Parameter dependent rough SDEs with applications to rough PDEs

Rough stochastic differential equations (rough SDEs), recently introduced by Friz, Hocquet and L\^e in arXiv:2106.10340, have emerged as a versatile tool to study "doubly" SDEs under partial conditioning (with motivation from pathwise filtering and control, volatility modelling in finance and mean-field stochastic dynamics with common noise ...). While the full dynamics may be highly non-Markovian, the conditional dynamics often are. In natural (and even linear) situations, the resulting stochastic PDEs can be beyond existing technology. The present work then tackles a key problem in this context, which is the well-posedness of regular solution to the rough Kolmogorov backward equation. To this end, we study parameter dependent rough SDEs in sense of $\mathscr{L}$-differentiability (as in Krylov, 2008). In companion works, we will show how this removes dimension-dependent regularity assumptions for well-posedness of the Zakai, Kushner-Stratonovich and nonlinear Fokker-Planck stochastic equations.