Robust filtering: Correlated noise and multidimensional observation

Abstract

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 $Y=\{Y_s,s\in[0,t]\}$. 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 $C([0,t],\mathbb{R}^d)$ 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 $\mathbf{Y}$ that consists of $Y$ and its corresponding L\'{e}vy area process, and we show that there exists a continuous map $\theta_t^f$, defined on a suitably chosen space of H\"{o}lder continuous paths such that $\pi_t(f)=\theta_t^f(\mathbf{Y})$, almost surely.Comment: Published in at http://dx.doi.org/10.1214/12-AAP896 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Kusuoka–Stroock gradient bounds for the solution of the filtering equation

“…Following from [5], under condition (16), the law of the signal X x t has a smooth density y ! p x t (y) with respect to the Lebesgue measure for all t > 0: 4 Under this assumption, we can deduce also deduce gradient estimates for the density y !…”

“…More precisely, the probability measureP is absolutely continuous with respect to P and its Radon-Nikodym derivative is given by dP dP where Z = fZ t ; t 0g is the exponential martingale de…ned in (4), that is,…”

“…Our present approach frequently makes use of the fact that we are dealing with the uncorrelated …ltering problem. Previous work using rough paths in the context of …ltering [4], also [7] considers the setting where the noises in the signal and observation are correlated.…”

The Inverse Problem for Rough Controlled Differential Equations

“…Theorem 9 (Theorems 6 and 7 in [15]). Under appropriate assumptions on the vector fields 2 , there exists a continuous deterministic function f on the rough path space such that…”

Semi-discretization for Stochastic Scalar Conservation Laws with Multiple Rough Fluxes

“…It is well known (see, for example, [10]) that the RDE above has a flow ψ z of solutions. The following is taken from Crisan, Diehl, Friz and Oberhauser [7,Lemma 13].…”