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
Best Answer
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.
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.