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Analysis of a stochastic difference equation: exit times and invariant distributions

Högnäs G., Jung B.

Fasciculi Mathematici

2010

Nr 44

69-74

The mean return time of a discrete Markov chain to a point x is the reciprocal of the invariant probability π(x). We revisit this classical theme to investigate certain exit times for stochastic difference equations of autoregressive type. More specifically, we will discuss the asymptotics, as 0, of the first time that the n-dimensional process ...[wzór] (where ξ1, ξ2, . . . is a sequence of i.i.d. random n-vectors) leaves a given neighborhood of the fixed point of the contraction f.

D-Step Ahead Kalman Predictor for Controlled Autoregressive Processes With Random Coefficients

Hilgert N., Vila J. P.

International Journal of Applied Mathematics and Computer Science

1999

Vol. 9, no 1

207-217

This paper deals with prediction of controlled autoregressive processes with additive white Gaussian noise and random coefficients adapted to an observation process. Our aim is twofold. We begin by extending to the standard Kalman predictor a result of Chen et al. (1989) on the optimality of the standard Kalman filter when applied to linear stochastic processes with almost surely finite random coefficients. We then show on an example how some particular nonlinear autoregressive processes can be embedded in these linear processes with random coefficients. Such nonlinear processes can then benefit from this optimal prediction, which is not provided by the usual extended Kalman predictor.