Wiener process and stochastic int

Question

Let $h:[0,1] \rightarrow \left\{-1,1 \right\}$. How to show that $X_t=(\int_0^th(s)dW_s)_{t \in [0,1]}$ is a Wiener process?

I know from the lecture that for every $h$ process $\int h \ dW_s$ is gaussian. Next $X_0=0$, and for every $r<t$ $Cov(X_t,X_r)=E(\int_0^rh^2ds)=E(\int_0^r1ds)=r$ Is it enough to say that $X_t$ is Wiener ?

Answer 1

You can use Levy's characterization of Brownian Motion, that is: a real valued stochastic process $(B_t)_{t\in[0,\infty)}$ with continuous sample paths and $B_0=0$ is a Brownian Motion if and only if it is a centered Gaussian process ($\mathbb{E}[B_t]=0$ $\forall t \ge0$) with covariance

$$ Cov[B_s,B_t]=\min(s,t). $$