Максимальное неравенство для косого броуновского движения

M. Zhitlukhin MAXIMAL INEQUALITY FOR SKEW BROWNIAN MOTION
Let $X_t^\alpha$ be a skew Brownian motion with parameter $\alpha\in(0,\,1)$ and $\tau$ be an arbitrary stopping time for $X_t^\alpha$. We prove the following maximal inequality: \[ \mathsf{E}[ \max_{s\le \tau} X_s^\alpha - \min_{s\le\tau} X_s^\alpha ] \le \sqrt{K_\alpha \mathsf{E}\tau}, \] where $K_\alpha$ is some constant dependent on $\alpha$. We find an explicit expression for $K_\alpha$ and also show that the inequality is strict in some sense.
This inequality can be viewed as a generalization of the known inequalities for a standard Brownian motion and its modulus.
A. Muravlev On some properties of the local time
Let $X = (X_t)_{t\ge0}$ be a one-dimensional regular diffusion on the interval $I \subset \mathbb{R}$ and $\mathcal{G}$ is its generating operator. For $\alpha > 0$ functions $\fpsi$ and $\fphi$ are the unique (up to a multiplicative constant) continuous solutions of the generalized differential equation
$$ \mathcal{G}u = \alpha u, $$
where $\psi_\alpha$ is increasing and $\varphi_\alpha$ is decreasing. We denote
$$ \fw(x) = \fpsi'(x) \fphi(x) - \fpsi(x) \fphi'(x), \quad \frho(x,y)=\fpsi(x) \fphi(y) - \fpsi(y) \fphi(x). $$
Let us consider the local time of the diffusion $X$ at the level $x$ $L(t,x)$. In this paper, we investigate the properties of $L(\tau_{ab},x)$, where
$$ \tau_{ab} = \inf \{ t \ge 0: X_t \not\in (a,b) \}.\\ $$

\begin{theorem} For </nomathmode><mathmode>$\alpha>0$, $\beta>0$ and $a\le x\le b$
$$ \Eb_x e^{-\alpha \tau_{ab} - \beta L(\tau_{ab},x)} = \frac{\frho(a,x) + \frho(x,b)}{\frho(a,b) - \frac{2\beta}{\fw(x)} \frho(a,x) \frho(x,b)}. $$
\end{theorem}
</mathmode><nomathmode>