A CONSTRUCTION OF THE ROUGH PATH ABOVE FRACTIONAL BROWNIAN MOTION USING VOLTERRA'S REPRESENTATION

Terry Lyons

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A CONSTRUCTION OF THE ROUGH PATH ABOVE FRACTIONAL BROWNIAN MOTION USING VOLTERRA'S REPRESENTATION DAVID NUALART AND SAMY TINDEL Abstract. This note is devoted to construct a rough path above a multidimensional fractional Brownian motion B with any Hurst parameter H ? (0, 1), by means of its representation as a Volterra Gaussian process. This approach yield some algebraic and computational simplifications with respect to [18], where the construction of a rough path over B was first introduced. 1. Introduction Rough paths analysis is a theory introduced by Terry Lyons in the pioneering paper [11] which aims to solve differential equations driven by functions with finite p-variation with p > 1, or by Hölder continuous functions of order ? ? (0, 1). One possible shortcut to the rough path theory is the following summary (see [7, 8, 9, 12] for a complete construction). Given a ?-Hölder d-dimensional process X = (X(1), . . . , X(d)) defined on an arbitrary interval [0, T ], assume that one can define some iterated integrals of the form Xnst(i1, . . . , in) = ∫ s≤u1<···

• surely


• rough paths


• volterra gaussian process


• n?-hölder continuous


• almost surely


• zc2 can


• analysis methods


• ft ?

Sujets

• Surely
• Almost surely

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• Publié par: pefav
• Nombre de lectures: 8
• Langue: English

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A CONSTRUCTION OF THE ROUGH PATH ABOVE FRACTIONAL BROWNIAN MOTION USING VOLTERRA’S REPRESENTATION

DAVID NUALART AND SAMY TINDEL

Abstract.is devoted to construct a rough path above a multidimensionalThis note fractional Brownian motionBwith any Hurst parameterH∈(0,1), by means of its representation as a Volterra Gaussian process. This approach yield some algebraic and computational simplifications with respect to [18], where the construction of a rough path overBwas first introduced.

1.Introduction

Rough paths analysis is a theory introduced by Terry Lyons in the pioneering paper [11] which aims to solve differential equations driven by functions with finitep-variation with p >1, or by Hölder continuous functions of orderγ∈(0,1) possible shortcut to the. One rough path theory is the following summary (see [7, 8, 9, 12] for a complete construction). Given aγ-Hölderd-dimensional processX= (X(1), . . . , X(d))defined on an arbitrary interval[0, T], assume that one can define some iterated integrals of the form X s n t (i 1 , . . . , i n ) = Z s≤u 1 <∙∙∙<u n ≤t dX u i 1 1 dX u i 2 2 ∙ ∙ ∙dX u i n n , for0≤s < t≤T,n≤ b1/γcandi 1 , . . . , i n ∈ {1, . . . , d} long as. AsXis a nonsmooth function, the integral above cannot be defined rigorously in the Riemann sense (and not even in the Riemann-Stieltjes sense ifγ≤1/2 it is reasonable to assume that). However, some elementsX n can be constructed, sharing the following three properties with usual iterated integrals (here and in the sequel, we denote byS k,T ={(u 1 , . . . , u n ) : 0≤u 1 < ∙ ∙ ∙< u n ≤T}thek th order simplex on[0, T]): (1)Regularity:Each component ofX n isnγ-Hölder continuous (in the sense of the Hölder norm introduced in (15)) for alln≤ b1/γc, andX s 1 t =X t −X s . (2)Multiplicativity: Letting(δX n ) sut :=X s n t −X s n u −X n ut for(s, u, t)∈ S 3,T , one requires

n−1 (δX n ) sut (i 1 , . . . , i n ) = X X s n u 1 (i 1 , . . . , i n 1 )X u n t −n 1 (i n 1 +1 , . . . , i n ).(1) n 1 =1 (3)Geometricity:For anyn, msuch thatn+m≤ b1/γcand(s, t)∈ S 2,T , we have: X s n t (i 1 , . . . , i n )X s m t (j 1 , . . . , j m ) = X X s n t +m (k 1 , . . . , k n+m ),(2) ¯ k∈Sh(¯ı,¯)

