Các câu hỏi phỏng vấn [18/8/2007]

[shinichi9htv]

nice post bro!
30/ you can define it from a complex error function
31/ Excellent! Now, let's compute

32/ E(|W_t|*N(a*W_t^2)

[nguyenxuanson]

hi Quan
30/ Error function is defined on real line, you answer has no sens becuase the way you prolong it will define the "complex error function". I don't know this function and do not know how you problong it i.e which are propert of this new abstract function? If you define it by the integration representation, it does not converge, you have to do something....

[shinichi9htv]

hi Son
I think you're confused between the error function and the normal cdf. The error function is defined by 2/sqrt(PI)int_0^z exp(-y^2) dy, hence it's well defined in the complex plane. Hope it helps

[shinichi9htv]

you're absolutely right Son, forget about my non-sense question

[nguyenxuanson]

I google and found only complex prolongation for erf(x) and it is defined in http://en.wikipedia.org/wiki/Error_function If you says: then you can defined the prolongation of N(x) to the complex plan and you can easily have the formula for it, something proportional to erf(1/sqrt(2))..

PS: in which context you need to evaluate N(i)?

[nguyenxuanson]

32/ not very smart solution
a/ check that the expectation is well defined (easy becasue the N(x) is bounded)
b/ express N(x) by erf(x)
c/ erf(x) has well defined Taylor expansion, on the whole real line
d/ integrate each term...easy part
e/ sum it up, you may simplify it somehow, have not checked

If you have smart solution, let me know
Son

[shinichi9htv]

Thanks Son for your solution,

I believe the solution is compact and beautiful (haven't checked either ). If you could give us your final result, that would be great

[nguyenxuanson]

integral by part is the other way, but i do not bother to do it as well. But what happen if we need to calculate
E(N(aW^{2}_{t}). Can we still do it?

[shinichi9htv]

One powerful method to solve this kind of problem is to use Itô lemma. (The key idea is that the first and second derivatives of N(x) are simple and easy to integrate) Once the solution is obtained, there might be a simpler solution to come