Approximations of the Brownian rough path with applications to stochastic analysis

By Peter K. Friz, Nicolas Victoir. 2003

77 citations recorded by the source corpus.

Abstract A geometric p-rough path can be seen to be a genuine path of finite p-variation with values in a Lie group equipped with a natural distance. The group and its distance lift ( R d , + , 0 ) and its Euclidean distance. This approach allows us to easily get a precise modulus of continuity for the Enhanced Brownian Motion (the Brownian Motion and its Levy Area). As a first application, extending an idea due to Millet and Sanz-Sole, we characterize the support of the Enhanced Brownian Motion (without relying on correlation inequalities). Secondly, we prove Schilder's theorem for this Enhanced Brownian Motion. As all results apply in Holder (and stronger) topologies, this extends recent work by Ledoux, Qian, Zhang [Stochastic Process. Appl. 102 (2) (2002) 265–283]. Lyons' fine estimates in terms of control functions [Rev. Mat. Iberoamericana 14 (2) (1998) 215–310] allow us to show that the Ito map is still continuous in the topologies we introduced. This provides new and simplified proofs of the Stroock–Varadhan support theorem and the Freidlin–Wentzell theory. It also provides a short proof of modulus of continuity for diffusion processes along old results by Baldi.

Subfields: Finance, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/d10a70f0780d6258f34480c80fba28d60aedf322

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Citations in the corpus

Multidimensional Stochastic Processes as Rough Paths: Theory and Applications (2010, methodology): “30) explain why martingale arguments (see also [65], [58], , [29]) are enough to discuss the step-2 case (  13) whereas equation (15.”
Differential Equations Driven by Π-Rough Paths (2012, background): “The rough path representation of an even larger class of Gaussian processes has been explored by Friz and Victoir [3, 4] and others.”
Rough path analysis via fractional calculus (2006, methodology): “We refer, for instance, to [1], [3], [4] and [7] for recent applications of the rough path analysis to the stochastic calculus with respect to the Wiener process.”
Ramification of rough paths (2006, background): “Geometric rough paths possess a rich structure and present nice connections with the geometry of certain Carnot groups [15] but there are situations where the geometric property is not natural, e.”
Differential equations driven by Gaussian signals (2007, background): “38]). Remark 6 Projection to g2 R d � yields to pleasant equality E(ln(Xs,t) |FA) = ln X A � which explains why martingale arguments [11], [ 7 ], [6], [4] are enough”
Non-degeneracy of Wiener functionals arising from rough differential equations (2007, background): “For instance, almost every joint realization of Brownian motion and L´evy’s area process is a geometric p -rough path [21, 6, 5].”
A (rough) pathwise approach to a class of non-linear stochastic partial differential equations (2009, background): “[33] in p -variation and [19] in Hölder rough path topology) it sufﬁces to recall that large deviation principles are stable under continuous maps Again, large deviation estimates for non-linear SPDEs in the small noise limit appear to be new and may be hard to obtain without rough paths theory.”
A note on the notion of geometric rough paths (2004, background): “0 sup D = (0= t0< < tl= 1) jD j l 1X i= 0 p Y (ti;ti+ 1)= 0: Reciprocally,we let n = fk2 n;k 2 f0; ;2ngg be a (very speci c for sim plicity) sequence ofsubdivisions of[0;1],whose m esh goes to 0.”
Rough path limits of the Wong–Zakai type with a modified drift term (2009, background): “t-Carath´eodory metric on G[p] Rd , i.e. kxkα-Ho¨l;[0,T] = sup 0≤s<t≤T d(xs,xt) |t −s α < ∞. For orientation, let us discuss the case α ∈(1/3,1/2), which covers Brownian motion (for details see [FV05],[Lyo98],[LCL07],[FV08b]). We realize G2 Rd as the set of all (a,b) ∈Rd ⊕Rd×d for which Sym(b)”
Semi-martingales and rough paths theory (2005, background): “The sequence (Xn)n∈N of semi-martingales with canonical decompositions Xn = Xn0 + M n + V n is said to satisfy the condition UCV on [0, T ] if (〈Mn〉T )n∈N is tight, (3) (Var1,[0,T ](V n))n∈N is tight.”
A variation embedding theorem and applications (2005, methodology): “Is is well-known, [5], that ‖B0,1‖ has a Gaussian tail and it follows that ‖B‖W δ,p has a Gaussian tail, provided p ≥ p0 (δ).”
On the Moments of the Modulus of Continuity of Itô Processes (2009, methodology): “Friz, P., and Victoir, N. 2005.”
Euler estimates for rough differential equations (2006, unknown): “‖x‖1/p-Hölder;[0,1] has Gauss tails since this is the case for all examples we have in mind: Enhanced Brownian motion B, see [9, 8], Enhanced Fractional Brownian Motion B and other Enhanced Gaussian processes [6, 10] and Enhanced Markov processes with uniformly elliptic generated in divergence form ”