Question

If X(t), t>=0 is a Brownian motion process with drift mu and variance sigma squared for which X(0)=0, show that -X(t), t>=0 is a Brownian Motion process with drift negative mu and variance sigma squared.

Answer 1

Solution:

X(t), t>=0 is a Brownian motion process with drift parameter and variance parameter Then, the process exhibits the following properties

1.

2. For s,t∈[0,∞) with s<t, the distribution of X(t)−X(s) is the same as the distribution of X(t−s)

3. X has independent increments. That is, for t1,t2,…,tn∈[0,∞) with t1<t2<⋯<tn, the random variables

X(t1), X(t2)−X(t1) ,…, X(tn)−X(tn−1) are independent.

are independent

Hence, -X(t) has independent increments.

4.

5. With probability 1, is continuous on With probability is continuous on [0,∞)

(as X is continuous random variable implies -X is also a continuous random variable)

Hence, -X(t) is a Brownian motion process with drift parameter and variance parameter