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_0in H, we take advantage of smoothing properties of the parabolic semigroup textrm{e}^{-t{mathcal {A}}}:,Hrightarrow H, tin mathbb {R}_+, generated by the Heston model, to derive the smoothness of the solution u(t) = textrm{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_0in H on the boundary behavior of u(t) for t>0.
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