Study with the several resources on Docsity
Earn points by helping other students or get them with a premium plan
Prepare for your exams
Study with the several resources on Docsity
Earn points to download
Earn points by helping other students or get them with a premium plan
Community
Ask the community for help and clear up your study doubts
Discover the best universities in your country according to Docsity users
Free resources
Download our free guides on studying techniques, anxiety management strategies, and thesis advice from Docsity tutors
A past paper from the university of cambridge's mathematical tripos exam, focusing on rough path theory and its applications. It includes four questions, covering topics such as defining free step-n nilpotent groups, identifying the step-2 nilpotent group with the heisenberg group, and investigating the relationship between lipschitz paths and the carnot-carathéodory distance. The document also includes a question on nested piecewise linear approximations to brownian motion and the rough path proof of the stroock-varadhan support theorem.
Typology: Exams
1 / 2
This page cannot be seen from the preview
Don't miss anything!
Tuesday 5 June 2007 9.00 to 11.
Attempt THREE questions.
There are FOUR questions in total.
The questions carry equal weight.
Cover sheet None Treasury Tag Script paper
1 (i) Define (GN^
Rd
, ⊗,−^1 , e), the free step-N nilpotent group over Rd, and give the
definition of the Carnot–Carath´eodory d distance on GN^
Rd
. What is a weak geometric p-rough path?
(ii) How can the step-2 nilpotent group over R^2 be identified with the 3-dimensional Heisenberg group H?
(iii) Since H ∼= R^3 we can equip H with the Euclidean distance inherited from R^3. Is a Lipschitz path in H relative to this Euclidean distance automatically a Lipschitz path relative to the Carnot–Carath´eodory distance on H?
2 Let x be a Lipschitz continuous Rd-valued path. Define SN (x)s,t , the step-
N signature of the path segment x|[s,t], as an element in a suitable tensor algebra over Rd. State and prove an algebraic relation between the step-N signature of the path segment x|[s,t] and the path segment x|[t,u] respectively. Show that the signature is invariant under reparametrisation of the path. More precisely, given ψ : [0, 1] → [0, 1] strictly increasing and continuously differentiable, show that
SN (x) 0 , 1 = SN (x ◦ ψ) 0 , 1.
3 Nested piecewise linear approximations to d-dimensional Brownian motion and their canonical area converge to Brownian motion and L´evy area in a rough path sense. Give a precise statement of this and sketch a proof with particular focus on martingale arguments.
4 Write an essay on the rough path proof of the Stroock–Varadhan support theorem. In particular, explain how the universal limit theorem is used.
Paper 32
