Heston随机波动率模型在零波动率处有边界迹
The Heston stochastic volatility model has a boundary trace at zero volatility
摘要：
本文在空间域（上半平面）$\mathbb{H} = \mathbb{R}\times (0,\infty)\subset \mathbb{R}^2$上建立了金融数学中Heston随机波动率模型的退化抛物问题在Hölder空间中的边界正则性结果.从非光滑初始数据$u_0\in H$出发，利用Heston模型生成的抛物半群$\mathrm{e}^{-t\mathcal{A}}\colon H\to H$，$t\in \mathbb{R}_+$的光滑性质，得到了所有$t>0$的解$u(t) = \mathrm{e}^{-t\mathcal{A}} u_0$的光滑性.在半平面$\mathbb{H}$的边界$\partial\mathbb{H} = \mathbb{R}\times \{ 0\}\subset \mathbb{R}^2$和无穷远点具有很弱的增长限制的Hilbert空间$H = L^2(\mathbb{H};\mathfrak{w})$中，得到了弱解的存在唯一性.我们研究了初始数据$u_0\in H$的边界行为对$t>0$的$u(t)$的边界行为的影响。
英文摘要：
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 $t>0$.
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