In: Statistics and Probability
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::
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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