Increasing process

An increasing process is a stochastic process...


 * $$(X_t)_{t \in M}$$

...where the random variables $$X_t$$ which make up the process are increasing almost surely and adapted:


 * $$0=X_0 \leq X_{t_1} \leq \cdots . $$

A continuous increasing process is such a process where the set $$M$$ is continuous.

Consider a stochastic process $$(\Chi_t)$$ satisfying $$X_t \leq X_s$$ a.s. for all $$t \leq s$$  My question is: Does there exist a modification $$\breve{X}$$ of ,$$X$$ which almost surely has increasing sample paths $$t \mapsto \breve{X}_t(\omega)$$?