# Change of Time in Stochastic Integral

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

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I think I have found an interesting article here : arxiv.org/PS_cache/arxiv/pdf/0906/0906.5385v1.pdf In particular lemma 2.3 seems to be the kind of result I was looking for –  The Bridge Aug 6 '10 at 13:22