Integration wrt to "absolute" Brownian motion (or a Levy process)

Question

I am trying the bound a term which looks like the following expression:

$$ \mathbb{E} \left| \int_{0}^t f(s) dL_s \right| $$

where $L_t$ is an $\alpha$-stable process with $\alpha >1$. I want to say something like

$$ \mathbb{E} \left| \int_{0}^t f(s) dL_s \right| \leq \int_{0}^t f(s) d|L|_s $$

but it is not clear to me what integration with respect to $d|L|_t$ would mean. Does it make sense to write the integration in the following form

\begin{align} \mathbb{E} \Biggl| \int_{0}^t f(s) dL_s \Biggr| =& \mathbb{E} \Biggl| \lim_{K\to \infty} \sum_{k=0}^{K-1} f \left( \frac{kt}{K} \right) \left(L_{\frac{(k+1)t}{K} } - L_{\frac{kt}{K} } \right) \Biggr| \\ \leq& \mathbb{E} \left[ \lim_{K\to \infty} \sum_{k=0}^{K-1} f \left( \frac{kt}{K} \right) \left|L_{\frac{(k+1)t}{K} } - L_{\frac{kt}{K} } \right| \right] \end{align}

then use this expression for the later computations? Does $d|L|_t$ has a name so that I can search for that keyword?

Answer 1

As mentioned here Which inequalities are there with stochastic integration?, triangle inequality bounds

$$|\int f dB|\leq \int |f| d|B|$$

are not possible in general because for example Brownian motion has infinite total variation.

But in terms of bounding $E|\int f dB|$, there are a couple of ways. More directly, if $f$ is deterministic, we use that

$$\int^t f dB\sim N(0,\sigma_{f}^{2}(t)),$$

where $\sigma_{f}^{2}(t):=\int_{0}^{t}f^{2}(s)ds$. So from here we can simply use Expected absolute value of normal distribution.

If $f$ is also bounded, we compare the integral to $\|f\|_{\infty} B_{t}\sim N(0,\|f^2\|_{\infty} t)$. Since $\|f^2\|_{\infty} t\geq \sigma_{f}^{2}(t)$, we get that

$$E|\int^t f dB|=\int P(|\int^t f dB|\geq x)dx\leq \int P( \|f\|_{\infty} |B_{t}|\geq x)dx\leq E\|f\|_{\infty} |B_{t}|.$$

You can also use Cauchy-Schwartz and Itô isometry, which works even for random $f$

$$E|\int^t f dB|\leq \sqrt{E\int^t f^{2}(s)ds}.$$

A similar bound can work for Lévy processes since they too satisfy a type of isometry. See here for general semimartingales https://almostsuremath.com/2010/03/29/quadratic-variations-and-the-ito-isometry/. For an Ito-Levy process $X_t$ with the dynamics given by $$ \mathrm{d}X_t = \sigma_t \mathrm{d}B_t + \int_{|z|<1}\gamma_t(z) \tilde{N}(\mathrm{d}t, \mathrm{d}z) + \int_{|z|\ge 1}\gamma_t(z) {N}(\mathrm{d}t, \mathrm{d}z), $$ where $B_t$ is a standard Brownian motion, $\tilde{N}(\mathrm{d}t, \mathrm{d}z)$ is the compensated Poisson random measure and ${N}(\mathrm{d}t, \mathrm{d}z)$ is the Poisson random measure. We have the following theorem (source: Th 1.17 of Applied Stochastic Control of Jump Diffusions by Agnès Sulem and Bernt Øksendal): $$ \mathbb{E}\left[X^2_T \right] = \mathbb{E}\left[ \int_0^T \left\{\sigma^2_t + \int_{\mathbb{R}} \gamma^2_t(z) \nu(\mathrm{d}z) \right\} \mathrm{d}t\right]. $$