Assignment 3-Martingales The stared items are optional. Due 4.4.2016
1. Let Xi be i.i.d random variables E[Xi ] = µ. Let Sn = S0 +
Pn
i=1
Xi . Show that:
(a) Sn − nµ is a martingale.
(b) Let ϕ(θ) = E[eθXi ]. Show that Mn =
eθSn
(ϕ(θ))n
is a martingale
2. Consider a game where you bet on 1 each time. With probability p 6= 1/2 you win 1
and with probability 1 − p you 1 lose 1. Let Sn be your fortune after n games. Find
for what values of z, such that the process z Sn is a martingale.
3. Consider the example of the gambler ruin problem presented in class. Assume that at
each game the gambler wins 1 w.p. 1/2 and loses 1 w.p. 1/2. She starts with fortune
0 < x < b, and exits the Casino if she gets bannkrupt or reaches b. Find the probability
that she will exit with b for 0 < x < b.
4. Let {Mn } be martingale. Let ξj = Mj+1 − Mj
(a) Find E[ξj ].
(b) Find E[ξi ξj ] for i 6= j.
5. Let S and T stopping time in some probability space (Ω, F, P), and filtration Fn . Show
that
(a) S + T is a stopping time.
(b) S ∧ T is a stopping time.
(c) S ∨ T is a stopping time.
6. Let Xn be a Markov chain with state space {0, 1, · · · , N }, and assume that Xn is also
a martingale. Let τ = min{n : Xn ∈ (0, N )}
(a) Show that 0 and N must be absorbing.
(b*) Assume that Px (τ < ∞) > 0 for all 1 ≤ x ≤ N − 1. Show that Px (τ < ∞) = 1
(c) Under the condition in (b) find the probability that Xτ = N (i.e. N is reached
before 0).You do not have to do (b) in order to solve this part.
7. Polya urn: Consider an urn containing at time 0 one red ball and one green ball. At
each time a ball is chosen randomly and it is returened with another ball with the same
color. Let Xn the fraction of the red balls at time n
Show that Xn is a martingale.
8. Consider the following process in discrete time:
Xn = Xn−1 + c − Zn
Assume that X0 = x is given. Let T = min{n : Xn ≤ 0} where Zi are i.i.d Normally
distributed such that c − Zn has normal distribution with mean µ and variance σ 2 . We
assume that µ > 0 (why?). This model presents the reserve of an insurance company
the premium income rate is c per time unit and Zn is the claim at time n.
1
(a) Show that E[eθ(c−Zn ) ] = eµθ+
(b) Show that Mn = eθXn −n(µθ+
σ2 θ2
2
σ2 θ2
)
2
is a martingale.
(c) Find θ = θ0 such that eθ0 Xn is a martingale.
(d) Show that E[eθ0 XT ∧n ] = eθ0 x .
(e*) Apply (d) to prove that P(T < ∞) ≤ eθ0 x .
9. Let N (t). be a Poisson process rate λ and Zi i.i.d independent of N (t) exponentially
distributed with parameter µ. Consider the risk process:
R(t) = R(0) + ct −
N (t)
X
Zj
j=1
where R0 = x (given). Assume that c > λ/µ (why?)
(a) Prove that E[eθR(t) ] = eϕ(θ)t , where ϕ(θ) = cθ − λ +
λµ
µ+θ
(b) Show that eθR(t)−tϕ(θ) is martingale.
(c) Find γ < 0 such that eγR(t) is a martingale.
(e*) let 0 < x < b. Apply (c) to obtain the probability that the process reach b before
ruin. (Hint: apply the memoryless property of the exponential distribution).
(f*) Let b → ∞ in (3) to obtain the ruin probability
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