Probability Theory and Stochastic Processes
LIST 6
(1) Let (Xn , Fn )n∈N be a martingale. Show that
(a) if X0 = E(X1 ) and F0 = {∅, Ω}, then (Xn , Fn )n∈N0 is also
a martingale.
(b) (Xn , σ(X1 , . . . , Xn ))n∈N is also a martingale.
(2) Let Y1 , Y2 , . . . be independent r.v.’s such that
P (Yn = an ) =
1
2n2
P (Yn = 0) = 1 −
P (Yn = −an ) =
where a1 = 2, an = 4
n−1
P
1
n2
1
2n2
aj . Decide if Xn and σ(X1 , . . . , Xn )
j=1
define a martingale when
n
P
(a) Xn =
Yj .
(b) Xn =
(c) Xn =
j=1
n
P
j=1
n
P
1
Y.
2j j
Yj2 .
j=1
(3) Let Y1 , Y2 , . . . be a sequence of iid rv’s such that P (Yn = 1) = p
and P (Yn = −1) = 1 − p. Decide if Xn and σ(Y1 , . . . , Yn ) define
a martingale when
n
P
(a) Xn =
Yj .
j=1
!2
n
P
(b) Xn =
Yj − n.
j=1
!
n
P
(c) Xn = (−1)n cos π
Yj .
j=1
(d) Xn =
1−p
p
Sn
where Sn =
n
P
Yj .
j=1
(4) Let Y1 , Y2 , . . . be a sequence of iid rv’s with Poisson distribution
and mean value λ. Consider also the sequence
Xn = Xn−1 + Yn − 1,
1
n ∈ N,
2
and X0 = 0. Find the values of λ for which Xn is a martingale, sub-martingale or super-martingale, with respect to the
filtration Fn = σ(Y1 , . . . , Yn ).
(5) Let (Xn , Fn )n∈N be a martingale and τ a stopping time. Determine E(Xτ ∧n ).
(6) Let Y1 , Y2 , . . . be a sequence of iid rv’s with distribution P (Yn =
1) = p and P (Yn = −1) = 1 − p where 0 < p < 1, and
n
P
Xn =
Yj . Compute E(τ ) for the stopping time
j=1
τ = min{n ≥ 1 : Xn = 1}
when
(a) p = 1/2. Hint: Use Wald’s equation.
(b) * p 6= 1/2. Hint: Use the optional stopping theorem for
Zn = [(1 − p)/p]Xn .