Some explicit conditions for maximal local diffusions in one-dimensional case
Ivo Vrkoč · Czechoslovak Mathematical Journal · 1971
The notions of maximal and strongly maximal matrix functions were defined in article [1] (see also Definition 1 in this article).The general criteria for maximality and strong maximality are given by Theorems 1 and 2 in [1] (Theorem 2 is also reformulated as Conclusion 1 in this article).These Theorems are valid if condition (A) given in [1] is fulfilled and in [1] there are also given more explicit assumptions under which condition (A) is fulfilled (see Lemma 4 and 5 in [l]).Due to recent results these assumptions can be simplified.Theorem 1 of this article concerns to this subject.The assumptions of Theorems 1 and 2 from [1] have no explicit form since we need to solve some parabolic equation and we must decide if the solution is convex in spatial variables.Only Theorem 3 from [1] (reformulated here as Theorem 2) has an explicit form, but the nonstochastic part a[t, x) of Ito stochastic equation has to be Hnear in x.We generahze this result to include nonlinear a(x) in one-dimensional case -see Theorems 3 and 4. Considering all this a question arises what conditions on a(x) are needed at all.Theorem 5 shows that even to constant b(x) there are a(x) such that b(x) is not maximal with respect to a(x).In § 8 it is made clear that Theorem 5 expresses some necessary condition.Nevertheless there is some gap between sufficient conditions of Theorems 3,4 and the necessary condition of Theorem 5.Example 1 in [1] shows that the unit matrix is not strongly maximal with respect to a(t, X, y) = 0 and with respect to Q = (O, L) x D where D is a square.A far going generalization of this result is Theorem 6.1. Definitions and notations.Let R" denote the n-dimensional EucHdean space with a norm I. |.Let ß = (0, L) x D be a given region in JR"+ i where Lis a positive number and D is a region in R".D denotes the closure of D and D the boundary of D. Put S = X D, Let ß be a set, J^ a cr-field of subsets of Q and P a probability measure on #".Random variables and processes may be considered as J^-measurable functions on Q.We assume that the structure of Q, ^, P enables us to express every random process as an J^-measurable function on Q.