Linear estimation of boundary value processes

Abstract

In this paper we discuss the problem of estimating processes by the innovations approach [6] and [7]. This boundary value processes in one or several dimensions. approach has some advantages over diagonalization. In The estimator dynamics are described, and by using particular, in constrast to diagonalization, no operaoperator transformations for these dynamics, several tor inversions are required in computing the estimate implementations are obtained which either diagonalize by triangularization. Having established the conditions or triangularize the linear least-squares estimator.for diagonalization we formulate matrix Riccati equaThese implementations enable us to compute the estimate tions which lead to stable diagonal forms for the esof the process by using two-filter type of smoothing timator for l-D boundary value processes. Finally, formulas, or more general smoothing formulas similar to questions of existence and uniqueness of diagonalizing those used for solving the smoothing problem for 1-D transformations for 2-D estimators are discussed. causal processes.2. where L is referred to as the formal adjoint differenand consequently an issue is the construction of tial operator [81, xb and Xb are elements of a Hilbert efficient methods to implement that solution. In space Hb of processes defined on Ag , and E is a mapp-[5] a detailed solution in the 1-D case is developed by ing from Hb into itself; E:HR*Hb. Yn particular, diagonalizing the dynamics of the smoother boundary these processes are defined Ehrough the action of an value differential equation (see also [2] for a brief operator A : L n() Hb, so that description). The computations involved in these reb 2 N sults relied heavily on the specific 1-D problem. In x = A x and X = abX (2) this paper we extend the ideas underlying that diagob b b nalization approach by describing the diagonalization The nature of Hb, Ab, and E all depend upon L and QN of estimator dynamics in an operator framework appliFor a discussion of Green's identity for ordinary cable to problems in several dimensions as well.differential operators see [9] and Chapter 3 of [10]; for elliptic, hyperbolic and parabolic second order After reviewing the form of the estimator solution partial differential operators see [8] and Chapter 7 in Section 2, we describe some equivalent dynamical reof [10].In this paper, we will restrict our discuspresentations for the differential operator description sions to operators L and regions QN that admit a Green's of the estimator dynamics in Section 3. The diagonal identity. form we seek, if it exists, is in the class of equivalent differential operator representations, and inThe boundary condition associated with L is deSection 4 we present the conditions which define the fined by a mapping V: class of transformation operators which lead to such forms. As an alternative to diagonalization, we out-V:Hb R(V) (3) line a method for triangularizing the dynamics which The work of these authors was supported by the National Science Foundation unde...

Weak Approximations. A Malliavin Calculus Approach

“…For example, Bossy and Talay [5] and [6] considered stochastic differential equations related to the Burgers and McKeanVlasov equations. These stochastic equations are of a type different from (1). In particular, the law of the solution process is also part of the coefficients in the equation.…”

“…Another direction of development has been to consider other stochastic differential equations of a type different from (1). For example, Bossy and Talay [5] and [6] considered stochastic differential equations related to the Burgers and McKeanVlasov equations.…”

“…. < t N = 1} be a partition of [0,1]. As before, we define the shift operator η(t) and we take π ≤ δ.…”

Stability, stochastic stationarity, and generalized Lyapunov equations for two-point boundary-value descriptor systems

“…Since the framework proposed in [22] and [23] for the solution of noncausal estimation problems is totally general, and is applicable to absolutely any model in any dimension, one of our objectives has been to find 1-D models which display as much noncausality as possible, so that estimation results developed for these models will be easy to transpose to higher dimensions. This has led us in [4] to examine estimation problems for TPBVDSs.…”

“…Another important motivation for the study presented here is our own work on linear estimation of noncausal stochastic processes in one or several dimensions [22]- [24]. Since the framework proposed in [22] and [23] for the solution of noncausal estimation problems is totally general, and is applicable to absolutely any model in any dimension, one of our objectives has been to find 1-D models which display as much noncausality as possible, so that estimation results developed for these models will be easy to transpose to higher dimensions.…”