On properties of solutions to Black–Scholes–Barenblatt equations

Abstract

This paper is concerned with the Black-Scholes-Barenblatt equation ∂ t u + r(x∂ x u-u) + G(x 2 ∂ xx u) = 0, where G(α) = 1 2 (σ 2-σ 2)|α| + 1 2 (σ 2 + σ 2)α, α ∈ 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.