Robust filtering: Correlated noise and multidimensional observation - Crisan, D; Diehl, J; Friz, P; Oberhauser, H (Institute of Mathematical Statistics, 2013)
In the late seventies, Clark [In Communication Systems and Random Process Theory (Proc. 2nd NATO Advanced Study Inst., Darlington, 1977) (1978) 721–734, Sijthoff and Noordhoff] pointed out that it would be natural for πt, 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 θft, defined on the space of continuous paths C([0,t],Rd) endowed with the uniform convergence topology such that πt(f)=θft(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 and 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 θft, defined on a suitably chosen space of Hölder continuous paths such that πt(f)=θft(Y), almost surely.