Integrate a random function

Let be a proability. Suppose is a random function, i.e. is a random variable for all . How can we evaluate

? In particular, we may assume , where the variance is a function of t.

[Bruno Luong]

To me integral of random is not mathematically defined (the function is NOT integrable).

In relation you might take a look of SDE

Answer 1

[Torsten]

Matlab tools can not be used for this problem because the integral is not simply a numerical value, but a random variable itself.

I think a statistics forum is the more appropriate place to ask your question.

[Shunan Sheng]

Noted. I just want to find a realization of that random variable, so I wonder if it is possible that some matlab function can help.

[Torsten]

Did you somewhere find a reference where e.g. Monto Carlo simulations are used to solve stochastic integrals ?

I have no experience in this field, but I have big doubts whether the naive approach (discretize the interval [0,T], generate random numbers in the discretization points according to f(t) and use "trapz", e.g. , to approximate the integral for this specific realization) makes sense.

Maybe you can find some references in the internet about "Numerical solution of stochastic integrals", if such numerical methods exist at all.

[Shunan Sheng]

Thanks for the suggestion. I will search for that direction.

[Bruno Luong]

You might read this https://en.wikipedia.org/wiki/Wiener_process

[Shunan Sheng]

Thanks!!