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5 Cubature on Wiener Space Quadrature rules replace Lebesgue measure λ on [0, 1] by a finite, convex linear combination of point masses, say µ = ai δxi , where weights (ai ) and points (xi ) are chosen such that all monomials (and hence all polynomials) up to degree N are correctly evaluated. In other words, one first computes the moments of λ, namely 1 xn dλ(x) = 0 1 , n+1 1 for all n ≥ 0. One then looks for a measure µ such that 0 xn dµ(x) = 1/(n + 1) for all n ∈ {0, 1, . . , N }. The same can be done on Wiener space: the monomial xn is then replaced by the n-fold iterated integrals (in the sense of Stratonovich), integration is on C [0, T ], Rd against standard d-dimensional Wiener measure.

More precisely, use the previous exercise to show that the sequence Bn = (B n , Bn ) is Cauchy in the sense that | n m a (B , B )|Lq →0 with n, m → ∞ . s. c) Show that B is the Lq -limit in α-H¨older rough path metric for all piecewise linear approximations, say B Dn , as long as mesh |Dn | → 0 with n → ∞. Show that the convergence is almost sure if |Dn | ∼ 2−n and also |Dn | ∼ 1/n. 18. We only sketch the main step in the proof of b). Without loss of generality, we set T = 1. The crux of the matter is to show that Bn0,1 converges in V ⊗ V .

Ab, it immediately follows that n Xs,t ≤ 2|t − s| , uniformly in n, s, t. In other words, supn X n 1/2 < ∞. The argument for the uniform bounds on Xs,t is similar. 14). On the other hand, we also have X˙ un ⊗ X˙ vn du dv ≤ X˙ n Xns,t = s 1 and the above bound for n2 |t − s| ≤ 1. c) The interpolation argument is left to the reader. 19 (Translation of rough paths). Fix α ∈ ( 13 , 12 ] and X = (X, X) ∈ C α [0, T ], Rd . For sufficiently smooth h : [0, T ] → Rd , the translation of X in direction h is given by def Th (X) = X h , Xh , where X h := X + h and t Xhs,t := Xs,t + t hs,r ⊗ dXr + s t Xs,r ⊗ dhr + s hs,r ⊗ dhr .