C^1,1 regularity for degenerate elliptic obstacle problems

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• 作者：Panagiota Daskalopoulos^a ; ^{pdaskalo@math.columbia.edu" class="auth_mail" title="E-mail the corresponding author} ; Paul M.N. Feehan^b ; ¹ ; ^{feehan@math.rutgers.edu" class="auth_mail" title="E-mail the corresponding author} ; ^{feehan@math.ias.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：primary ; 35J70 ; 35J86 ; 49J40 ; 35R45 ; secondary ; 35R35 ; 49J20 ; 60J60
• 刊名：Journal of Differential Equations
• 出版年：2016
• 出版时间：15 March 2016
• 年：2016
• 卷：260
• 期：6
• 页码：5043-5074
• 全文大小：786 K

文摘

The Heston stochastic volatility process is a degenerate diffusion process where the degeneracy in the diffusion coefficient is proportional to the square root of the distance to the boundary of the half-plane. The generator of this process with killing, called the elliptic Heston operator, is a second-order, degenerate-elliptic partial differential operator, where the degeneracy in the operator symbol is proportional to the distance to the boundary of the half-plane. In mathematical finance, solutions to the obstacle problem for the elliptic Heston operator correspond to value functions for perpetual American-style options on the underlying asset. With the aid of weighted Sobolev spaces and weighted Hölder spaces, we establish the optimal C^1,1$C^{1,1}$ regularity (up to the boundary of the half-plane) for solutions to obstacle problems for the elliptic Heston operator when the obstacle functions are sufficiently smooth.