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The questions and solutions for attempt three of the rough path theory and applications exam for the m.phil. In statistical science program. Topics such as lipschitz continuous paths, stepn signatures, controlled ordinary differential equations, free step-n nilpotent group, chow's theorem, enhanced brownian motion, and the stroock-varadhan support theorem. Students are required to attempt three questions out of four, with equal weight.
Typology: Exams
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Monday 5 June, 2006 1.30 to 3.
Attempt THREE questions. There are FOUR questions in total.
The question carry equal weight.
Cover sheet None Treasury Tag Script paper
1 (i) 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]. Show that the path t 7 → SN (x) 0 ,t solves a controlled ordinary differential equation driven by x.
(ii) What is meant by pathlevel solution to a Rough Differential Equation (RDE)? Use Davie’s lemma to prove existence of a pathlevel RDE solution.
2 Define (GN^
Rd
, ⊗, −^1 , e), the free step-N nilpotent group over Rd. State Chow’s Theorem and prove it in the special case of N = 2 and d = 2. [Hint: draw a picture.] A path with values in the step−2 nilpotent group over R^2 is given by
yt = exp
0 t −t 0
Compute y− s 1 ⊗ yt and discuss the H¨older regularity of y with respect to the Carnot- Caratheodory metric. For what p is y a weak geometric p-rough path?
3 (i) Let d > 2 be an integer. In the context of a d-dimensional standard Brownian motion B =
B^1 , B^2 ,... , Bd
, define Enhanced Brownian Motion. Show that there is a modification of Enhanced Brownian Motion, denoted by B, so that for any fixed α ∈ [0, 1 /2), ‖B‖α−H¨older;[0,1] < ∞ a.s.
[Integrability properties of L´evy’s area and scaling properties of Enhanced Brownian Motion may be assumed.]
(ii) Now consider the case of a 2-dimensional standard Brownian motion B = (β, β˜).
Let B(n) =
β (n) , β˜ (n)
be the dyadic piecewise linear approximation of level n to B,
that is, B (n)k/ 2 n equals Bk/ 2 n for all k = 0,... , 2 n^ and B (n) is affine linear on each interval [(k − 1) / 2 n, k/ 2 n] , k = 1,... , 2 n. You may assume without proof that (a) for all t ∈ [0, 1], B (n)t = E
Bt|Bi/ 2 n^ ; i = 0,... , 2 n
∫ (^) t
0
β (n) d β˜ (n) = E
[∫ (^) t
0
βd β˜|Bi/ 2 n^ ; i = 0,... , 2 n
and (b) the r.v. ‖B‖α−H¨older;[0,1] has finite moments of all orders. Explain how martingale arguments can be used to prove that for all t ∈ [0, 1], B (n)t ≡ S 2 (B (n))t → Bt a.s. and supn ‖B (n)‖α−H¨older < ∞ a.s..
Rough Path Theory and Applications
