Is this process a martingale I was solving some practice problems in stochastics and faced the following exercise:
Given Brownian motion $W(t)$ and a stochastic process $B(t)$ defined as:
$$B(t) =
\begin{cases}
W(t),  & \text{if $0 \le t < 1$} \\
tW(1), & \text{if $1 \le t < \infty$}  \\
\end{cases}$$
Answer the following:


*

*Is $B(t)$ a martingale?

*Compute $QV_2(B)$


I have never faced a problem in this form before, thus I am slightly confused, so could you help me on it?
My thoughts:


*

*Speaking about 1, is it correct to show that $tW(1)$ is not a martingale and using this fact state that $B(t)$ is not a martingale?

*Well, actually I have never seen such notation, but I guess the question is to compute the quadratic variation, so how should one do it for this sort of processes?


Thank you in advance.

Hi your first intuition is correct. 
Formally you could write for example to hsow the statement that for $t>s>1$ : 
$E[W_t | \mathcal{F}_s]\not=W_s$
For your second question it is more a direct application of your course if you want my opinion. For $t<1$ Have you seen what the Quadratic variation a Brownian motion is ? 
For t>1, you can check that a finite variation process that is continuous has null Quadratic variation.
Best regards
