Question

Let Be be Brownian motion and fix to > 0. Prove that By: = Bto+t - Blo; t o is a Brownian motion.

Help please!

Answer 1

Solcelion: According to the definition of Bodinian motion A motion Bę ý said to be Brownian if it Satis - tice the following four conditions: 1. $ By: tet} the independent incoements ctor any actictac... <tr Btz-Ot, 1 Btz-Bez, .... Ben Beni Qre Independent ). 2. Bta - Bt, ~ N Co, ta-t,) bor any ty, ta ET with t, 2tz 3. Br = 0 almost Surely 4. The trajectories & BEITET are continway. so according to our problem, we can show that B Satistues the above fover conditions. 4. Since By is a Brownian motion, Bt hay indepen- dent increment. Therefore Bt has independent incre- ment, too Bt - B = Blotto - Big Brott, & Bro = Btott, - Blott, which ý independent OF Brotta - Blott, & t,, ta, tz ET 2. Bt - Bt, ~ " (o. ta-ti) bor any ti, taET with ticts because Bt - Bt - Blotta Blotti (ortat) Scanned with CamScanner

mo u 2 50 = Boto - Bro =0 almost surely, ut since the to ajectiories ty BECW), TET, are continuouy for any we2 tu (w), LET, olare continuous too. More