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Advanced probability exercises covering various topics such as martingales, poisson random measure, and brownian motion. The exercises involve proving theorems, finding distributions, and working with stochastic processes. Students are expected to have a solid understanding of probability theory and stochastic processes to solve these problems.
Typology: Exams
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Monday 5 June, 2006 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 In this exercise, we consider a filtered probability space (Ω, F, (Fn, n > 0), P ), and all definitions are understood with respect to this filtered space.
a) Let (Xn, n > 0) be a submartingale which is bounded in L^1.
(i) Prove that for every n > 0, the sequence (E[X p+ |Fn], p > n) is increasing and converges to an a.s. limit Mn.
(ii) Show that (Mn, n > 0) is a non-negative martingale which is bounded in L^1 , and conclude that Xn can be written in the form Mn − Yn, where (Yn, n > 0) is a non-negative supermartingale which is bounded in L^1.
b) Let (Xn, n > 0) be a supermartingale which is bounded in L^1. Show that Xn can be written in the form Mn + Yn, where (Mn, n > 0) is a uniformly integrable martingale, and (Yn, n > 0) is a supermartingale with limit 0 when n → ∞.
Advanced Probability
3 Let (Mt, t > 0) be a continuous-time martingale with respect to a filtered space (Ω, F, (Ft, t > 0), P ), such that (Mt, t > 0) is a non-negative process with continuous paths, and which converges a.s. to 0 as t → ∞. Let M ∗^ = supt> 0 Mt. We use the notation P (A|G) = E[ (^1) A|G], where A is an event and G a sub-σ-algebra of F.
a) Show that for every x > 0,
P (M ∗^ > x|F 0 ) = 1 ∧ (M 0 /x).
[Hint: Use the stopped martingale (Mt∧Tx , t > 0), where Tx = inf{t > 0 : Mt > x}.]
b) Deduce that M ∗^ has the same distribution as M 0 /U , where U is uniform on [0, 1] and independent of M 0.
c) Let (Bt, t > 0) be a Brownian motion started at B 0 = a > 0. Give the distribution of sup 06 t 6 T 0 Bt, where T 0 = inf{t > 0 : Bt = 0}.
4 State and prove the reflection principle for the standard 1-dimensional Brownian motion.
Let (Bt, t > 0) be a standard 1-dimensional Brownian motion defined on some probability space (Ω, F, P ). Use the reflection principle to show that St = sup 06 s 6 t Bs has the same law as |Bt| for every t > 0.
Let a < b < c < d be non-negative real numbers. Show that
sup a 6 t 6 b
Bt = sup c 6 t 6 d
Bt
5 Let (Bt, t > 0) be a standard 1-dimensional Brownian motion. For x ∈ R, let Tx = inf{t > 0 : Bt = x}. Fix a, b > 0, and let T = Ta ∧ T−b.
By considering processes of the form (exp(λBt − λ^2 t/2), t > 0), or otherwise, prove that for every λ ∈ R,
E(e−λ
(^2) T / 2 (^1) {T =Ta}) =
sinh(λb) sinh(λ(a + b))
and that
E(e−λ
(^2) T / 2 ) =
cosh(λ(a − b)/2) cosh(λ(a + b)/2)
Advanced Probability
6 a) Write carefully the definition of a Poisson random measure on a measurable space (E, E), with σ-finite intensity μ(dx).
b) Fix d > 1, and let λ(dx) be Lebesgue measure on Rd. We let B(0, r) be the open Euclidean ball in Rd^ with centre 0 and radius r > 0, and we let vd = λ(B(0, 1)).
Let M (dx) be a Poisson random measure on Rd^ with intensity λ(dx). If f is a non-negative measurable function and ν is a non-negative measure, we let ν(f ) =
f dν.
(i) Let R = sup{r > 0 : M (B(0, r)) = 0}. Show that the distribution of R has a density and compute it.
(ii) Let Nr = M (B(0, r)) for r > 0. Let f : Rd^ → R+ be a continuous function with compact support. Compute E[Nr exp(−M (f ))].
(iii) Show that the two quantities
E[exp(−M (f ))|Nr > 1] and
E[Nr exp(−M (f ))] P (Nr > 1)
have the same limit as r ↓ 0, and compute this limit.
Advanced Probability
