0
$\begingroup$

We know that if $W_t$ is a Brownian motion, $W_{t+t_0}-W_{t_0}$ is one too.

Does the "converse" holds : Let $t_0$ be a positive number. I have a Brownian motion $W_t$ and I seek another Brownian motion, $W^*$ such that $W_t=W^*_{t+t_0}-W^*_{t_0}$ does such Brownian motion exists ?

$\endgroup$
1
$\begingroup$

Start with a Brownian motion $Y_t$, where the processes $W$ and $Y$ are independent. Take $$ W^*_t = \cases{ Y_t & for $0 \le t \le t_0$\cr Y_{t_0} + W_{t-t_0} & for $t > t_0$\cr} $$

| cite | improve this answer | |
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.