Analogues of Conditional Wiener Integrals with Drift and Initial Distribution on a Function Space

作者：Cho Dong Hyun^*

来源：Abstract and Applied Analysis, 2014, 2014: 916423.

DOI：10.1155/2014/916423

Let C[0,T] denote a generalized Wiener space, the space of real-valued continuous functions on the interval [0,T], and define a stochastic process Z : C[0,T]x[0,T] -> R by Z(x,t) = integral(t)(0) h(u)dx(u)+x(0)+a(t), for x is an element of C[0,T] and t is an element of [0,T], where h is an element of L-2[0,T] with h not equal 0 a.e. and a is a continuous function on [0,T]. Let Z(n): C[0,T] -> Rn+1 and Z(n+1): C[0,T] -> Rn+2 be given by Z(n)(x) = (Z(x,t(0)), Z(x,t(1)), ... , Z(x,t(n))) and Z(n+1)(x) = (Z(x,t(0)), Z(x,t(1)), ... , Z(x,t(n)), Z(x,t(n+1))), where 0 = t(0) < t(1) < ... < t(n) < t(n+1) = T is a partition of [0,T]. In this paper we derive two simple formulas for generalized conditional Wiener integrals of functions on C[0,T] with the conditioning functions Z(n) and Z(n+1) which contain drift and initial distribution. As applications of these simple formulas we evaluate generalized conditional Wiener integrals of the function exp{integral(T)(0) Z(x,t)dm(L)(t)} including the time integral on C[0,T].

• 出版日期2014