Question

Show that the inverted process X = (X_t)_t≥0 defined by X_0=0 and W_t=tW(1/t) for t>0 is also a standard Brownian motion.

Answer 1

To show that the inverted process X = (Xt)t≥0 defined by X0=0 and Wt=tW(1/t) for t>0 is also a standard Brownian motion, we need to demonstrate that it satisfies the properties of a standard Brownian motion.

Step-by-step explanation:

To show that the inverted process X = (Xt)t≥0 defined by X0=0 and Wt=tW(1/t) for t>0 is also a standard Brownian motion, we need to demonstrate that it satisfies the properties of a standard Brownian motion:

1. X0 = 0
2. Xt is continuous.
3. Xt has independent increments.
4. Xt has normally distributed increments.
5. Xt has stationary increments.
6. Xt has zero mean.
7. Xt has quadratic variation equal to t.

By verifying each of these properties, we can conclude that the inverted process X is also a standard Brownian motion.