数传在 容易看出{Zm}是关于Fn的鞅，即对于所有的n 1951 E[Zn+1 Fn]Zn 特别地，E[Z]=0.此外，如果假定E[B]<∞，则 2 Var[n]=EZh=∑EB] i=1 事实上， 5/50 Z=∑BX好+2 BiBiXiXj i=1 1≤i<j≤n 再注意到X=1,如果i<J,则B,X,B都是F-1可测 的，且X独立于F-1,得 E BiBXiXi=EE BiBiXiXi Fi-1=E BiBiXiE(Xi=0 GoBack FullScreen Close Quit5/50 kJ Ik J I GoBack FullScreen Close Quit N¥w—{Zn}¥'uFnß=Èu§kn E[Zn+1|Fn] = Zn 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−1ß E[BiBjXiXj] = E[E[BiBjXiXj|Fj−1]] = E[BiBjXiE(Xj)] = 0