  Processes which arise as solutions to stochastic differential equations
involving the local time (SDELTs), such as skew Brownian motion, are frequent
sources of inspiration in theory and applications. Existence and uniqueness
results for such equations rely heavily on the It\^o-Tanaka formula. Recent
interest in time-inhomogeneous SDELTs indicates the need for comprehensive
existence and uniqueness results in the time-dependent case, however, the
absence of a suitable time-dependent It\^o-Tanaka formula forms a major
barrier. Rigorously developing a two-parameter integral with respect to local
time, known as the local time-space integral, we connect together and extend
many known formulae from the literature and establish a general time-dependent
It\^o-Tanaka formula. Then, we prove the existence of a unique strong solution
for a large class of time-inhomogeneous SDELTs.
