Discrete-time Gauss-Markov processes with fixed reciprocal dynamics

Motivated by a problem considered earlier by Schrodinger [1]{[2], Jamison [3]{[4] and others, we examine in this paper the construction of Gauss-Markov processes with xed reciprocal dynamics. Given the class of reciprocal processes speci ed by a second-order model, a procedure is described for constructing a Markov process in the class with preassigned marginal probability densities at the end points. The problem of changing the nal density of a Gauss-Markov process while remaining in the same reciprocal class is also examined, and is interpreted in terms of an estimation problem.

[1]  B. Jamison,et al.  Reciprocal Processes: The Stationary Gaussian Case , 1970 .

[2]  J. Zambrini.  New probabilistic approach to the classical heat equation , 1988 .

[3]  A. Laub.  A schur method for solving algebraic Riccati equations , 1978, 1978 IEEE Conference on Decision and Control including the 17th Symposium on Adaptive Processes.

[4]  Ruggero Frezza,et al.  Gaussian reciprocal processes and self-adjoint stochastic differential equations of second order , 1991 .

[5]  A. Beurling,et al.  An Automorphism of Product Measures , 1960 .

[6]  W. Wonham.  Linear Multivariable Control: A Geometric Approach , 1974 .

[7]  L. Ljung,et al.  Scattering theory and linear least squares estimation: Part II: Discrete-time problems , 1975, 1975 IEEE Conference on Decision and Control including the 14th Symposium on Adaptive Processes.

[8]  A. Krener,et al.  Modeling and estimation of discrete-time Gaussian reciprocal processes , 1990 .

[9]  J. Willems.  Least squares stationary optimal control and the algebraic Riccati equation , 1971 .

[10]  L. Ljung,et al.  Scattering theory and linear least squares estimation , 1976 .

[11]  G. Picci,et al.  On the Stochastic Realization Problem , 1979 .

[12]  E. Wong.  Representation of Martingales, Quadratic Variation and Applications , 1971 .

[13]  Paolo Dai Pra,et al.  A stochastic control approach to reciprocal diffusion processes , 1991 .

[14]  Thomas Kailath,et al.  Discrete-time complementary models and smoothing , 1989 .

[15]  J. Pearson.  Linear multivariable control, a geometric approach , 1977 .