We prove existence (and simpleness) of the trace for both forward and backward Loewner chains under fairly general conditions on semimartingale drivers. As an application, we show that stochastic Komatu-Loewner evolutions SKLE$_{α,b}$ are generated by curves. As another application, motivated by a question of A. Sepúlveda, we show that for $α>3/2$ and Brownian motion $B$, the driving function $|B_t|^α$ generates a simple curve for small $t$. On a related note we also introduce a complex variant of Bessel-type SDEs and prove existence and uniqueness of strong solution. Such SDEs appear naturally while describing the trace of Loewner chains. In particular, we write SLE$_κ$, $κ<4$, in terms of stochastic flow of such SDEs.