STATF407 Stochastic Models: second part, sheet 1
Ex 1. Let X1 , XP
2 , . . . be i.i.d., with P [Xi = 1] = p and P [Xi = −1] = q = 1 − p.
n
Let Yn := i=1 Xi be a partial sum of the Xi ’s. Show that
• when q = p, Yn2 − n is a martingale with respect to the filtration (An =
σ(X1 , . . . , Xn ))n≥1
Ex.2. Let X1 , X2 , . . . be independent with E[Xi ] P
= 0 for all i = 1, 2,P. . . and
n
n
2
2
V ar(Xi ) = σi2 , i = 1, 2, . . . and put Sn =
i=1 Xi and Tn =
i=1 σi .
Show that Sn2 − Tn2 is a martingale with respect to (An = σ(X1 , . . . , Xn ))n≥1 .
Ex 3. Let (Xn )n≥0 be a sequence of random variables with finite means and satisfying for n ≥ 1,
E(Xn+1 |X0 , X1 , ..., Xn ) = aXn + bXn−1
where 0 < a, b < 1 and a + b = 1.Find a value of α for which the process
(Sn )n≥0 defined by Sn = αXn + Xn−1 , n ≥ 1, is a martingale with respect to
the sequence (Xn ).
Ex 4. Proof that if (Yn ) is a martingale w.r.t. a filtration (An ), and if (Yn ) is
uniformly integrable, then
lim E[Yn IT >n ] = 0
n→∞
Ex 5. We consider the doubling strategy, for which the winnings of the gambler
after n steps are
n
X
Yn =
2i−1 Xi
i=1
where the Xi ’s are i.i.d. with P[Xi = 1] = P[Xi = −1] = 0.5. Let T =
inf n ∈ N : Xn = 1. Show that (Yn ) is a martingale and that T is a stopping
time w.r.t. (Xn ).
Ex 6. We consider the symmetric random walk, with initial wealth Y0 = k for
some k ∈ N. Verify that the assumptions of the optional stopping theorem
are satisfied for the stopping time T = inf{n ∈ N : Yn ∈ {0, m}}.
Ex 8. Let T1 and T2 be two stopping times with respect to a filtration (Fn ). Show
that T1 + T2 , max(T1 , T2 ) and min(T1 , T2 ) are stopping times
Ex 9. (Doob decomposition). Let (Ω, A, P) be a probability space endowed with a
filtration F = (Fn ). An F-adapted sequence An is an increasing predictable
process if A0 = 0 and for any n ≥ 0, An ≤ An+1 and An+1 is Fn -mesurable.
Let Xn be a submartingale.
• Show that the process An defined by
A0 = 0,
An+1 = An + E[Xn+1 − Xn kFn ]
is an increasing predictable process.
• Show that the process
Mn = Xn − An
is a martingale
• Show that the decomposition
Xn = Mn + An
is unique.
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