اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدFree boundary regularity in the parabolic fractional obstacle problem
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The parabolic obstacle problem for the fractional Laplacian naturally arises in American option models when the assets prices are driven by pure jump Levy processes. In this paper we study the regularity of the free boundary. Our main result establishes that, when $s>frac12$, the free boundary is a $C^{1,alpha}$ graph in $x$ and $t$ near any regular free boundary point $(x_0,t_0)in partial{u>varphi}$. Furthermore, we also prove that solutions $u$ are $C^{1+s}$ in $x$ and $t$ near such points, with a precise expansion of the form [u(x,t)-varphi(x)=c_0bigl((x-x_0)cdot e+a(t-t_0)bigr)_+^{1+s}+obigl(|x-x_0|^{1+s+alpha}+ |t-t_0|^{1+s+alpha}bigr),] with $c_0>0$, $ein mathbb{S}^{n-1}$, and $a>0$.
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