Hi everyone,
Let's be given $I(0,t)$ a Stochastic Integral with respect to a local martingale $ M_t$ of the form :
$I(0,t)=\int_0^t h(s_-)dM_s$ with $h\in L(M)$ (for example $h$ is an adapted locally bounded processes)
Now, if I want to make a change of variable with respct to time with a given increasing (càdlàg, or continuous or why not even smooth) fonction $g:[0,t]\to [0,t]$ then is it possible to express $I(0,t)$ with respect to the time changed local martingale $M_{g(s)}$ ?
I would expect something like : $I(0,t)=\int_{0}^{t}h(g(s)-)dM_{g(s)}$
The thing is that : -I am not quite sure about this result -Not sure about the way to prove it if it is true -Not sure about what condition on $g$ to impose
I would be pleased with answers or references for special cases like Brownian Motion Integrator or Lévy Processes, or with specific conditions on $g$.
Best Regards
