Random Oper. & Stock. Equ., Vol. 1, No. 1, pp. 47-55 (1993) © VSP 1993 A. V. ILCHENKO Department of Mechanics and Mathematics, Kyjiv University, 25201 7 Kyjiv, Ulrraine Received for ROSE 22 September 1991 Abstract--In this paper the limit behaviour of the solutions of the system dx(t) = (Ax(t) + ()) dt + (Bx(t) + «(O) dw(t) is investigated for t --»· oo if the system dy(t) = Ay(t) at + By(t) dw(t) is stable with probability 1. Let us consider the system dx(t) = (Ax(t) + ,,(t)) at + (Bx(t) + tf (t)) d»(*), where A and are ( n)-matrices, (1) are vector- columns, w(t) is the one-dimensional Wiener process, ty(0) = 0. As it is known from , the solution of the system (1) may be represented in the form x(t) = H'x + Hi ('}-1[(3) + ()\ as + *0 (^ dw(S), JQ JQ where HQ is the matrix-solution of the system d#o = £ dt + BHl dw(t), E is the identity matrix. Assuming that () is a smooth function and det ^ 0, we have IVans/ated by the author A. V. Ilchenko Then /* Jo Hl f (H'}'1 \(-A + 2)-*(3) +
Random Operators and Stochastic Equations – de Gruyter
Published: Jan 1, 1993
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