$\mathfrak{X}$PDE for $\mathfrak{X}\in \{\mathrm{BS},\mathrm{FBS},\mathrm{P}\}$: a rough volatility context

Abstract

Recent mathematical advances in the context of rough volatility have highlighted interesting and intricate connections between path-dependent partial differential equations and backward stochastic partial differential equations. In this note, we make this link precise, identifying the slightly obscure random field introduced in [Pricing options under rough volatility with backward SPDEs, C. Bayer; J. Qiu and Y. Yao. SIFIN, 13(1), 179-212 (2022)] as a pathwise derivative of the value function.