zbMATH — the first resource for mathematics

Almost periodic solutions of affine Ito equations. (English) Zbl 0692.60045
Consider the affine Ito type stochastic differential equation \[ (1)\quad dx(t)=[A_ 0(t)x(t)+f_ 0(t)]dt+\sum^{m}_{j=1}[A_ j(t)x(t)+f_ j(t)]dw_ j(t),\quad t\in R, \] where \(A_ j:R\to R^ d\otimes R^ d\), \(f_ j:R\to R^ d\), \(0\leq j\leq m\), are measurable and bounded. The authors discussed the problem of existence of almost periodic solutions of (1). It is proved that if the linear part of the affine equation is exponentially stable in mean square then the unique continuous \(L^ 2\)- bounded solution of the affine system has almost periodic one-dimensional distributions. An analogous result is shown for the asymptotic almost periodic case.
Reviewer: B.G.Pachpatte

60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
Full Text: DOI
[1] Dorogovtsev A., Teoria Verojatn.i Matem. Statistica 39 pp 47– (1988)
[2] Fink A.M., Almost Periodic Differential Equations (1974) · Zbl 0325.34039
[3] Friedman A., Stochastic Differential Equations and Applications 1 (1975) · Zbl 0323.60056
[4] Halanay A., Časopis pró pestovani matematiky 111 pp 127– (1986)
[5] Halanay A., Stochastics 21 pp 287– (1987)
[6] DOI: 10.1080/00207178808906017 · Zbl 0642.93068 · doi:10.1080/00207178808906017
[7] Halanay, A. and Morozan, T. 1988. Tracking almost periodic signals under white noise perturbations. Proc. of the Sixth IFIP Conference on Stochastic Systems and Optimization. 1988, Warsaw. to appear · Zbl 0693.93071
[8] Ichikawa, A. 1986. Bounded solutions and periodic solutions of a linear stochastic evolution equation. Proc of the Fifth Japan-USSR Symposium on Probability Theory. 1986, Kyoto.
[9] Ikeda N., Stochastic Differential Equations and Diffusion Processes (1981) · Zbl 0495.60005
[10] Khasminskii R., Stability of Differential Equations under Random Perturbation (1969)
[11] Levitan B.M., Almost Periodic Functions and Differential Equations (1978) · Zbl 0414.43008
[12] DOI: 10.1080/07362998608809081 · Zbl 0583.60055 · doi:10.1080/07362998608809081
[13] Morozan T., Studii si Cercetari Matematice 38 pp 523– (1986)
[14] Morozan T., Revue Roumaine Math. Pures et AppL
[15] Romisch W., Lect. Notes in Control and Information Sciences 96 (1986)
[16] Skorokhod A., Studies in the Theory of Random Processes (1965) · Zbl 0146.37701
[17] Varsan C., Tohoku Math. J (1965)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.