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A high-order and fast scheme with variable time steps for the time-fractional Black-Scholes equation
  • Kerui Song,
  • Pin Lyu
Kerui Song
Southwestern University of Finance and Economics

Corresponding Author:[email protected]

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Pin Lyu
Southwestern University of Finance and Economics
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Abstract

In this paper, a high-order and fast numerical method is investigated for the time-fractional Black-Scholes equation. In order to deal with the typical weak initial singularities of the solution, we construct a finite difference scheme with variable time steps, where the fractional derivative is approximated by the nonuniform Alikhanov formula and the sum-of-exponentials (SOE) technique. In the spatial direction, an average approximation with fourth-order accuracy is employed. The stability and the convergence with second-order in time and fourth-order in space of the proposed scheme are religiously derived by the energy method. Numerical examples are given to demonstrate the theoretical statement.
23 Oct 2021Submitted to Mathematical Methods in the Applied Sciences
23 Oct 2021Submission Checks Completed
23 Oct 2021Assigned to Editor
30 Oct 2021Reviewer(s) Assigned
23 Nov 2021Review(s) Completed, Editorial Evaluation Pending
10 Feb 2022Editorial Decision: Revise Major
05 May 20221st Revision Received
09 May 2022Submission Checks Completed
09 May 2022Assigned to Editor
09 May 2022Review(s) Completed, Editorial Evaluation Pending
09 May 2022Editorial Decision: Revise Major
12 May 20222nd Revision Received
13 May 2022Submission Checks Completed
13 May 2022Assigned to Editor
14 May 2022Reviewer(s) Assigned
22 Jul 2022Review(s) Completed, Editorial Evaluation Pending
23 Jul 2022Editorial Decision: Accept
30 Jan 2023Published in Mathematical Methods in the Applied Sciences volume 46 issue 2 on pages 1990-2011. 10.1002/mma.8623