If I find out that a process $X_t$ has the same characteristic of a Brownian motion, namely: $B_0=0$ a.s., stationary and independent increment, $B_t-B_s$ is $N(0,t-s)$ and continuous sample path a.s.
1) Can I conclude that $X_t$ is a modification of the Brownian motion?
2) What if $X_t$ has all the property of the B.M. except a.s. continuous sample path?
If I can't conclude that is a modification (namely question 1) is false), how could I show that a process with that characteristic is a modification of a B.M.?
If this is a silly question please let me know anyway.
1) It's the definition of a standard BM.
2) $(X_t)$ in this case is called a pre-Brownian Motion. There exists a modification of $(X_t)$ having continuous sample paths (e.g., Thereom 2 on page 14 here).