Consider a continuous martingale $(X_t)_{t≥0}$ starting at 0 and define the associated Local Time at $a$:
$$L^a_t=\lim_{ϵ→0} {1 \over 2ϵ}∫_0^t 1_{[a−ϵ,a+ϵ]}(X_s)ds.$$
Is it possible for the local time $L^a_t$ to have a derivative in the spatial variable $a$?
Under what conditions is the function $g(a) = E[L^a_t] $ differentiable?