The Heston stochastic volatility model has a boundary trace at zero volatility
Abstract
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 halfplane) $\mathbb{H} = \mathbb{R}\times (0,\infty)\subset \mathbb{R}^2$. 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 halfplane $\mathbb{H}$. We investigate the influence of the boundary behavior of the initial data $u_0\in H$ on the boundary behavior of $u(t)$ for $t>0$.
 Publication:

arXiv eprints
 Pub Date:
 April 2020
 arXiv:
 arXiv:2004.00444
 Bibcode:
 2020arXiv200400444A
 Keywords:

 Mathematics  Analysis of PDEs;
 Primary 35B65;
 35K65;
 Secondary 35K15;
 91G80
 EPrint:
 48 pages