ROBUST FILTERING: CORRELATED NOISE AND MULTIDIMENSIONAL OBSERVATION

• 成果类型: Article
• 署名作者: Crisan, D.; Diehl, J.; Friz, P. K.; Oberhauser, H.
• 署名单位: Imperial College London; Technical University of Berlin; Leibniz Association; Weierstrass Institute for Applied Analysis & Stochastics
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/12-AAP896
• 发表日期: 2013
• 页码: 2139-2160
• 关键词: continuity

摘要

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 pi(t), the solution of the stochastic filtering problem, to depend continuously on the observed data Y = {Y-s, s is an element of [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 theta(f)(t), defined on the space of continuous paths C([0, t], R-d) endowed with the uniform convergence topology such that pi(t) (f)= theta(f)(t) (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 Levy area process, and we show that there exists a continuous map theta(f)(t), defined on a suitably chosen space of Holder continuous paths such that pi(t)(f) = theta(f)(t)(Y), almost surely.