It was not out of the

probability space," he added.

Given a

probability space ([OMEGA], A, P) and a hyper-measurable structure (P(X), B) on X, a neutrosophic random set on X (see [8]) is defined to be a triple [xi] := ([[xi].sub.T], [[xi].sub.I], [[xi].sub.F]) in which [[xi].sub.T], [[xi].sub.I] and [[xi].sub.F] are mappings from [OMEGA] to P(X) which are A-B measurables, that is,

We always assume that ([OMEGA], F, P) is a completed

probability space, W =: {[W.sub.t] : t [greater than or equal to] 0} is a real-valued Brownian motion defined on ([OMEGA], F, P), and {[F.sub.t] : t [greater than or equal to] 0} is natural filtration generated by the Brownian motion W; i.e., for any t [greater than or equal to] 0

Convergence of an approximation to a strong solution on a given

probability space was established by Gyongy and Krylov in [16] using coupling.

(ii) Let ([OMEGA], A, P) be a

probability space; that is, [OMEGA] alone is called the sample space, A is a [sigma]-algebra on [OMEGA], and P is a probability measure on ([OMEGA], A).

Till the end of the section let ([OMEGA], S, P) be a

probability space (i.e., [OMEGA] is a nonempty set, S is a [sigma]-algebra of subsets of [OMEGA], and P : S [right arrow] [0,1] is a probability measure).

where [a.sub.i](t) = [a.sub.i](t; w) :]0, +[infinity][[x[OMEGA] [right arrow] R, 1 [less than or equal to] i [less than or equal to] 3 and f (x) = f (x; w) : R x [OMEGA] [right arrow] R are s.p.'s, defined in a complete

probability space ([OMEGA], F, P), that satisfy certain hypotheses that will be specified later.

Let ([mathematical expression not reproducible]) be a complete

probability space. Here, [mathematical expression not reproducible] is a cadlag, [mathematical expression not reproducible] is the time limit for uncertain economic system, and Q is the equivalent martingale measure.

We start by building a

probability space on the initiative knowledge, and we will see how beliefs vary.

The fundamental ingredients of probability theory are the

probability space ([OMEGA], [summation], [mu]) and the abelian von Neumann algebra [L.sup.[infinity]]([OMEGA], [mu]), where [OMEGA] is a set, [summation] is [sigma]-algebra of measurable subsets of [OMEGA], [mu] is a probability measure, and [L.sup.[infinity]]([OMEGA], [mu]) is the set of all essentially bounded measurable functions.

The Birkhoff-Kinchin Ergodic Theorem (See Billingsley [10]) states that if [theta] is a measure preserving transformation defined on a

probability space ([OMEGA], F, P) and X : ([OMEGA], F) [right arrow] ([R.sup.d], B([R.sup.d])) is an integrable random vector, then