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The fourth attempt of questions from a m. Phil. Exam in statistical science, focusing on stochastic calculus and applications. The questions involve topics such as martingales, stochastic differential equations, quadratic variation, and the itô integral. Students are expected to demonstrate their understanding of these concepts through problem-solving.
Typology: Exams
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Wednesday 5 June 2002 9 to 12
Attempt FOUR questions There are six questions in total
The questions carry equal weight
1 Consider the stochastic differential equation in R
dXt = σ(Xt)dBt, X 0 = 0
where B is a Brownian motion in R. Assume that X is a solution and that σ is a bounded measurable function on R. Write v = σ^2 and set
Zt = X t^2 −
∫ (^) t
0
v(Xs)ds.
Show that both X and Z are martingales.
Let now X be a pure jump Markov process in R, starting from 0 and having L´evy kernel K. Assume that ∫
R
K(x, dy) = λ(x),
R
yK(x, dy) = 0,
R
y^2 K(x, dy) = v(x)
where λ and v are bounded functions on R. Set
Zt = X t^2 −
∫ (^) t
0
v(Xs)ds.
Show that both X and Z are martingales.
2 Let B be a Brownian motion in R^2 with |B 0 | = 1. Set Mt = log |Bt| and, for r ∈ (0, 1) and R ∈ (1, ∞), set
T = T (R, r) = inf{t ≥ 0 : |Bt| ∈ {r, R}}.
Show that M is a local martingale, at least up to the first time B hits 0. Hence show that E(MT ) = 0 and deduce that Bt 6 = 0 for all t > 0 almost surely. Show further that M is not a martingale. [You may wish to use, with justification, following inequality:
M (^) T^2 (2,0)∧t 6 (log 2)^2 + M (^) t^2 1 |Bt| 61 / 2 .]
6 State Girsanov’s theorem and deduce the Cameron–Martin formula.
Let B be a Brownian motion in R, starting from 0. For which of the following processes is its distribution on C(R+, R) given by a density with respect to Wiener measure? Justify your answers.
(a) Bt + t, (b) Bt − (t ∧ 1),
(c) 2Bt,
(d) Bt − (t ∧ 1)B 0.
