How to code integral of product of lognormal variables?

I am trying to code integral of a product of variables which follow different distributions. Here's a simple example:

where b is a scalar (let's say 0.65), ϵ and z are lognormal variables, and σ is a variable that follows Uniform distribution. Here, is a particular value that σ takes which comes from solving another equation not shown here. How can I code this expression? How can I handle a variable integral limit in such cases ( in this example)?

Answer 1

[Askic V]

Please have a look at how this example would be solved in Matlab:

syms x y z;
f = @(x,y,z) 2*x.*exp(y).*sin(z); % define the input function
Q = integral3(f,1,2,0,1,0,pi)

Q = 10.3097

[VS]

Thanks @Askic V. I knew about integral3 but wasn't sure if it would do what I need it to do. Does it matter for numerical integration what distribution the variables follow? Is there a way to do this without using symbolic toolbox because rest of my code will not use it and I have to run this code with another plugin where I am not sure if symbolic integral will work. And how does one handle variable integral limit?

[Torsten]

I'd first try with the symbolic toolbox since when working with distributions, numerical integration often gives bad results. If the symbolic integration is successful, you can copy the "solution" as is into your code and quit symbolic computation for the rest of your coding.

[VS]

Thanks @Torsten. Is there a way to handle variable integral limit in multiple integration? I have multiple integrals whose lower or upper limit is a variable that I obtain by solving another equation.

[Torsten]

Yes, define the integral as a function of the limit and evaluate the integral when the limit becomes known as a numerical value.

Or use symbolic computations with the limit as a symbolic variable. If you are lucky, the symbolic toolbox will be able to compute the integral involving the symbolic limit variable.

syms a b c x lb ub
f = a*x^2+b*x+c;
F = int(f,x,lb,ub)

F =