Sergey V. Lototsky

Nonlinear Filtering Revisited: A Spectral Approach

The objective of this paper is to develop an approach to nonlinear filtering based on the Cameron--Martin version of Wiener chaos expansion. This approach gives rise to a new numerical scheme for nonlinear filtering. The main feature of this algorithm is that it allows one to separate the computations involving the observations from those dealing only with the system parameters and to shift the latter off-line.

A Sobolev space theory of SPDEs with constant coefficients on a half line

Equations of the form

Stochastic Differential Equations: A Wiener Chaos Approach

du=(a^{ij}u_{x^{i}x^{j}} +D_{i}f^{i})\,dt+\sum_{k}(\sigma^{ik}u_{x^{i}} +g^{k})\,dw^{k}_{t}

Wiener chaos solutions of linear stochastic evolution equations

are considered for t > 0 and

ASYMPTOTIC PROPERTIES OF THE MAXIMUM LIKELIHOOD ESTIMATOR FOR STOCHASTIC PARABOLIC EQUATIONS WITH ADDITIVE FRACTIONAL BROWNIAN MOTION

x\in\bR^{d}_{+}

Dirichlet problem for stochasticparabolic equations in smoothdomains

. The unique solvability of these equations is proved in weighted Sobolev spaces with fractional positive or negative derivatives, summable to the power

STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS DRIVEN BY PURELY SPATIAL NOISE

p\in[2,\infty)

Recursive nonlinear filter for a continuous discrete-time model: separation of parameters and observations

.

Spectral asymptotics of some functionals arising in statistical inference for SPDEs

A new method is described for constructing a generalized solution for stochastic differential equations. The method is based on the Cameron-Martin version of the Wiener Chaos expansion and provides a unified framework for the study of ordinary and partial differential equations driven by finite- or infinite-dimensional noise with either adapted or anticipating input. Existence, uniqueness, regularity, and probabilistic representation of this Wiener Chaos solution is established for a large class of equations. A number of examples are presented to illustrate the general constructions. A detailed analysis is presented for the various forms of the passive scalar equation and for the first-order It\^{o} stochastic partial differential equation. Applications to nonlinear filtering if diffusion processes and to the stochastic Navier-Stokes equation are also discussed.

STATISTICAL INFERENCE FOR STOCHASTIC PARABOLIC EQUATIONS: A SPECTRAL APPROACH

A new method is described for constructing a generalized solution of a stochastic evolution equation. Existence, uniqueness, regularity and a probabilistic representation of this Wiener Chaos solution are established for a large class of equations. As an application of the general theory, new results are obtained for several types of the passive scalar equation.