How to compute this integral numerically ?

Dear guys,

working on option pricing problems, I have come up with one integral ( details see the function h(x) on the attached file) that I find a lot of difficulties to compute . It would be great if you could have a look and give me some ideas on it.

Thanks,
 

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Dear Alex

since the upper/lower limits of the integrals are not constant so Monte carlo cannot be applied ( I should say it is very hard to apply )
 
How about reduction to an initial value problem for an ODE and then solving a problem on a new domain using a Runge–Kutta type method?
 
Hey, why not just using rectangal approximation using mid-point or lower-point? I don't see any problem, since most variables are given, so you just need to calculate each integral using rectangle approximation starting from the very inside one. But since X is random variable, you need to run more loops for values of X and then take average.
 
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