In a lemma in my graduate level course on financial mathematics uses the fact that integral of a progressive portfolio process(which is almost surely lower bounded i.e it is admissible) $\theta_t$ wr.t to a Q-local martingale is a Q-local martingale. Why is this true? In the lecture we just talked about integration wrt to the Brownian Motion which is ofcourse a (local) martingale but how could be generalize it to the set of integrands which are local martingales. I understand that this might not be a precise question. But I would really appreciate if you could just direct me to some book or notes where I could read about it
2026-03-30 22:40:37.1774910437
A question on integration wr.t to a local martingale
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Every continuous local martingale is a continuous time change of a Brownian Motion in a possibly extended probability space. So, modulo some technicalities, all the properties obtained from integration w.r.t. Brownian Motion translates, one way or another, to properties when integrating w.r.t. to continuous time local martingales.