What represent the stochastic integral?

Question

I don't really understand what represent intuitively the stochastique integral : What is intuitively $$\int_S^T X_tdB_t \ \ ?$$ Where $(B_t)$ is the Brownian motion. An interpretation would have been to use Stiljes integral, but since the Brownian motion has not bounded variation, the Stiltjes integral is not well defined. So why do we want to create such an "integral" of the form $\int X_s dB_s$ ? Because as Ito integral or Stronovitch integral is defined, it's rather different than a generalisation of Riemann integral (since it's defined as $$\lim_{n\to \infty}\sum_{k}X_{\xi} (B_{t_{k+1}}-B_{t_k}),$$ and $\xi$ can't be unspecified in $[t_k,t_{k+1}]$ contrary to Stiltjes integral). So why using the symbol of integral ?

Answer 1

We specify $\xi_i$ (in the case of the Ito integral) as the lower limit of each subinterval. We then, with some loss of generality specifiy the integral as the limit of the "fineness" of the partitions, so we get something $$ \int_a^b X_sdB(s)= \lim \sum_{i=1}^n X_{t-i}(Bt_i-Bt_{i-1}). $$ The fact that the Brownian motion is of infinte variance is not important, it works because we require $X_t$ to be adapted! The limit is also in prob and not pathwise. I would guess that the integral notation comes from the striking resemblance to Darboux integrals. It gives us a nice way to handle "derivatives" of functions of adapted processes as well, in analogy with integrals.