The Heston stochastic volatility model has a boundary trace at zero volatility

Abstract

We establish boundary regularity results in H\"older 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 $t>0$.