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.