[SOLVED] Amath 562 homework assignment 5

1. Please give an integer-valued stochastic Levy process ´ (ηt)t≥0,
ηt ∈ Z = {· · · , −2, −1, 0, 1, 2, · · · },
which, furthermore, is a martingale.2. We have discussed in the class that “independent, stationary increments”, under certain
conditions, gives rise to the Gaussian nature of Brownian motion due to the central limit
theorem: The item 3 in MLN’s Definition 7.2.1. Discuss why the Levy process as defined ´
in Definition 10.1.1, item 3 cannot be improved to a Gaussian distribution?3. A probability distribution is infinitely divisible if it can be expressed as the probability
distribution of the sum of an arbitrary number of independent and identically distributed
(i.i.d.) random variables. It turns out, every infinitely divisible probability distribution
corresponds to a Levy process. ´(a) Show that normal distribution on R is infinitely divisible.
(b) Show that Poisson distribution on N is infinitely divisible.
(c) The pdf of Cauchy distribution is
fX(x) = γ
π(γ
2 + x
2
)
.Show its characteristic function ϕX(u) := Ee
iuX
is e
−γ|u|
. What are its expected value
and its variance?(d) A real-valued random variable X is infinitely divisible if and only if its characteristic
function ϕX(u) is of the form e
ψ(u)
, with
ψ(u) = iµu −
1
2
σ
2u
2 +
Z
R

e
iuz − 1 − iuz1|z|<1

ν(dz). (1)We see that when the ν(dz) = 0, the ϕX(u) corresponds to the normal distribution
N (µ, σ2
). Taking Eq. 1 as given, with µ = σ = 0, find the ν(dz) such that X has a
Cauchy distribution.4. Let Pt be a Poisson process with rate λ. Introducing “compensated Poisson process”
P˜
t = Pt − E[Pt
].(a) Show that its characteristic function has the form
E
h
e
iuP˜t
i
= e
tψ(u)
.
(b) Give the expression for ψ(u).5. Professor Matt Lorig’s notes, exercise 10.1.