Stochastic Algebraic Equations

Consider the stochastic algebraic equation

517

M. Pandey et al. (eds), Advances in Engineering Structures, Mechanics & Construction, 517-525. © 2006 Springer. Printed in the Netherlands.

(a + r (Z)) X = Y, (1)

where a denotes an (d, d) real-valued deterministic matrix, Z and Y are R"-valued and Rd-valued random variables defined on some probability spaces (^1, F1, Pi) and (^2, F2, P2), respectively, that may or may not coincide, and r is an (d, d) real-valued matrix whose entries depend on Z. It is assumed that (1) the random variables Y and Z are independent and are in L2, (2) the function r (■) is measurable, so that r (Z) and a + r (Z) are random variables on (^i, Fi, Pi), and (3) the operator a + r(Z) is bounded almost surely (a. s.), that is, a + r (Z(«1)) is bounded for all «1 є Nb

where Ni є Fi and Pi(Ni) = 0.

If det(a + r (Z)) = 0 a. s., then Equation (1) has the unique solution

X(«1, o>2) = (a + r(Z(a>1 )))-1 Y(«2) = в(Z(«) Y(«2) a. s., (2)

which is a random variable on the product probability space (^ x ^2, F1 ® F2, P1 ® P2), where ^ x ^2, F1 ® F2, and P1 ® P2 denote the product sample space, the a – field generated by the measurable rectangles (A1 x A2}, A1 є F1, A2 є F2, and the extension of the set function ^(A1 x A2) = P1(A1) P2(A2) defined for A1 є F1 and A2 є F2 to F1 ® F2, respectively.