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Numerical approximation of a time-fractional Black-Scholes equation

机译:时间分数Black-Scholes方程的数值逼近

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In this paper a time-fractional Black-Scholes equation is examined. We transform the initial value problem into an equivalent integral-differential equation with a weakly singular kernel and use an integral discretization scheme on an adapted mesh for the time discretization. A rigorous analysis about the convergence of the time discretization scheme is given by taking account of the possibly singular behavior of the exact solution and first-order convergence with respect to the time variable is proved. For overcoming the possibly nonphysical oscillation in the computed solution caused by the degeneracy of the Black-Scholes differential operator, we employ a central difference scheme on a piecewise uniform mesh for the spatial discretization. It is proved that the scheme is stable and second-order convergent with respect to the spatial variable. Numerical experiments support these theoretical results. (C) 2018 Elsevier Ltd. All rights reserved.
机译:在本文中,研究了时间分数的Black-Scholes方程。我们将初值问题转换为具有弱奇异核的等效积分微分方程,并在自适应网格上使用积分离散化方案进行时间离散化。通过考虑精确解的可能奇异行为,对时间离散化方案的收敛性进行了严格的分析,并证明了时间变量的一阶收敛性。为了克服由Black-Scholes微分算子的退化引起的计算解中可能的非物理振动,我们在分段均匀网格上采用中心差分方案进行空间离散化。证明了该方案是稳定的,并且相对于空间变量是二阶收敛的。数值实验支持了这些理论结果。 (C)2018 Elsevier Ltd.保留所有权利。

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