MATH 452/STAT 552
Assignment 5
Due: December 2nd, 2020 at the beginning of class.
Submission: Upload your solutions of Exercises 5.1–5.5 to Crowdmark as one PDF
file. (Do not hand in any solution to the Supplementary Exercises.)
Exercise 5.1. Given a PDF f on [0,∞), the renewal equation for the corresponding
renewal function m(t) [1, (7.5) on p.435] states that
m(t) =
∫ t
0
f(x)dx+
∫ t
0
m(t− x)f(x)dx, t ≥ 0.
From this identity, the following identity for the Laplace transform of m(t) was shown
in class: ∫ ∞
0
e−µtm(t)dt =
µ−1
∫∞
0
e−µtf(t)dt
1− ∫∞
0
e−µtf(t)dt
, µ ∈ (0,∞). (5.1)
Find the Laplace transform of the PDF of a hyperexponential distribution: f(x) =∑n
j=1 Pjλje
−λjx, x > 0, where (Pj) is a probability vector and λj > 0 [1, p.302]. Then
use your result and (5.1) to get the explicit solution of the Laplace transform of the
corresponding renewal function.
Exercise 5.2. (1) Find the Laplace transform of the ruin probability R(x) defined
on [1, p.489] by applying the derivation of (5.1) to [1, (7.53) on p.490]. Express your
solution explicitly in R(0), λ, c, and the Laplace transform of F (x).
(2) Apply your solution in (1) to the case where F is the CDF of an exponential
random variable with mean λ.
Exercise 5.3. Let B and B′ be two independent one-dimensional standard Brownian
motions such that B0 = B′0 = 0. Define a stochastic process (Wt;−∞ < t < ∞) by
setting Wt = Bt if t ≥ 0 and Wt = B′−t if t < 0. Show that W has independent
increments: for all integers n ≥ 3 and −∞ < t1 < t2 < · · · < tn <∞,
Wt2 −Wt1 , Wt3 −Wt2 , · · · ,Wtn −Wtn−1 are independent.
In your verification, it is enough to include 0 as one of the time points tj. Moreover,
show that Wt+h −Wt ∼ N (0, |h|), for all t, h ∈ R.
For the next two exercises, recall that with respect to a one-dimensional standard
Brownian motion B (starting from zero) and a ∈ R, we write Ta for the first hitting
time of a.
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Exercise 5.4. Fix a > 0. Show that by [1, (10.6) on p.643], the two random variables
Ta and (a/B1)2 have the same CDF.
Exercise 5.5. This exercise aims for an alternative proof of the identity in [1, p.644]
by using martingales. Read [1, p.674] for the definition of martingales, Exercise 17
and the martingale stopping theorem.
(1) Show that
aP(Ta < Tb) + bP(Tb < Ta) = 0, ∀ a < 0 < b. (5.2)
(2) Show that for a < 0 < b, ({Ta < Tb} ∪ {Tb < Ta}){ ⊂ {Tb = ∞}, and then use
the explicit distribution of max0≤s≤tBs to show that P(Tb =∞) = 0. Therefore,
P(Ta < Tb) + P(Tb < Ta) = 1. (5.3)
(3) Conclude this exercise by using (5.2) and (5.3) to find P(Ta < Tb) and P(Tb <
Ta).
Supplementary Exercises: 2, 5 [1, Chapter 7]. 1, 3, 5, 10, 17, 18 [1, Chapter 10].
References
[1] Ross, S. (2019). Introduction to Probability Models. 12th edition. Academic Press.
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