This is a cross-post of my unanswered (more than a week) question on Math.SE. Since it covers topics from my graduate-level course on stochastic processes, I thought it might be appropriate to try to ask it here.

Prove or Disprove:A process $(X_t)_{t \ge 0}$ is a (continuous) local martingale if and only if it can be represented in the form: $$\int_0^t \xi dB = \large B_{\int_0^t \xi_s^2 ds} $$ where the quadratic variation $\langle X \rangle_t=\int_0^t \xi_s^2 ds$ for some increasing finite variation process $\xi$ (whose paths are therefore absolutely continuous with respect to Lebesgue measure).In particular, a (continuous) local martingale is a deterministic time change of Brownian motion if and only if its quadratic variation is a deterministic function (and absolutely continuous with respect to Lebesgue measure).

Basically I want to know how much the theory of stochastic integration/local martingales can be reduced to the study of Brownian motion and finite variation processes *only*.

Is this the same thing as saying thing that all local martingales are Ito processes and vice versa?

**EDIT:** See here for my latest attempt to solve this problem using Stephan Sturm's answer below.

**Related:** this question covers a lot of similar ground, although it does not address how that specific representation can be made to relate to time changes. https://math.stackexchange.com/questions/1457270/representation-theorem-for-local-martingales

This question on MathOverflow seems somewhat similar in spirit: The only continuous martingales with stationary increments are Brownian motions

Also, and most importantly, these two blog posts by George Lowther (who answered the above linked MO question) are the primary basis for my question, because I think the answer might be contained somewhere within the two posts, but I am having difficulty putting everything together.

https://almostsure.wordpress.com/2010/04/20/time-changed-brownian-motion/ https://almostsure.wordpress.com/2010/04/13/levys-characterization-of-brownian-motion/

**Motivation:** Basically I am trying to devise as simple of an introduction to the concept of local martingale as possible for my study guide for stochastic integration, to be found here.

I find the most commonly used characterization, as a localization of a martingale, to be unsatisfactory for several reasons.

First, it affords none of the intuition developed from studying martingales in the discrete-time case, since in the discrete time case, local and regular martingales coincide. Hence, for someone learning for the first time, it is not immediately clear why the two concepts shouldn't coincide in continuous time, or why localization of processes should be at all interesting, when it is not at all interesting in discrete time.

Second, the commonly used intuition that "a martingale is just an integrable local martingale" is incorrect, as Rick Durrett found out the hard way after publishing the first edition of his textbook on stochastic calculus. In fact the necessary and sufficient condition (Theorem 2.5 on p. 41) is $$\mathbb{E}\left( \underset{0 \le s \le t}{\sup} |X_s| \right) < \infty $$ which is a lot less elegant than would be desired.

Third, and along the lines of the second objection, as noted on p. 123 in Revuz and Yor, local martingales are *much more general* than martingales, not just "a little more", making the name "local martingale" something of a misnomer. Revuz and Yor list at least two integrability classes of stochastic processes, $(D)$ and $(DL)$ on p. 124 which one presumably would think of before thinking of the concept of local martingale.

Fourth and finally, local martingales as a concept were introduced in order to generalize the integral with respect to Brownian motion developed by Ito, *not* to generalize martingales, so I think it would be more appropriate to characterize them in terms of stochastic integration. Both my professor, who mentioned Kunita and Watanabe's work, as well as Revuz and Yor in their remark preceding Theorem 1.8 on p.124, state that local martingales as a concept were introduced in order to advance the theory of stochastic integration.

By the Bachteler-Dellacherie theorem, which tells us that the most general class of "good" stochastic integrands is equal to a local martingale up to (plus) a finite variation process, if we were able to characterize local martingales entirely in terms of Brownian motion and finite variation processes, I think it would be an impressive conceptual achievement. Maybe not a practical one, but a conceptually elegant one, which is what interests me at the moment.