The equation is a deterministic equality for every measurable bounded function B:[0,1]\to R by a theorem on existence of the Stieltjes integral, since \psi_n is of bounded variation. For simple explanation see e.g.

https://faculty.etsu.edu/gardnerr/4217/notes/6-3.pdf

Thus, there is no need to refere to Ito stochastic integration. To the contrary, I think that the picture presents an introduction to the integral developed on next pages, which is supposed to be obvious - at least to those familiar with Rieman-Stieltjes integral. The formula could be seen as an application of a deterministic integral wrt to random function (more precisely - it is defined almost surely).

Dear Prof Domsta

Thanks for helping for example in picture bellow

we have P and we compute integral. In this case I need answering for B like P

@ Majid Erfanian

Sory, but I have no idea about the questions:

1. what is \psi(2i-1/m, 2j-1/m)?

2. what is h_l ?

3. what is right: P(h(t)) or (P(h))(t)

3. what does it mean that you need answering for B like P?

Regards, Joachim

Dear prof Domsta

Thanks for answering

"h_l" is Haar Wavelet please looking

Article Rationalized Haar wavelet bases to approximate solution of n...

now we need compute B like P