# Why does it hold true that $\theta_{t} d\overline{X}_{t}$ is a local $Q$ martingale if $\overline{X}$ is a local $Q$ martingale

I am learning from Bernt Oksendal's Stochastic Differential Equations and on page 276 Lemma 12.1.6, it is stated that:

The existence of an equivalent martingale measure $$Q$$ on the discounted price process $$\overline{X}$$ implies that the market is arbitrage-free.

A statement in the proof has come up, that I cannot justify (I will only look at dimension $$1$$ for simplicity of notation):

Let $$\theta$$ be an admissible trading strategy, then we have $$d\overline{V}_{t}^{\theta}=\theta_{t}d\overline{X}_{t}$$ which is clear by the self-financing property. Next, since $$\overline{X}$$ is a local $$Q$$ martingale, it must follow that $$\overline{V_{t}}=\overline{V_{0}}+\int_{0}^{t}\theta_{s}d\overline{X}_{s}$$ is also a local $$Q$$ martingale $$(*)$$.

This last sentence is the part I do not understand. The assumptions that could be relevant are:

$$dX_{t}=\mu_{t} dt+\sigma _{t}dB_{t}$$

$$d\overline{X_{t}}=\overline{X}_{t}((\mu_{t}-r_{t})dt+\sigma_{t}dB_{t})$$

and $$\theta$$ is admissible if it is self-financing, lower bounded and

$$\int_{0}^{T}\sigma^{2}_{t}\theta_{t}^{2}dt< \infty \; a.s.$$ where $$\sigma$$ is fixed, progressive and satisfies $$\int_{0}^{T}\sigma^{2}_{t}dt< \infty\; a.s.$$

Now I have

What I know thus far: If $$\Phi$$ is a progressive process such that $$\int^{T}_{0}(\Phi_{t})^{2}dt<\infty\; a.s.$$, then for a brownian motion $$B$$, the stochastic integral:

$$\int_{0}^{t}\Phi_{s}dB_{s}$$ is a local martingale on $$[0,T]$$.

Now to prove $$(*)$$ I think I need to show $$\int^{T}_{0}(\Phi_{t})^{2}dt<\infty\; a.s.$$ , which I have been unable to do thus far. And even then, $$\overline{X}$$ is certainly not necessarily a Brownian motion, so I am lost as to how I can justify why $$\theta_{t}d\overline{X}_{t}$$ is indeed a local $$Q$$ martingale. Any ideas that I am missing?