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1.令（; F; P）为i.i.d的概率空间。随机变量（Yn）n¸1是

k] =¾2

Xn =
ÃX
k = 0
k
！2
·n¾2
;

mar wrt（Fn）n¸0。
2.令M =（Mn）n¸0为w（Fn）n¸0。令（Gn）n¸0为
（Fn）n¸0，即Gn½Fn； 8n。对于每个n，Dene Xn = E [MnjGn]。证明（Xn）n¸0是一个
ting wrt（Gn）n¸0。
3.令（Xn）n¸0为w（Fn）n¸0。证明对于任何整数k·l·m

E [（Xm¡Xl）Xk] = 0：
4.令X1; X2; ：：：是独立的，并且均匀分布，其中Á（µ）= E（expfµX1g）<
1.进一步，令（Fn）n¸0为（Xn）n¸0的自然比率，Sn = S0 + X1 + ::: + Xn。

i）（Mn）n¸0与
Mn = expngSng
Á（µ）n

ii）（Zn）n¸0，其中Zn = expfµSng; µ> 0；是关于（Fn）n¸0的子集市

1. Let (­;F; P) be a probability space on which the i.i.d. random variables (Yn)n¸1 are
de¯ned. Suppose E[Yk] = 0 and E[Y 2
k ] = ¾2
and let Y0 = 0. Let
Xn =
Ã n X
k=0
Yk
!2
¡ n¾2
;
for each n ¸ 0. Let (Fn)n¸0 be the natural ¯ltration of (Yn)n¸0. Show that (Xn)n¸0 is
a martingale wrt (Fn)n¸0.
2. Let M = (Mn)n¸0 be a martingale wrt (Fn)n¸0. Let (Gn)n¸0 be a sub¯ltration of
(Fn)n¸0, i.e Gn ½ Fn; 8n. De¯ne Xn = E[MnjGn], for each n. Show that (Xn)n¸0 is a
martingale wrt (Gn)n¸0.
3. Let (Xn)n¸0 be a martingale wrt (Fn)n¸0. Show that for any set of integers k · l · m
the di®erence Xm ¡ Xl is uncorrelated with Xk. That is,
E[(Xm ¡ Xl)Xk] = 0:
4. Let X1;X2; : : : be independent and identically distributed with Á(µ) = E(expfµX1g) <
1. Further, let (Fn)n¸0 be the natural ¯ltration of (Xn)n¸0, and Sn = S0+X1+: : :+Xn.
Show that:
i) (Mn)n¸0 with
Mn = expfµSng
Á(µ)n
is a martingale with respect to (Fn)n¸0.
ii) (Zn)n¸0 with Zn = expfµSng; µ > 0; is a submartingale with respect to (Fn)n¸0
if the random variables Xn are standard-normal distributed. Calculate E[Zn].

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