Before revision:  The main difficulty is that these filters are valued in infinite-dimensional space: it satisfy a stochastic partial differential equations named "Kushner-Stratonovich equations".
After revision:  Adapting the stochastic non-linear filtering, we show that under some assumptions on the model coefficients, the estimation of the filters depend on a priorimodels for the trend and the stochastic volatility. Moreover, these filters satisfy a stochastic partial differential equations named "Kushner-Stratonovich equations".