Let $W(t)$ be a Wienner process. Let's define local time as
$$L_t(x) = \lim_{\epsilon \rightarrow0^+}\frac{1}{2 \epsilon}\int_0^t \chi_{\{|W(s) - x| \le\epsilon\}} ds$$
I want to check weather this object is supermartingale, martingale or submartingale.
The hint is to use Tanak's formula:
$$|W(t) - t|= |x| + \int_0^t\text{sign}(W(s) - x)dW(s)+L_t(x)$$
I tried to do it by definition i.e. to show that $E[L_{t+h} \mid \mathbb{F}_t] = L_t$, but I had no idea how to rewrite this conditional expected value. My second idea was more tricky - to use Ito formula ale see what's the expression in $dt$, however I also end up with nothing and with no idea how the Tanak's formula can be used for simplicity. Could you please give me a hint in which direction should I follow and in which moment Tanak's formula should be used ?
EDIT
I just have several questions about things that I don't fully get.
By facts that $W$ is a local martingale and $\text{sign}(W-x)$ is bounded predictable process you concluded that $H_t$ is local martingale. Could you please tell me to which exactly theorem you are referring to ? I'm afraid that I don't know such.
The second this is that you are rewriting expected value of quadratic variation of $H$:
$$E[(\int_0^t\text{sign}(W_s - x)dWs)^2] = E[\int_0^t \text{sign}(W_s -x)^2dt]$$
This equality is just direct application of Ito Isometry theorem. Now, because $\text{sign}(W_s - x)$ takes value $1$ almost everywhere (it doesn't take it for such $x: W_s = x$) we have that $E[H] = t$. But how this proves that $H_t$ is a martingale ? I was seeking for such theorem but wasn't able to find.
The correct Tanak's formula is given by $$|W_t - x|= |x| + \int_0^t\text{sign}(W_s - x)\Bbb dW_s+L_t(x).$$ As $W$ is a local martingale and $\text{sign}(W-x)$ is a bounded predictable process the stochastic integral $H_t:=\int_0^t\text{sign}(W_s - x)\Bbb dW_s$ is a local martingale. By using the generalized Itô-isometrie ($[.]$ denotes the quadratic variation) $$\forall t\geq0:\Bbb E[H]_t=\Bbb E\int_0^t\text{sign}(W_s - x)^2 \Bbb ds=t<\infty, $$ $H$ is a martingale. Furthermore $|W - x|$ is a submartingale, as by Jensen's inequality follows $$\Bbb E\big(|W_t - x|\big|{\cal F_s}\big)\geq \Bbb |E\big(W_t - x\big|{\cal F_s}\big)|=|W_s - x|. $$ Altogether $$L_t(x)= |W_t - x| - \int_0^t\text{sign}(W_s - x)\Bbb dW_s-|x|$$ is for each $x$ the difference of a submartingale and a martingale, thus it is a submartingale.