l****e
发帖数: 1718 | 1 int^0_T(Bt^3)dBt,这个是normal吗？知道expectation是0， var是int^0_T(BT^6)dt,
但不会算这个var，请大牛指教。谢谢 |
x******a
发帖数: 6336 | 2 yes,
不用算，用定义,不是int_0^T EB_t^6\,dt? |
c**********e
发帖数: 2007 | |
l****e
发帖数: 1718 | 4 can you elaborate how you got this answer? thanks.
【在 c**********e 的大作中提到】
: 15/4 T^4.
|
x******a
发帖数: 6336 | 5 E(B_t^6)=15t^3.
【在 l****e 的大作中提到】
: can you elaborate how you got this answer? thanks.
|
l****e
发帖数: 1718 | 6 真的很菜，我就是stuck在这一步，不会算这个期望。。。
【在 x******a 的大作中提到】
: E(B_t^6)=15t^3.
|
x******a
发帖数: 6336 | 7 from either the fact that exp(sB_t-s^2t/2) is a martingale
or direct calculation, we can get
Eexp(sB_t)=exp(s^2t/2)=\sum_0^\infty s^(2n)*(t/2)^n/n!.
On the other hand,
Eexp(sB_t)=sum_0^\infty (s^nEB_t^n)/n!.
Compare the coefficients of the two series,
EB_t^(2n)/(2n)! = (t/2)^n/n!,
EB_t^(2n)=(2n)!(t/2)^n / n!.
n=3 ==> EB_t^6=... |
p******5
发帖数: 138 | 8 Is it true that any Ito integral has expected mean 0? |
x******a
发帖数: 6336 | 9 no
【在 p******5 的大作中提到】
: Is it true that any Ito integral has expected mean 0?
|
k*******d
发帖数: 1340 | 10 Why it is normal distributed? The integrand is not deterministic, I don't
see it immediately why it should be normal distributed.
Recall that \int_0^T B(t) dB(t) = 1/2 * (B(T)^2 - T), which is not normal
distributed. |
m*********g
发帖数: 646 | 11 anything without a drift part is a martingale. check Ito lemma
【在 p******5 的大作中提到】
: Is it true that any Ito integral has expected mean 0?
|
p*****k
发帖数: 318 | 12 kiteflied, i think you are right that it is not normal, though it's not needed
in order to apply ito isometry to get the variance. |
u******s
发帖数: 157 | 13 Is it the integrand has to be finite?
Can you give some intuition on counter examples? Thanks.
【在 x******a 的大作中提到】
: no
|
