On Multistep Stabilizing Correction Splitting Methods with Applications to the Heston Model

Abstract

In this note we consider splitting methods based on linear multistep methods and stabilizing corrections. To enhance the stability of the methods, we employ an idea of Bruno & Cubillos [5] who combine a high-order extrapolation formula for the explicit term with a formula of one order lower for the implicit terms. Several examples of the obtained multistep stabilizing correction methods are presented, and results on linear stability and convergence are derived. The methods are tested in the application to the well-known Heston model arising in financial mathematics and are found to be competitive with well-established one-step splitting methods from the literature.2000 Mathematics Subject Classification: 65L06, 65M06, 65M20.The following examples fit in the framework outlined in the previous section, with coefficients in the prediction stage obtained by extrapolation. The examples are described by specifying k and the coefficients a " pa 1 , . . . , a k q, b " pb 1 , . . . , b k q, θ " b 0 of

Convergence in the maximum norm of ADI-type methods for parabolic problems

“…This assumption was also useful in [9,Theorem 3.2] in order to prove convergence for linear multistep methods with stabilizing corrections applied to split ODEs. Although (39) is closely related to (27), the proof of convergence presented in [1] does require the assumption on power-boundedness for the stability matrix R, which, as far as we are aware, has not been shown for m ≥ 3 so far, whereas our proof does not require such an assumption.…”

Operator splitting schemes for American options under the two-asset Merton jump-diffusion model

“…The stabilizing correction two-step Adams-type (SC2A) scheme is a prominent member of the class of stabilizing correction multistep methods that has been investigated by Hundsdorfer & in 't Hout [13] for the numerical solution of PDEs. Its direct adaptation to PIDEs has been studied by Boen & in 't Hout [3] and treats the integral part and mixed derivative term jointly in a two-step Adams-Bashforth fashion.…”

“…Its direct adaptation to PIDEs has been studied by Boen & in 't Hout [3] and treats the integral part and mixed derivative term jointly in a two-step Adams-Bashforth fashion. The combination with IT(κ) splitting yields: Here the coefficients are pb 1 1 , b 1 2 q "`3 2 ,´1 2˘a nd pb 0 1 , b 0 2 q "`3 2´θ ,´1 2`θ˘a nd, following [13], we select θ " 3 4 . The SC2A-IT(κ) method also requires just one evaluation of the integral part per time step.…”

Operator splitting schemes for the two-asset Merton jump–diffusion model

“…In Hundsdorfer & in 't Hout [18] a novel class of stabilizing correction multistep methods has recently been investigated for the numerical solution of PDEs. We select here a prominent member of this class, the two-step Adams-type scheme called SC2A, and consider its direct adaptation to PIDEs:…”

“…Method (4.7) is also of order two for any value θ. Following [18] we take θ " 3 4 , which is motivated by stability and accuracy results. For starting (4.7), the one-step method (4.5) is used with θ " 1 3 to define V 1 .…”