Brownian Motions and Martingales

Question

State whether each of the statement is true or false.

• If $Z$ is a standard normal random variable then $X_t = Z\sqrt t$ is a Brownian motion.
• If $W_t$ and $W_t'$ are two independent Brownian motions and $\rho$ is a fixed constant such that $|\rho|<1$, then $X_t = \rho W_t+ W'_t \sqrt{1-\rho%2}$ is a brownian motion.
• If $W_t$ is a Brownian motion then $X_t = \exp\left(\alpha W_t - \alpha^2t/2\right)$ is a martingale, where $\alpha$ is a fixed constant.

Any help is appreciated! As a recall, the conditions for a brownian motion are :

1. $W_0=0$

2. The sample path of $W$ is continuous and nowhere differentiable.

3. For $s\leqslant t$, $W_t-W_s$ is independent of all $W_u$ for $u\leqslant s$ and $W_t-W_s \sim N(0, t-s)$.

Answer 1

The first one is clearly false because $(\sqrt t - \sqrt s) Z$ is not independent from $\sqrt s Z$. The second is also false because $E[X_t^2] = \rho^2 t + (1-\rho) t \neq t$. For the last it is correct and to prove it use the characteristic function of normal distribution variable.