168 CHAPTER 5. BROWNIAN MOTION
whereK 1 ,K 2 , andK 3 are constants that do not depend onx. For example,
K 1 is the product of 1/
√
2 πsfrom thefs(x) term, and 1/√
2 π(t−s) from
theft−s(B−x) term, times the 1/ft(B) term in the denominator. TheK 2
term multiplies in an exp(−B^2 /(2(t−s))) term. TheK 3 term comes from
the adjustment in the exponential to account for completing the square. We
know that the result is a conditional density, so theK 3 factor must be the
correct normalizing factor, and we recognize from the form that the result is
a normal distribution with meanBs/tand variance (s/t)(t−s).
Corollary 7.The conditional density ofX(t)fort 1 < t < t 2 givenX(t 1 ) =
AandX(t 2 ) =Bis a normal density with mean
A+ ((B−A)/(t 2 −t 1 ))(t−t 1 )and variance
(t 2 −t)(t−t 1 )/(t 2 −t 1 )
Proof.X(t) subject to the conditionsX(t 1 ) =AandX(t 2 ) = Bhas the
same density as the random variableA+X(t−t 1 ), under the condition
X(t 2 −t 1 ) =B−Aby condition 2 of the definition of Brownian motion.
Then apply the theorem withs=t−t 1 andt=t 2 −t 1.
Sources
The material in this section is drawn from A First Course in Stochastic
Processesby S. Karlin, and H. Taylor, Academic Press, 1975, pages 343–345
andIntroduction to Probability Modelsby S. Ross.
Problems to Work for Understanding
- LetW(t) be standard Brownian motion.
(a) Find the probability that 0< W(1)<1.
(b) Find the probability that 0< W(1)<1 and 1< W(2)−W(1)<
3.
(c) Find the probability that 0< W(1)<1 and 1< W(2)−W(1)< 3
and 0< W(3)−W(2)< 1 /2.- LetW(t) be standard Brownian motion.