I am trying to solve the following problem (exercise 4.19 from Shreve's Stochastic Calculus for Finance, Vol. 2):
Let $W(t)$ be a Brownian motion, and define $$ B(t) = \int_0^t \mathrm{sign}(W(s))dW(s). $$ Show that $B(t)$ is a Brownian motion.
I have to check four things:
- $B(0) = 0$.
- $B(t)$ is a continuous function of $t$.
- For any three distinct $t_1> t_2> t_3$, $B(t_1) - B(t_2)$ is independent of $B(t_2) - B(t_3)$.
- For any two distinct $t_1> t_2$, $B(t_1) - B(t_2)$ is distributed as a normal random variable with mean 0 and variance $t_1 - t_2$.
The first requirement, $B(0)=0$ is clear from the definition.
To show that $B(t)$ is continuous, I would start writing the Ito integral for a simple stochastic process: $$ B(t) = \sum_{j=0}^{n-1} \mathrm{sign}(W(t_j))(W(t_{j+1}) - W(t_j)), $$ where $t_0=0, \dots, t_n=t$ are a partition of the interval $[0,t]$. I would check continuity of $B(t)$ in this way: $$ \begin{array} \lim\limits_{\epsilon\to0} |B(t+\epsilon) - B(t)| &= |\mathrm{sign}(W(t))(W(t + \epsilon) - W(t))| \\ &= |W(t+\epsilon) - W(t)| &=0, \end{array} $$ where the last equality follows by the continuity of $W(t)$.
To check independence of the increments of $B(t)$, I would just write down: $$ B(t_1) - B(t_2) = \int_{t_2}^{t_1} \mathrm{sign} W(s) dW(s) $$ and $$ B(t_2) - B(t_3) = \int_{t_3}^{t_2} \mathrm{sign} W(s) dW(s), $$ and since the first integral depends only on the history of $W(t)$ between $t_2$ and $t_1$, whereas the second integral depends only on the history of $W(t)$ between $t_3$ and $t_2$, by the independence of the increments of $W(t)$ it follows that also the increments of $B(t)$ are independent random variables.
Finally, to check that the increments of $B(t)$ are normally distributed, I would attempt a computation of the moment generating function of $\Delta(t_2, t_1)=B(t_1) - B(t_2)$. $$ \begin{array} M_{\Delta(t_2,t_1)}(u) &= \mathbb{E}[e^{u\Delta(t_2,t_1)}] \\ &= \mathbb{E}\left[e^{u \int_{t_2}^{t_1} \mathrm{sign} W(s) dW(s)} \right], \end{array} $$ where the expectation is done among all possible paths of $W(s)$ between $t_2$ and $t_1$.
My question is: are my proofs of the first three points correct? How can I prove the fourth one?
Edit: related question, but not exactly the same as mine (I am asking if I can prove the normality of the increments by explicitly computing a moment generating function, and if the other three points of the exercise are correctly proven).
$B(t)$ is a continuous local martingale with $B(0)=0$ and quadratic variation process given by $$ [B(t)]=\int_0^t [\operatorname{sgn}(W(s))]^2\,ds=\int_0^t 1\{W(s)\ne 0\}\,ds=t \quad\text{a.s.} $$ Therefore, $B(t)$ is a standard BM by Levy's characterisation theorem.