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The fourth attempt of a m.phil. Exam in advanced probability theory. The exam covers various topics including martingales, brownian motion, and poisson random measures. The questions involve proving theorems, showing convergence, and calculating probabilities. The document also includes instructions for the exam, such as the number of questions to attempt and the stationery requirements.
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Monday 6 June, 2005 9 to 12
Attempt FOUR questions. There are SIX questions in total.
The questions carry equal weight.
Cover sheet None Treasury Tag Script paper
1 (a) Let M = (Mn)n≥ 0 be a discrete-time random process, which is integrable, and adapted to a filtration (Fn)n≥ 0. Show that the following are equivalent:
(i) M is a martingale,
(ii) E(MT ) = E(M 0 ) for all bounded stopping times T. (b) Assume that M is a martingale and that Mn → M∞ a.s. as n → ∞. State an additional condition, expressible in terms of the laws μn(dx) = P(Mn ∈ dx), which would allow us to conclude that E(MT ) = E(M 0 ) for all, possibly infinite, stopping times T.
(c) Let (Zn)n≥ 1 be a sequence of independent N(0, 1) random variables and let (an)n≥ 1 be a sequence of real numbers. Set M 0 = 0 and define
Mn =
∑^ n
k=
akZk, n ≥ 1.
By consideration of characteristic functions, or otherwise, show that Mn converges a.s. only if
k=1 a
2 k <^ ∞. (d) Under what additional conditions if any on the sequence (an)n≥ 1 can we conclude that E(MT ) = 0 for T = inf{n ≥ 0 : Mn ≥ 1 }?
2 (a) State the almost-sure martingale convergence theorem.
(b) Let f : [0, 1] → R be a Lipschitz function and define for n ∈ N, k ∈ { 0 , 1 ,... , 2 n^ − 1 } and ω ∈ [k 2 −n, (k + 1)2−n),
Xn(ω) = 2n{f ((k + 1)2−n) − f (k 2 −n)}.
Show that, for a suitable probability space (Ω, F, P) and a suitable filtration (Fn)n≥ 0 , the sequence (Xn)n∈N may be considered as a martingale.
(c) Deduce that there exists a bounded measurable function f˙ : [0, 1] → R such that, for all a, b ∈ [0, 1] with a ≤ b, we have
∫ (^) b
a
f^ ˙ (x)dx = f (b) − f (a).
4 (a) Let B = (Bt)t≥ 0 be a Brownian motion in Rd, d ≥ 3, starting from x. Fix ε > 0 and set T = inf{t ≥ 0 : |Bt| ≤ ε}.
Assume that |x| > ε. Show that
Px(T < ∞) = (ε/|x|)d−^2.
(b) For t > 0 and x, y ∈ Rd, set
p(t, x, y) = (2πt)−d/^2 e−|x−y|
(^2) / 2 t .
By evaluating the integral
0
Rd
p(t, x, y)p(s, 0 , y)dyds
in two different ways, establish the identity
∫
Rd
p(t, x, y)|y|^2 −ddy = cd
t
p(s, 0 , x)ds,
where cd is given by
cd =
0
(2πs)−d/^2 e−^1 /^2 sds.
(c) Show that, for x 6 = 0, as ε → 0, we have
ε^2 −dPx(T ≤ t) → cd
∫ (^) t
0
p(s, 0 , x)ds.
5 (a) Let W be a Brownian motion in Rn, n ≥ 1, starting from 0, and let U be a random variable in Rn^ which is uniformly distributed on the unit ball {|x| ≤ 1 } and is independent of W. Set T = inf{t ≥ 0 : |Wt| = |U |}. Show that WT has the same distribution as U.
(b) Suppose now that W starts from a general point x in some connected open set D in Rn. Set gD (x) = Ex(TD ), x ∈ D,
where TD = inf{t ≥ 0 : Wt 6 ∈ D}. Show that if gD (x) < ∞ for some x ∈ D then gD (y) < ∞ for all y ∈ D.
(c) For n = 1, 2 , 3 and for D = Dn = (0, ∞)n, determine whether gD is finite.
6 (a) Let μ be a Poisson random measure on R × (0, ∞) with intensity
ν(dy, dt) = K(dy)dt = c|y|−^2 dydt,
where c ∈ (0, ∞) is determined by
2 c
0
(1 − cos z) z^2
dz = 1.
Set
Xt =
(0,t]×{|y|≤ 1 }
y(μ − ν)(dy, ds) +
(0,t]×{|y|> 1 }
yμ(dy, ds).
Explain why these integrals are well-defined in spite of the fact that
∫
{|y|≤ 1 }
yK(dy) =
{|y|> 1 }
yK(dy) = ∞.
(b) Write down the characteristic function of X 1 and hence obtain the density function of X 1.
(c) Fix α ∈ (0, ∞) and set X t( α)= αXαt. Show that the processes X(α)^ and X have the same distribution.
