Give 2 different continuous martingales with independent inc

Give 2 different continuous martingales with independent increment

(not multiples of each other)

Solution

every continuous martingale XX with stationary independent increments is a Brownian motion or, to be precise, X=X0+BtX=X0+Bt for a standard Brownian motion BB and constant . This is because any such process is a Lévy process, and Brownian motions (possibly with drift) are the only continuous Lévy processes. This is standard, and most decent book on continuous-time stochastic processes should show this. I also have a proof of this on my blog here.

However, you do not mention the necessary condition that the increments of XX are independent. So, the statement in the question is false. There do in fact exist continuous martingales with stationary increments which are not Brownian motions. You can take

Xt=t0Y,dBXt=0tY,dB

for a standard Brownian motion BB and independent stationary process YY. Then XX has stationary increments, but is not a Brownian motion unless YY is constant. For example, YY could be an Ornstein-Uhlenbeck process started in its stationary distribution.