Approximations to optimal nonlinear filters

Let the signal and noise processes be given as solutions to nonlinear stochastic differential equations. The optimal filter for the problem, derived elsewhere, is usually infinite dimensional. Several methods of obtaining possibly useful finite dimensional approximations are considered here, and some of the special problems of simulation are discussed. The numerical results indicate a number of useful features of the approximating filters and suggest methods of improvement. The paper is concerned with problems where the noise and nonlinear effects are much too large for the use of "linearization" methods, which for the simulated problem, at least, were useless.

[1]  D. Widder,et al.  The Laplace Transform , 1943, The Mathematical Gazette.

[2]  L. Kantorovich,et al.  Approximate methods of higher analysis , 1960 .

[3]  R. L. Stratonovich.  CONDITIONAL MARKOV PROCESSES , 1960 .

[4]  R. E. Kalman,et al.  New Results in Linear Filtering and Prediction Theory , 1961 .

[5]  L. Sirovich.  Formal and Asymptotic Solutions in Kinetic Theory , 1963 .

[6]  Henry Cox,et al.  On the estimation of state variables and parameters for noisy dynamic systems , 1964 .

[7]  W. Wonham.  Some applications of stochastic difierential equations to optimal nonlinear ltering , 1964 .

[8]  Karl Johan Åström,et al.  On a first-order stochastic differential equation , 1965 .

[9]  H. Kushner.  On the Differential Equations Satisfied by Conditional Probablitity Densities of Markov Processes, with Applications , 1964 .

[10]  Robert H. Kraichnan,et al.  Direct-Interaction Approximation for Shear and Thermally Driven Turbulence , 1964 .

[11]  R. Bucy.  Nonlinear filtering theory , 1965 .

[12]  E. Wong,et al.  On the Convergence of Ordinary Integrals to Stochastic Integrals , 1965 .

[13]  R. Sridhar,et al.  Sequential estimation of states and parameters in noisy non-linear dynamical systems , 1965 .

[14]  John Martin Cortham Clark.  The representation of non-linear stochastic systems with applications to filtering , 1966 .

[15]  Rangasami Sridhar,et al.  Sequential Estimation of States and Parameters in Noisy Nonlinear Dynamical Systems , 1966 .

[16]  H. Kushner.  Dynamical equations for optimal nonlinear filtering , 1967 .

[17]  A. Shiryaev.  Addendum: On Stochastic Equations in the Theory of Conditional Markov Processes , 1967 .