Numerical Double Integral in Matlab

I have to do the following integration in matlab

b is log normally distributed and I have a vector of 100 random number for b call b_rand.

a is a vector of numbers that denotes a Markovian transition probability matrix

[0.1680 0.4098 1.0000 2.4400 5.9537]

I wrote the code as follows, but I am not confident I am doing the right thing. Could someone help?

    inner_term = W_t./b
        outer_term = a
        final_term = inner_term.*outer_term
        integral = mean(mean(outer_term)

[Walter Roberson]

b being log-normally distributed is not compatible with doing an integration with respect to db .

If you have sufficient specific b values then potentially you could do an approximation of the integral using trapz()

[Prerna Mishra]

I have a vector of 100 values for b:

1.8352

1.4867

1.5380

2.3116

1.2344

3.0172

1.1257

0.6769

2.3303

3.0776

2.6294

2.3909

0.6308

1.5471

1.5040

2.0812

1.7468

1.6081

1.5864

2.7595

1.9314

2.9608

2.8772

2.5619

4.0533

5.7075

1.3209

3.0450

1.2633

2.8767

2.1860

1.5720

0.7313

3.5476

1.4644

2.7337

2.0794

1.1249

0.7128

1.5698

2.9628

0.7943

1.5670

1.5715

1.4503

0.9092

1.9785

2.2779

1.7367

2.7557

3.3738

1.0204

4.4439

1.7279

4.4764

0.6940

3.0422

2.4732

1.8714

1.3300

3.0580

1.2416

2.8013

0.8385

4.4282

1.9283

1.1553

0.7967

2.5670

5.5474

1.1477

1.8341

2.0979

2.8339

1.2942

1.6179

2.9810

1.6754

1.6775

2.7661

1.4816

1.3977

1.0378

1.0305

3.1925

2.0005

1.3732

0.8174

0.9816

1.9778

2.0038

0.7942

3.0344

2.4299

2.0695

2.5443

2.1619

1.1240

3.8042

1.2711

Would I need to make a nested call to trapz?

[Torsten]

@Prerna Mishra

Do you have a link on how to evaluate stochastic integrals ? Not with trapz - that's for sure.

[Prerna Mishra]

@Torsten Not sure what you mean by link.

[Torsten]

I mean if you know of a numerical method on how to evaluate stochastic integrals and if you know a web page where this is explained.

[Prerna Mishra]

No, I dont know. If I have a sufficiently long vector of log normally distributed random variables, it is possible to use the trapeziod rule, i.e say if I have a vector of 100 variables with the lbounds being the lowest and highest value in the vector.

[Torsten]

If I have a sufficiently long vector of log normally distributed random variables, it is possible to use the trapeziod rule, i.e say if I have a vector of 100 variables with the lbounds being the lowest and highest value in the vector.

Sorry, but this is nonsense. How does the fact that the values for b are generated from a lognormal distribution influence the integral ? b is the independent, not the dependent variable. If you use the trapezoidal rule, you will get the same value as if you use a uniform grid between b_l and b_h.

And how can you Markovian transition probability matrix have values greater than 1 ?

[Walter Roberson]

you will not get the same value, but for a sufficiently dense random sample the value will approach what you would get with a comparably dense uniform grid.

Answer 1

[Chunru]

It seems that the integration is separatable into two 1-D integrations:

inner_term = W_t./b
outer_term = a
final_term = inner_term.*outer_term
integral = trapz(b, inner_term) * trapz(a, outer_term)

[Prerna Mishra]

It is separable, yes. I will try it out this way.