To motivate stochastic integration, the book I am working on start with $H$ a stochastic process that is simple :
$H_t = \sum_{i\geq 1}H_{t_{i-1}}1_{[t_{i-1}, t_i)}(t)$ where $0= t_0 < t_1 < ... $ and $t_n\to\infty$ as $n\to\infty$.
Then it is said that "it is natural to set : $\int_{0}^{t} H_s dW_s = \sum_{i\geq 1}H_{t_{i-1}}(W_{t\wedge t_i} - W_{t\wedge t_{i-1}})$"
Clearly, this is not at all natural for me, someone knows why it should be ?
I suppose that $W_t$ is a brownian motion and the fact we are talking about a simple stochastic process makes me think of the construction of the Lebesgue integral.
But here I don't see any measure to "weight" $H$ so the intuition one cane have in Lebesgue integral is, at least for me and until now, inaccurate here.
Also, since $H$ is a stochastic process (and depends also on $t$) this integral has nothing to do with what we know with random variable and the fact that, if $X$ is integrable, $\mathbb{E}[X]$ is totally deterministic while the integral above does not remove this "random" dimension since the right hand side is clearly random
Thank you a lot !