On properties of solutions to Black–Scholes–Barenblatt equations

This paper is concerned with the Black–Scholes–Barenblatt equation \(\partial _{t}u+r(x\partial _{x}u-u)+G(x^{2}\partial _{xx}u)=0\), where \(G(\alpha )=\frac{1}{2}(\overline{\sigma}^{2}-\underline{\sigma}^{2})|\alpha |+\frac{1}{2}(\overline{\sigma}^{2}+\underline{\sigma}^{2})\alpha \), \(\alpha \in \mathbb{R}\). This equation is usually used for derivative pricing in the financial market with volatility uncertainty. We discuss a strict comparison theorem for Black–Scholes–Barenblatt equations, and study strict sub-additivity of their solutions with respect to terminal conditions.