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

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