y 容易看出{Zn}是关于Fn的鞅，即，若m<n,则 E[Zn Fm]Zm 特别地，E[Z]=0.此外，如果假定E[B1<o∞，则 Var[Zn]=E[Z]=E[B] 2=1 事实上， 5/47 Z=∑BX?+2∑ BiBiXiXi i=1 1≤i<j≤n 再注意到X=1,如果i<j,则B,X,B,都是Fj-1可测 的，且X独立于F)-1,于是由定理？？，得 EBiBjXiXil EE BiBjXiXiF1]=E[BiBiXiE(Xi]=0 GoBack FullScreen Close Quit5/47 kJ Ik J I GoBack FullScreen Close Quit N¥w—{Zn}¥'uFnß=ßem < nßK E[Zn|Fm] = Zm AO/, E[Zn] = 0. d ßXJb½E[B2 n ] < ∞ßK V ar[Zn] = E[Z 2 n ] = X n i=1 E[B 2 i ] Ø¢˛ß Z 2 n = X n i=1 B 2 i X2 i + 2 X 1≤i<j≤n BiBjXiXj 25øX2 i = 1ßXJi < jßKBi, Xi, Bj—¥Fj−1åˇ ßÖXj’·uFj−1ßu¥d½n??ß E[BiBjXiXj] = E[E[BiBjXiXj|Fj−1]] = E[BiBjXiE(Xj)] = 0