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Quant版 - 弱问两个问题 (stochastic calculus)
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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.

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
i******d
发帖数: 54
7
according to Fubini theorem

\

【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?

1 (共1页)
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