Page 349 - Applied Probability

15. Diffusion Processes
338
11. Suppose the transformed Brownian motion with infinitesimal mean α
2
and infinitesimal variance σ described in Example 15.2.2 has α ≥ 0.
If c = − and d< ∞, then demonstrate that equation (15.12) has
solution
2
2
w(x)= e γ(d−x) for γ = α − √ α +2σ θ .
σ 2
Simplify w(x) when α = 0, and show by differentiation of w(x)with
respect to θ that the expected time E(T) to reach the barrier d is
infinite. When α< 0, show that
2α (d−x)
Pr(T< ∞)= e σ 2 .
(Hints: The variable γ is a root of a quadratic equation. Why do we
−θT
discard the other root? In general, Pr(T< ∞) = lim θ↓0 E e .)
12. In Problem 11 find w(x) and E(T) when c is finite. The value α< 0
is allowed.
13. Prove that the linear density f ij0 + f ij1 x is nonnegative throughout
the interval (a ij ,a i,j+1 ) if and only if its center of mass c ij lies in the
middle third of the interval. (Hint: Without loss of generality, take
a ij =0.)

15.11 References

[1] Breze´zniak Z, Zastawniak T (1999) Basic Stochastic Processes.
Springer-Verlag, New York

[2] Chung KL, Williams RJ (1990) Introduction to Stochastic Integration,
2nd ed. Birkh¨auser, Boston

[3] Crow, JF, Kimura M (1970) An Introduction to Population Genetics
Theory. Harper & Row, New York

[4] Ewens, W.J. (1979). Mathematical Population Genetics. Springer-
Verlag, New York

[5] Fan R, Lange K (1999) Diffusion process calculations for mutant
genes in nonstationary populations. In Statistics in Molecular Biol-
ogy and Genetics. Institute of Mathematical Statistics, Lecture Notes-
Monograph Series 33, Edited by Seillier-Moiseiwitsch F, The Institute
of Mathematical Statistics and the American Mathematical Society,
38-55

[6] Fan R, Lange K, Pe˜na EA (1999) Applications of a formula for the
variance function of a stochastic process. Stat Prob Letters 43:123–130