Question

Let c > 0 be a given, fixed number. Show that the process X = (Xₜ)ₜ≥0 defined by Xₜ = Wₜ - c - Wc is a standard Brownian motion.

Answer 1

The process X being a standard Brownian motion can be demonstrated by verifying all properties of Brownian motion and applying a translation to X. The translated process X' starts at 0 and retains all properties of standard Brownian motion, making the original process also a standard Brownian motion.

Step-by-step explanation:

The student asks how to show that the process X = (Xₜ)ₜ≥0 defined by Xₜ = Wₜ - c - Wc is a standard Brownian motion, given that c > 0 is a fixed number. To show that X is a standard Brownian motion, we need to verify that it satisfies the defining properties of a Brownian motion:

1. The process X starts at 0, i.e., X0 = 0.
2. The increments of X are independent.
3. The increments of X are normally distributed with mean 0 and variance t for any increment of t.
4. The paths of X are continuous with probability 1.

By definition of X, we have X0 = W0 - c - Wc. Since W0 is always 0 for any Brownian motion, this expression simplifies to X0 = -c - Wc, which means X does not start at 0. However, this can be adjusted by considering the translated process X' = X + c + Wc. This new process X' does start at 0 and the increments X'ₜ - X'₝s = (Wₜ - c - Wc) - (W₝s - c - Wc) for s < t are independent and normally distributed, since they correspond to increments of W, which is a standard Brownian motion. The paths of X' are also continuous since they are equivalent to the paths of W. Therefore, X' is a standard Brownian motion and by the translation invariance property of Brownian motion, so is X.