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The instructions and questions for paper 33 of an m.phil. In statistical science exam, focusing on stochastic calculus and applications. Topics include local martingales, itô's lemma, lévy's characterization of brownian motion, and markov jump processes. Students are required to solve problems involving stochastic differential equations, cauchy problems, and martingales.
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Friday 6 June 2003 9 to 12
Attempt FOUR questions.
There are six questions in total. The questions carry equal weight.
1 (a) Let M and A be continuous adapted processes. What does it mean to say that M is a local martingale? What does it mean to say that A has finite variation? Show that if M = A then Mt = M 0 for all t > 0. You may assume without proof that the total variation process of A is continuous and adapted.
(b) Let f : R^2 → R be a continuous function with continuous derivatives of first and second order. Set Xt = f (Mt, At), t > 0. Show, stating clearly any standard theorems to which you appeal, that X may be expressed as the sum of a continuous local martingale M f^ and a continuous adapted process Af^ of finite variation, with Af 0 = 0.
(c) Consider the case where M is a Brownian motion and where At = t for all t > 0. Determine for which functions f we have Aft = 0 for all t > 0 almost surely.
2 (a) Let M be a continuous local martingale and let H be a locally bounded previsible process. Show that the Itˆo integral H · M has quadratic variation process H^2 · [M ], where [M ] is the quadratic variation process of M. Any localization argument you use should be set out in detail. Standard properties of the Itˆo isometry defining the Itˆo integral may be assumed without proof.
(b) State and prove L´evy’s characterization of Brownian motion. The one- dimensional case will suffice.
(c) Which of the following processes, defined in terms of two independent Brownian motions B and W , are themselves Brownian motions?
(i) Xt =
∫ (^) t 0 sgn(Bs)dBs, (ii) Yt = eWt^ cos Bt, (iii) Zt = Yτt , τt = inf{u > 0 :
∫ (^) u 0 e
2 Ws (^) ds = t}.
Justify your answers.
3 Compute, for each t > 0, the distribution of Xt and Yt, given by the stochastic differential equations dXt =σ dBt − λXtdt, X 0 = 1, dYt =dWt + (2Yt)−^1 dt, Y 0 = 1.
Here B and W are independent Brownian motions; σ and λ are constants, with σ > 0. You may assume that the distribution of Y is determined by the distribution of W and its stochastic differential equation. You may express your answers in terms of the distributions of suitable functions of Gaussian random variables.
Paper 33
