stochastic integration: int_0^t s W_s ds

Chemist_MZ 发表于 2013-8-25 11:32
I am sorry that I did something wrong, since there is some very complicated issue when change the  ...

You can solve it without Ito isometry

xuruilong100 发表于 2013-8-26 09:58
another approach to solve it
A^2 = 2int_0^t A_s dA
       = 2int_0^t A_s s W_s ds

Wow, you killed it nicely!

xuruilong100 发表于 2013-8-26 09:58
another approach to solve it
A^2 = 2int_0^t A_s dA
       = 2int_0^t A_s s W_s ds

ok, that's cool.

A colleague of mine showed me another approach, which is another cool way to kill it.

E(A^2) = E(int_0^t u W_u du int_0^t s W_s ds)
           = E(int_0^t   int_0^t  u W_u s W_s du ds) --- now double integral on a square where 0 < u, s < t
we can break the square integral into two triangle integrals, where u > s and s < u, respectively, that is,
E(A^2) = E(int_0^t   int_0^s u W_u s W_s du ds) + E(int_0^t   int_0^u u W_u s W_s ds du)
           = int_0^t   int_0^s u s E(W_u W_s) du ds + int_0^t   int_0^u u s E(W_u W_s) ds du
           = int_0^t   int_0^s u s u du ds + int_0^t   int_0^u u s s ds du
           = int_0^t s int_0^s u^2 du ds  + int_0^t u int_0^u s^2 ds du
           = int_0^t s s^3/3 ds + int_0^t u u^3/3 ds
           = t^5/15 + t^5/15
           = 2 t^5/15
Similar to xuruilong100' approach in a way

shelf317 发表于 2013-8-31 03:39
A colleague of mine showed me another approach, which is another cool way to kill it.

E(A^2) = E( ...

I am glad to see this heat discussion.