Runge-Kutta theory for Volterra and Abel integral equations of the second kind

Lubich, Christian

doi:10.1090/s0025-5718-1983-0701626-6

Cited by 110 publications

(26 citation statements)

References 10 publications

(14 reference statements)

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“…We have seen in Lemma 5.4 that u → f (u, t) is analytic for any fixed t. But it is also analytic w.r.t. t for fixed u: From Theorem 1 in Lubich [27], itself based on earlier work by Miller and Feldstein [28], it follows that f (u, ·) is analytic on the whole interval (0,T α (u)). By Hartogs's theorem (Theorem 1.2.5 in [22]), we conclude that the bivariate function f (·, ·) is continuous.…”

Section: Explosion Time In the Rough Heston Modelmentioning

confidence: 93%

Moment explosions in the rough Heston model

Gerhold

Gerstenecker

Pinter

2019

Decisions Econ Finan

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We show that the moment explosion time in the rough Heston model [El Euch, Rosenbaum 2016, arXiv:1609.02108] is finite if and only if it is finite for the classical Heston model. Upper and lower bounds for the explosion time are established, as well as an algorithm to compute the explosion time (under some restrictions). We show that the critical moments are finite for all maturities. For negative correlation, we apply our algorithm for the moment explosion time to compute the lower critical moment. arXiv:1801.09458v3 [q-fin.MF]

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Section: Explosion Time In the Rough Heston Modelmentioning

confidence: 93%

Moment explosions in the rough Heston model

Gerhold

Gerstenecker

Pinter

2019

Decisions Econ Finan

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“…First of all, this is a quite unrealistic assumption since, as any researcher in this field should know, the solution of a FDE (and consequently the rhs of the FDE) does in general not have such kind of smoothness (e.g. see [3,12,14]). However, and this is more important, why the authors consider the third derivative?…”

Section: Analysis Of the Methodsmentioning

confidence: 99%

Neglecting nonlocality leads to unreliable numerical methods for fractional differential equations

Garrappa

2019

Communications in Nonlinear Science and Numerical Simulation

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In the paper titled "New numerical approach for fractional differential equations" by A. Atangana and K.M. Owolabi [Math. Model. Nat. Phenom., 13(1), 2018], it is presented a method for the numerical solution of some fractional differential equations. The numerical approximation is obtained by using just local information and the scheme does not present a memory term; moreover, it is claimed that third-order convergence is surprisingly obtained by simply using linear polynomial approximations. In this note we show that methods of this kind are not reliable and lead to completely wrong results since the nonlocal nature of fractional differential operators cannot be neglected. We illustrate the main weaknesses in the derivation and analysis of the method in order to warn other researchers and scientist to overlook this and other methods devised on similar basis and avoid their use for the numerical simulation of fractional differential equations.

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“…For previous work on Abel-type equations with singular kernels, we refer the reader to [6,21,22,42]. These papers, however, are mostly concerned with implicit marching schemes.…”

Section: The Robin Problem In One Dimensionmentioning

confidence: 99%

Explicit unconditionally stable methods for the heat equation via potential theory

Barnett

Epstein

Greengard

et al. 2019

Pure Appl. Analysis

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We study the stability properties of explicit marching schemes for second-kind Volterra integral equations that arise when solving boundary value problems for the heat equation by means of potential theory. It is well known that explicit finite difference or finite element schemes for the heat equation are stable only if the time step ∆t is of the order O(∆x 2 ), where ∆x is the finest spatial grid spacing. In contrast, for the Dirichlet and Neumann problems on the unit ball in all dimensions d ≥ 1, we show that the simplest Volterra marching scheme, i.e., the forward Euler scheme, is unconditionally stable. Our proof is based on an explicit spectral radius bound of the marching matrix, leading to an estimate that an L 2 -norm of the solution to the integral equation is bounded by c d T d/2 times the norm of the right hand side. For the Robin problem on the half space in any dimension, with constant Robin (heat transfer) coefficient κ, we exhibit a constant C such that the forward Euler scheme is stable if ∆t < C/κ 2 , independent of any spatial discretization. This relies on new lower bounds on the spectrum of real symmetric Toeplitz matrices defined by convex sequences. Finally, we show that the forward Euler scheme is unconditionally stable for the Dirichlet problem on any smooth convex domain in any dimension, in L ∞ -norm.

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Runge-Kutta theory for Volterra and Abel integral equations of the second kind

Abstract: Abstract. The present paper develops the local theory of general Runge-Kutta methods for a broad class of weakly singular and regular Volterra integral equations of the second kind. Further, the smoothness properties of the exact solutions of such equations are investigated.

Cited by 110 publications

References 10 publications

Moment explosions in the rough Heston model

Moment explosions in the rough Heston model

Neglecting nonlocality leads to unreliable numerical methods for fractional differential equations

Explicit unconditionally stable methods for the heat equation via potential theory

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