Is this process a martingale

Question

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:

1. Is $B(t)$ a martingale?
2. 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:

1. 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?
2. 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.

Answer 1

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