论文标题
赫斯顿随机波动率模型的边界迹线在零波动率下
The Heston stochastic volatility model has a boundary trace at zero volatility
论文作者
论文摘要
我们在霍尔德空间中建立边界规律性结果，用于在空间域（上半平面）$ \ mathbb {h} = \ mathbb {r MathBB {r} \ times（0，\ infty）\ subset \ supset \ subbb^2 $ {r的2.2 $ {r的2.2 $ {r {r} r} r {r {r。从h $中的非平滑初始数据$ u_0 \开始，我们利用了抛物线抛物线派的属性的平滑属性$ \ mathrm {e}^{ - t \ mathcal {a}}} \ colon h \ to h $ to h $，$ t \ in \ mathbb in \ mathbb {r}+$ $+$ y+smoty y he+smoty y he+smopery of here $ \ Mathrm {e}^{ - T \ Mathcal {a}} u_0 $ for All $ t> 0 $。弱解决方案的存在和独特性是在希尔伯特空间中获得的，$ h = l^2（\ mathbb {h}; \ mathfrak {w}）$，在无穷大的生长限制和边界$ \ partial \ partial \ mathbb {h} = h} = \ mathbb {r} \ Mathbb {r}^2 $的半平面$ \ Mathbb {h} $。我们研究了初始数据$ u_0 \ in H $的边界行为对$ u（t）$ for $ t> 0 $的边界行为的影响。
We establish boundary regularity results in Hölder spaces for the degenerate parabolic problem obtained from the Heston stochastic volatility model in Mathematical Finance set up in the spatial domain (upper half-plane) $\mathbb{H} = \mathbb{R}\times (0,\infty)\subset \mathbb{R}^2$. Starting with nonsmooth initial data $u_0\in H$, we take advantage of smoothing properties of the parabolic semigroup $\mathrm{e}^{-t\mathcal{A}}\colon H\to H$, $t\in \mathbb{R}_+$, generated by the Heston model, to derive the smoothness of the solution $u(t) = \mathrm{e}^{-t\mathcal{A}} u_0$ for all $t>0$. The existence and uniqueness of a weak solution is obtained in a Hilbert space $H = L^2(\mathbb{H};\mathfrak{w})$ with very weak growth restrictions at infinity and on the boundary $\partial\mathbb{H} = \mathbb{R}\times \{ 0\}\subset \mathbb{R}^2$ of the half-plane $\mathbb{H}$. We investigate the influence of the boundary behavior of the initial data $u_0\in H$ on the boundary behavior of $u(t)$ for $t>0$.
