I have some (beginner) questions related to terms of the form $\int_{0}^{t}X_{s}\:ds$, because I am dealing with the following problem: I would like to obtain a decent estimate (polynomial in $t$ and $\vert y\vert$, if possible) for an expression like
$$\mathbb{E}\left[\exp{\left(\int_{0}^{t}\log{(\vert W_{s}-y\vert)}\:ds\right)}\right],$$ where $W_{s}$ is a d-dimensional Brownian motion and $y\in \mathbb{R}^{d}$.
1.)If we assume the process $X_{t}$ to have continuous paths, this term should be well defined as a Riemann or Lebesgue integral, path-wise at least (right?).
2.)If so would it make sense to apply, for example, a version of Jensen's inequality (pathwise) and write the following: Let $g$ be a convex function, then for every $\omega$ $$g\left(\frac{t}{t}\int_{0}^{t}X_{s}\:ds\right)\leq \frac{1}{t} \int_{0}^{t}g(tX_{s})\:ds.$$ From this, follows $$\mathbb{P}\left(g\left(\int_{0}^{t}X_{s}\:ds\right)>x\right)\leq\mathbb{P}\left(\frac{1}{t}\int_{0}^{t}g(tX_{s})\:ds>x\right)$$
for all $x\in \mathbb{R}$. Hence $$\mathbb{E}\left[g\left(\int_{0}^{t}X_{s}\:ds\right)\right]\leq \mathbb{E}\left[\frac{1}{t}\int_{0}^{t}g(tX_{s})\:ds\right].$$
In the special case above this would mean $$\mathbb{E}\left[\exp{\left(\int_{0}^{t}\log{(\vert W_{s}-y\vert)}\:ds\right)}\right]\leq \mathbb{E}\left[\exp{\left(\log{\left(t\int_{0}^{t}\vert \frac{W_{s}-y}{t}\vert\:ds\right)}\right)}\right]\leq\mathbb{E}\left[\int_{0}^{t}\vert W_{s}-y\vert\:ds\right].$$
Is this correct or did I make a stupid mistake anywhere?
3.) How "strong" are pathwise estimates, are there cases where pathwise estimates may lead one "intuitively" in the wrong direction?