w*****n 发帖数: 9 | 1 【 以下文字转载自 Mathematics 讨论区 】
发信人: weljohn (a P), 信区: Mathematics
标 题: 弱问两个问题
发信站: BBS 未名空间站 (Sun Feb 24 21:07:11 2008), 转信
B(t)是brownian motion, normally distributed as N(0,t)
请问如何计算 E(B(t)B(s)) 和 Var(integral of B(t)dt)
多谢 |
z*g 发帖数: 110 | 2 the above answers where wrong.
1. the answer is min(s,t).
2. the answer is integral_integral_min(s,t)_ds_dt. and I think it is T^3/3. |
w*****n 发帖数: 9 | 3 can you show more details? thanks
【在 z*g 的大作中提到】 : the above answers where wrong. : 1. the answer is min(s,t). : 2. the answer is integral_integral_min(s,t)_ds_dt. and I think it is T^3/3.
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i******d 发帖数: 54 | 4 Shreve's book has discussions on these. |
i******d 发帖数: 54 | 5 i also saw second one on heard on the street. |
Q***5 发帖数: 994 | 6
(1) By definition of BM, E((B(t)-B(s))^2) = t-s (assume t>s)
So E(B(t)^2)-2E(B(t)B(s))+E(B(s)^2) = t-s, and we know E(B(s)^2) =s, hence
the result: E(B(t)B(s)) = s
(2) I guess this one use some kind of change of oder of integeration:
\int_0^T B_t dt = \int_0^T \int_0^t d(B_s) dt = \int_0^T\int_s^T dt dB_s = \
int_0^T (T-s)dB_s
The variance of this random variable is \int_0^T (T-s)^2ds
Can some experts shed light on the condition of changing order of
integeration in stochastic calculus?
【在 w*****n 的大作中提到】 : can you show more details? thanks
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i******d 发帖数: 54 | 7 according to Fubini theorem
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【在 Q***5 的大作中提到】 : : (1) By definition of BM, E((B(t)-B(s))^2) = t-s (assume t>s) : So E(B(t)^2)-2E(B(t)B(s))+E(B(s)^2) = t-s, and we know E(B(s)^2) =s, hence : the result: E(B(t)B(s)) = s : (2) I guess this one use some kind of change of oder of integeration: : \int_0^T B_t dt = \int_0^T \int_0^t d(B_s) dt = \int_0^T\int_s^T dt dB_s = \ : int_0^T (T-s)dB_s : The variance of this random variable is \int_0^T (T-s)^2ds : Can some experts shed light on the condition of changing order of : integeration in stochastic calculus?
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