Large deviations in the theory of nonlinear filtering and nilpotent Lie algebras (Q1332272)

Let \((X^ \varepsilon, Y^ \varepsilon)\) denote the signal-observation couple solving the Stratonovich SDE \[ \begin{aligned} dX^ \varepsilon_ t &= \varepsilon \sum_ j \widetilde{A}_ j (X^ \varepsilon_ t)\circ d\widetilde{B}_ t^{\varepsilon,j}+ A_ 0 (X^ \varepsilon_ t)dt+ \sum_ k A_ k (X^ \varepsilon_ t)\circ dY_ t^{\varepsilon,k},\\ dY_ t^ \varepsilon &= \Gamma(X^ \varepsilon_ t)dt+ dB_ t; \qquad X^ \varepsilon_ 0= x^ \varepsilon; \qquad Y^ \varepsilon_ 0=0.\end{aligned} \] If the Lie algebra generated by the vector fields \(A_ k\) is nilpotent, the conditional probabilities of the signal process \(X^ \varepsilon_ t\) (given the observation \(Y^ \varepsilon_ t\)) obey a large deviation principle. This result is applied to the theory of nonlinear filtering extending an earlier result by Doss.