Parameter estimation in linear filtering

Suppose on a probability space ([Omega], F, P), a partially observable random process (xt, yt), t >= 0; is given where only the second component (yt) is observed. Furthermore assume that (xt, yt) satisfy the following system of stochastic differential equations driven by independent Wiener processes (W1(t)) and (W2(t)): dxt-[beta]xtdt+dW1(t), x0=0, dyt=[alpha]xtdt+dW2(t), y0=0; [alpha], [beta][infinity](a,b), a>0. We prove the local asymptotic normality of the model and obtain a large deviation inequality for the maximum likelihood estimator (m.l.e.) of the parameter [theta] = ([alpha], [beta]). This also implies the strong consistency, efficiency, asymptotic normality and the convergence of moments for the m.l.e. The method of proof can be easily extended to obtain similar results when vector valued instead of one-dimensional processes are considered and [theta] is a k-dimensional vector

The nonlinear filtering problem for the unbounded case

AbstractThe finitely additive nonlinear filtering problem for the model yt = ht(Xt)+et is solved when the function h is unbounded and satisfies no growth conditions whatever

Spectral theory of stationary H-valued processes

AbstractFor weakly stationary stochastic processes taking values in a Hilbert space, spectral representation and Cramér decomposition are studied. Using these ideas and the moving average representation for such processes established earlier by the authors, some necessary and sufficient spectral conditions for such stochastic processes to be purely nondeterministic are given in both discrete and continuous parameter cases

Supports of Gaussian measures

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A Concentration Inequality for the Sum of Independent Symmetrically Distributed Random Variables

1 online resource (PDF, 6 pages

The filtering equations revisited

The problem of nonlinear filtering has engendered a surprising number of mathematical techniques for its treatment. A notable example is the change-of--probability-measure method originally introduced by Kallianpur and Striebel to derive the filtering equations and the Bayes-like formula that bears their names. More recent work, however, has generally preferred other methods. In this paper, we reconsider the change-of-measure approach to the derivation of the filtering equations and show that many of the technical conditions present in previous work can be relaxed. The filtering equations are established for general Markov signal processes that can be described by a martingale-problem formulation. Two specific applications are treated

Multiplicity and Representation Theory of Purely Non-deterministic Stochastic Processes

1 online resource (PDF, 54 pages

Semi-Groups of Isometries and the Representation and Multiplicity of Weakly Stationary Stochastic Processes

1 online resource (PDF, 27 pages

Stochastic Differential Equations in Statistical Estimation Problems

1 online resource (PDF, 31 pages