Question

If X(t), t ≥ 0 is a Brownian motion process with drift parameter μ and variance parameter σ² for which X(0)=0, show that -X(t), t ≥ 0 is a Brownian motion process with drift parameter -μ and variance parameter σ².

Answer 1

ANSWER::

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

1.

Prob[X(0)=0] =1 P[-X(0)=0]=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)

the distribution of [-X(t)]-[-X(s)] is same as -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.

-X(t1), -[X(t2)−X(t1)] ,…, -[X(tn)−X(tn−1)] are independent

Hence, -X(t) has independent increments.

4.

X(t) ~N(, ) for t∈[0,∞). -X(t) ~N(-, )

5.

With probability 1, tX(t) is continuous on [0,∞) With probability 1, t-X(t) 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

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