# Example of a bounded simple process $H_t$ that changes value only once such that $\int_0^t H_s dB_s$ doesn't have normal distribution?

I am currently studying for an exam, and in studying one of the examples I am trying to construct is a bounded simple process $H_t$ that changes value only once such that$$\int_0^t H_s\,dB_s$$does not have a normal distribution. However, I was not able to come up with anything. Could someone please help?

EDIT: Progress so far. How about any $H_s = \textbf{1}_{s < T}$, with $T$ random? But I do not know how to go about showing that this works...

• Please add the [self-study] tag & read its wiki. – gung - Reinstate Monica Sep 18 '15 at 15:37
• What about $T = \inf\{s|B_s>K\}$ with $K$ real? – RUser4512 Sep 18 '15 at 15:44

If $T$ is allowed to depend on $B$, you could use $T_K=\inf\{t|B_t>K\}$. Your indicator becomes equivalent to $H_s=1_{\sup_{u<s}B_u<K}$.
The integral itself becomes:$$\int_0^t H_s\,dB_s=B_{\min(t,T_K)}$$
For a non-random $T$ you can split the integral and get a sum of two Gaussians. This is clearly Gaussian.
For a random $T$, say with finite support, you can condition on $T$ to get the previous result. Marginalizing on $T$, you will be getting a mixture of Gaussian.