Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userRobust filtering: Correlated noise and multidimensional observation
135
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic 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 $Y={Y_s,sin[0,t]}$. 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.
rate research
Read More
The objective in stochastic filtering is to reconstruct information about an unobserved (random) process, called the signal process, given the current available observations of a certain noisy transformation of that process. Usually X and Y are modeled by stochastic differential equations driven by a Brownian motion or a jump (or Levy) process. We are interested in the situation where both the state process X and the observation process Y are perturbed by coupled Levy processes. More precisely, L=(L_1,L_2) is a 2--dimensional Levy process in which the structure of dependence is described by a Levy copula. We derive the associated Zakai equation for the density process and establish sufficient conditions depending on the copula and $L$ for the solvability of the corresponding solution to the Zakai equation. In particular, we give conditions of existence and uniqueness of the density process, if one is interested to estimate quantities like P( X(t)>a), where a is a threshold.
In this paper we study zero-noise limits of $alpha -$stable noise perturbed ODEs which are driven by an irregular vector field $A$ with asymptotics $% A(x)sim overline{a}(frac{x}{leftvert xrightvert })leftvert xrightvert ^{beta -1}x$ at zero, where $overline{a}>0$ is a continuous function and $beta in (0,1)$. The results established in this article can be considered a generalization of those in the seminal works of Bafico cite% {Ba} and Bafico, Baldi cite{BB} to the multi-dimensional case. Our approach for proving these results is inspired by techniques in cite% {PP_self_similar} and based on the analysis of an SDE for $tlongrightarrow infty $, which is obtained through a transformation of the perturbed ODE.
We consider the filtering of continuous-time finite-state hidden Markov models, where the rate and observation matrices depend on unknown time-dependent parameters, for which no prior or stochastic model is available. We quantify and analyze how the induced uncertainty may be propagated through time as we collect new observations, and used to simultaneously provide robust estimates of the hidden signal and to learn the unknown parameters, via techniques based on pathwise filtering and new results on the optimal control of rough differential equations.
We study the problem of pathwise stochastic optimal control, where the optimization is performed for each fixed realisation of the driving noise, by phrasing the problem in terms of the optimal control of rough differential equations. We investigate the degeneracy phenomenon induced by directly controlling the coefficient of the noise term, and propose a simple procedure to resolve this degeneracy whilst retaining dynamic programming. As an application, we use pathwise stochastic control in the context of stochastic filtering to construct filters which are robust to parameter uncertainty, demonstrating an original application of rough path theory to statistics.
This paper suggests a new approach to error analysis in the filtering problem for continuous time linear system driven by fractional Brownian noises. We establish existence of the large time limit of the filtering error and determine its scaling exponent with respect to the vanishing observation noise intensity. Closed form expressions are obtained in a number of important special cases.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
