Rough stochastic filtering

The theory of rough stochastic differential equations (arXiv:2106.10340) is applied to revisit classical problems in stochastic filtering. We provide rough counterparts to the Kallianpur-Striebel formula and the Zakai and Kushner-Stratonovich equations, seen to coincide with classical objects upon randomization. We follow Crisan-Pardoux (arXiv:2411.11125) in doing so in a correlated diffusion setting. Well-posedness of the (rough) filtering equation is seen to hold under dimension-independent regularity assumption, in contrast to the stochastic case.