In this paper, we present and analyse a class of "filtered" numerical schemes for second order Hamilton-Jacobi-Bellman equations.Our approach follows the ideas introduced in B.D. Froese and A.M. Oberman, Convergent filtered schemes for the Monge-Ampère partial differential equation, SIAM J. Numer. Anal., 51(1):423--444, 2013, and more recently applied by other authors to stationary or time-dependent firstorder Hamilton-Jacobi equations.For high order approximation schemes (where "high" stands for greater than one), the inevitable loss of monotonicity prevents the use of the classical theoretical results for convergence to viscosity solutions. The work introduces a suitable local modification of these schemes by "filtering" themwith a monotone scheme, such that they can be proven convergent and still show an overall high order behaviour for smooth enough solutions.We give theoretical proofs of these claims and illustrate the behaviour with numerical tests from mathematical finance, focussing also on the use of backward differencing formulae for constructing the high order schemes.
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机译:在本文中,我们提出并分析了一类用于二阶Hamilton-Jacobi-Bellman方程的“滤波”数值格式。我们的方法遵循B.D.中引入的思想。 Froese和A.M. Oberman,Monge-Ampère偏微分方程的收敛滤波方案,SIAM J. Numer。 Anal。,51(1):423--444,2013,最近被其他作者应用到平稳或时间依赖的一阶Hamilton-Jacobi方程。对于高阶逼近方案(其中“高”表示大于一个) ,不可避免地失去了单调性,因此无法使用经典的理论结果来收敛到粘度溶液。这项工作通过用单调方案“过滤”它们来介绍这些方案的合适的局部修改,这样它们可以被证明是收敛的,并且对于足够光滑的解决方案仍然显示出整体的高阶行为。我们给出了这些主张的理论证据并举例说明了行为。通过数学金融的数值测试,还集中在使用向后微分公式来构造高阶方案上。
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