Which inequalities are there with stochastic integration? Which inequalities can I use with stochastic integration?
For example, with the standard lebesgue integral we have $$\left|\int_\Omega f(x) dx\right| \le M |\Omega|$$ 
(where $M$ is the maximum of $|f|$ on $\Omega$ is exists)
Also, $$\left|\int_\Omega f(x) dx\right| \le \int_\Omega |f(x)| dx$$
Plus the ever-useful Holder and Jensen inequalities
Which of these inequalities are still valid if for $$\int_0^\infty H_s dM_s$$ where $M$ is a local martingale and $H \in L^2_{\text{loc}}(M)$? Or maybe there are other type of inequalities in this case?
Can you provide a good reference to this topic, which I imagine is pretty well known? Feel free to take as many additional restriction you want on $H$ and $M$ (also the extreme of integration can very well be taken finite). A special case of interest is $M_t = W_t$ the Brownian Motion
 A: Neither of them holds, in general, for stochastic integrals. 
The trouble already starts if you consider measures which need not to be non-negative, i.e. signed measures. For a signed measure $\mu: (\Omega,\mathcal{A}) \to \mathbb{R}$ we cannot expect that the triangle inequality
$$\left| \int f(x) \, \mu(dx) \right| \leq \int |f(x)| \, \mu(dx) \tag{1}$$ holds. To see this, just consider the case that $f$ is an elementary function, i.e. $f$ is of the form
$$f(x) = \sum_{j=1}^n c_j 1_{A_j}(x).$$
Then $(1)$ reads
$$\left| \sum_{j=1}^n c_j \mu(A_j) \right| \leq \sum_{j=1}^n |c_j| \mu(A_j).$$
Note the right-hand side does not even need to be non-negative (but the left-hand side is), so this doesn't make any sense. With the same reasoning, we find that
$$\left| \int f(x) \, \mu(dx) \right| \leq \|f\|_{L^{\infty}} \mu(\Omega) \tag{2}$$
does, in general, not hold true for signed measures.  However, one can show that
$$|\mu|(A) := \sup \left\{ \sum_{n \in \mathbb{N}} |\mu(A_n)|; A_n \in \mathcal{A} \, \text{disjoint}, \bigcup_{n \in \mathbb{N}} \subseteq A \right\}$$
defines a non-negative measure, the so-called total variation norm, and that
$$\left| \int f \, d\mu \right| \leq \|f\|_{\infty} |\mu|(\Omega)$$
and
$$\left| \int f \, d\mu \right| \leq \int |f| \, d|\mu|.$$
These two are the natural generalizations of $(1)$ and $(2)$ for signed measures. 
Since stochastic integrals are "randomized" signed measures, the situation becomes even more complicated. For example if $(M_t)_{t \geq 0}$ is a Brownian motion, then the stochastic integral
$$\int_0^t H_s \, dM_s$$
is not a pointwise integral and this means that we cannot simply use the above considerations for fixed $\omega$. Additionally, there is the trouble that the Brownian motion has infinite total variation, so, as far as I can see, there is no chance to get such inequalities for stochastic integrals with respect to Brownian motion.
Very important inequalities for stochastic integrals (with respect to martingales) are e.g. 


*

*Doob's inequality 

*the Burkholder-Davis-Gundy inequality


but they don't provide any pointwise estimates.
The only exception I can think of are processes with bounded variation. In this case, we can define the stochastic integrals as a Riemann-Stieltjes integral and obtain similar estimates as for signed measures. This works in particular for processes with non-decreasing sample paths, e.g. subordinators.
