It is well-known that well-posedness of a martingale problem in the class of continuous (or r.c.l.l.) solutions enables one to construct the associated transition probability functions. We extend this result to the case when the martingale problem is well-posed in the class of solutions which are continuous in probability. This extension is used to improve on a criterion for a probability measure to be invariant for the semigroup associated with the Markov process. We also give examples of martingale problems that are well-posed in the class of solutions which are continuous in probability but for which no r.c.l.l. solution exists.

We give a construction of an explicit mapping

$$\Psi: D([0,\infty),\mathbb{R})\to D([0,\infty),\mathbb{R}),$$

where $D([0,\infty), \mathbb{R})$ denotes the class of real valued r.c.l.l. functions on $[0,\infty)$ such that for a locally square integrable martingale $(M_t)$ with r.c.l.l. paths,

$$\Psi(M.(\omega))=A.(\omega)$$

gives the quadratic variation process (written usually as $[M,M]_t$) of $(M_t)$. We also show that this process $(A_t)$ is the unique increasing process $(B_t)$ such that $M_t^2-B_t$ is a local martingale, $B_0=0$ and

$$\mathbb{P}((\Delta B)_t=[(\Delta M)_t]^2, 0 < \infty)=1.$$

Apart from elementary properties of martingales, the only result used is the Doob’s maximal inequality. This result can be the starting point of the development of the stochastic integral with respect to r.c.l.l. martingales.