Conditions for strong maximality of local diffusions in multi-dimensional case
Ivo Vrkoč · Czechoslovak Mathematical Journal · 1978
Introduction.Let an Itô stochastic differential equation dx == a{t, x)ât + Б(г, x) dw be given in a region Q, Q = (0, L) x D where D is a region in the n-dimensional Euclidean space, the n-dimensional vector function a(^t, x), the matrix function Б(г, x) of the type n x n and the region D fulfil conditions guaranteeing the existence and unicity of solutions, w(t) is an n-dimensional Wiener process.Denote by x(t, XQ) the solution of the Itô equation fulfilling the initial condition x(0, XQ) -Xo (XQ being a deterministic value) and by Р(Б, a, XQ, ß) the probability that the solution x(^, Xo) leaves the region D during the time interval }} .The matrix function B{\, x) or A{t, x) (Л(г, x) = JB(r, x) B^(t, x) where Б^ is the transposed matrix) is called strongly maximal with respect to a(t, x) and Q if P{B, a, Xo, Q) ^ P{B\ a, Xo, Ô) for all XQE D and for all matrix functions B'(t, x) fulfilling the conditions guaranteeing the existence and unicity and such thatThis definition was used in the papers [1], [2], [5] with the following conditions guaranteeing existence and unicity: i) a(t, x), J5(f, x) are Holder continuous in t; ii) a(^t, x), P(f, x) are Lipschitz continuous in x; iii) A[t, x) = B[t, x) B^(f, x) is uniformly positive definite in Q; iv) the region D is bounded and has the outside strong sphere property [4].If the matrix functions B{t, x) and Ä{t, x) are diagonal at every point [t, x] G ß,