturbed by smooth additive Gaussian noise in a spatial domain with smooth boundary in dimension d ≤ 3, and study the semidiscretization in time of the equation by an implicit Euler method. We show that the method converges pathwise with a rate O(Δt γ) for any γ < ½. We also prove that the scheme converges uniformly in the strong L p -sense but with no rate given.

scholarly journals FULLY DISCRETE SCHEMES FOR THE SCHRÖDINGER EQUATION: DISPERSIVE PROPERTIES

2007 ◽  
Vol 17 (04) ◽  
pp. 567-591 ◽  
Author(s):  
LIVIU I. IGNAT
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Numerical Solutions ◽  
Continuous Model ◽  
Local Smoothing ◽  
Euler Approximation ◽  
One Dimensional ◽  
Fully Discrete ◽  
Backward Euler ◽  
The One

We consider fully discrete schemes for the one-dimensional linear Schrödinger equation and analyze whether the classical dispersive properties of the continuous model are presented in these approximations. In particular, Strichartz estimates and the local smoothing of the numerical solutions are analyzed. Using a backward Euler approximation of the linear semigroup we introduce a convergent scheme for the nonlinear Schrödinger equation with nonlinearities which cannot be treated by energy methods.

Robustness of the Ewens sampling formula

1995 ◽  
Vol 32 (03) ◽  
pp. 609-622 ◽  
Author(s):  
Paul Joyce
Keyword(s):  
Mutation Rate ◽  
Direct Result ◽  
Ewens Sampling Formula ◽  
Asymptotically Normal ◽  
Sampling Formula ◽  
Selective Neutrality ◽  
Infinite Alleles Model ◽  
Asymptotically Normal Estimator

Under the assumptions of the neutral infinite alleles model, K (the total number of alleles present in a sample) is sufficient for estimating θ (the mutation rate). This is a direct result of the Ewens sampling formula, which gives a consistent, asymptotically normal estimator for θ based on K. It is shown that the same estimator used to estimate θ under neutrality is consistent and asymptotically normal, even when the assumption of selective neutrality is violated.

Non-local Degenerate Diffusion Coefficients Break Down the Components of Positive Solutions

2020 ◽  
Vol 20 (1) ◽  
pp. 19-30
Author(s):  
M. Delgado ◽  
C. Morales-Rodrigo ◽  
J. R. Santos Júnior ◽  
A. Suárez
Keyword(s):  
Fixed Point ◽  
Diffusion Coefficient ◽  
Positive Solutions ◽  
Diffusion Coefficients ◽  
Degenerate Diffusion ◽  
Nonlinear Elliptic Problems ◽  
Nonlinear Elliptic ◽  
Non Local ◽  
Local Term ◽  
The Continuum

AbstractThis paper deals with nonlinear elliptic problems where the diffusion coefficient is a degenerate non-local term. We show that this degeneration implies the growth of the complexity of the structure of the set of positive solutions of the equation. Specifically, when the reaction term is of logistic type, the continuum of positive solutions breaks into two disjoint pieces. Our approach uses mainly fixed point arguments.

Robustness of the Ewens sampling formula

1995 ◽  
Vol 32 (3) ◽  
pp. 609-622 ◽  
Author(s):  
Paul Joyce
Keyword(s):  
Mutation Rate ◽  
Direct Result ◽  
Ewens Sampling Formula ◽  
Asymptotically Normal ◽  
Sampling Formula ◽  
Selective Neutrality ◽  
Infinite Alleles Model ◽  
Asymptotically Normal Estimator

Under the assumptions of the neutral infinite alleles model, K (the total number of alleles present in a sample) is sufficient for estimating θ (the mutation rate). This is a direct result of the Ewens sampling formula, which gives a consistent, asymptotically normal estimator for θ based on K. It is shown that the same estimator used to estimate θ under neutrality is consistent and asymptotically normal, even when the assumption of selective neutrality is violated.

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