Robust filtering: Correlated noise and multidimensional observation

In the late seventies, Clark [In Communication Systems and Random Process Theory (Proc. 2nd NATO Advanced Study Inst., Darlington, 1977) (1978) 721.734, Sijthoff & Noordhoff] pointed out that it would be natural for .t, the solution of the stochastic filtering problem, to depend continuously on the observed data Y={Ys,s.[0,t]}. Indeed, if the signal and the observation noise are independent one can show that, for any suitably chosen test function f, there exists a continuous map .ft, defined on the space of continuous paths C([0,t],Rd) endowed with the uniform convergence topology such that .t(f)=.ft(Y), almost surely; see, for example, Clark [In Communication Systems and Random Process Theory (Proc. 2nd NATO Advanced Study Inst., Darlington, 1977) (1978) 721.734, Sijthoff & Noordhoff], Clark and Crisan [Probab. Theory Related Fields 133 (2005) 43.56], Davis [Z. Wahrsch. Verw. Gebiete 54 (1980) 125.139], Davis [Teor. Veroyatn. Primen. 27 (1982) 160.167], Kushner [Stochastics 3 (1979) 75.83]. As shown by Davis and Spathopoulos [SIAM J. Control Optim. 25 (1987) 260.278], Davis [In Stochastic Systems: The Mathematics of Filtering and Identification and Applications, Proc. NATO Adv. Study Inst. Les Arcs, Savoie, France 1980 505.528], [In The Oxford Handbook of Nonlinear Filtering (2011) 403.424 Oxford Univ. Press], this type of robust representation is also possible when the signal and the observation noise are correlated, provided the observation process is scalar. For a general correlated noise and multidimensional observations such a representation does not exist. By using the theory of rough paths we provide a solution to this deficiency: the observation process Y is .lifted. to the process Y that consists of Y and its corresponding Lévy area process, and we show that there exists a continuous map .ft, defined on a suitably chosen space of Hölder continuous paths such that .t(f)=.ft(Y), almost surely.