Date: August 28, 2009. 2000Mathematics Subject Classification.60H05, 60H07, 60G15. Key words and phrases.Rough paths theory; Fractional Brownian motion; Multiple stochastic integrals. 1

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DAVID NUALART AND SAMY TINDEL

where, for two tuples¯ı, ¯,Σ (ı¯,) r the set of permutations of the indices ¯ stands fo contained in(ı¯, ¯), and Sh(ı¯, ¯)is a subset ofΣ (¯ı,¯) defined by: Sh(ı¯, ¯=)  σ∈Σ (¯ı,¯) ;σdoes not change the orderings ofı¯and¯  . We shall call the family{X n ;n≤ b1/γc}a rough path overX(it is also referred to as the signature ofXin [7]). Once a rough path overXis defined, the theory described in [7, 8, 12] can be seen as a procedure which allows us to construct, starting from the family{X n ;n≤ b1/γc}, the complete stack{X n ;n≥1}. Furthermore, with the rough path overXin hand, one can also define rigorously and solve differential equations driven byX. The above general framework leads thus naturally to the question of a rough path construction for standard stochastic processes. The first example one may have in mind concerning this issue is arguably the case of ad-dimensional fractional Brownian motion (fBm)B= (B(1), . . . , B(d))with Hurst parameterH∈(0,1) is a Gaussian process. This with zero mean whose components are independent and with covariance function given by E(B t (i)B s (i 2 1 =))  t 2H +s 2H − |t−s| 2H  . ForH= 2 1  variance of the increments is thenthis is just the usual Brownian motion. The given by E  (B t (i)−B s (i)) 2  = (t−s) 2H ,(s, t)∈ S 2,T , i= 1, . . . , d, and this implies that almost surely the trajectories of the fBm areγ-Hölder continuous for anyγ < H, which justifies the fact that the fBm is the canonical example for a rough path construction. The first successful rough path analysis forBhas been implemented in [5] by means of a linearization of the fBm path, and it leads to the construction of a family{B 1 ,B 2 ,B 3 } satisfying (1) and (2), for anyH >1/4 Some(see also [6] for a generalized framework). other constructions can be found in [13, 15] by means of stochastic analysis methods, and in [17] thanks to complex analysis tools. In all those cases, the barrierH >1/4remains, and it has long been believed that this was a natural boundary, in terms of regularity, for an accurate rough path construction. Let us describe now several recent attempts to go beyond the thresholdH= 1/4. First, the complex analysis methods used in [16] allowed the authors to build a rough path above a processΓcalled analytic fBm, which is a complex-valued process whose real and imaginary parts are fBm, for any value ofH∈(0,1) should be mentioned. It however that<Γand=Γare not independent, and thus the arguments in [16] cannot be extrapolated to the real-valued fBm. Then, a series of brilliant ideas developed in [18, 19] lead to the rough path construction in the real-valued case. We will try now to summarize briefly, in very vague terms, this series of ideas (see Section 3 for a more detailed didactic explanation): (i)Consider a smooth approximationB n,ε ofB n , satisfying relation (1), which may diverge asε→0wheneverH <1/4 one can decompose. Then,B s n, t ε asB s n t ε, =A s nε, t +C s nε, t , where C n,ε is the increment of a functionf, namelyC s nε, t =f t −f s , andA n,ε is obtained as a boundary term in the integrals definingB n,ε  explained in Section 3, a typical example. As of such a decomposition is given (forn= 2), up to approximations, byA s 2 t =−X s ⊗δX st andC 2 st = R t s X u ⊗dX u , and in this casef t = R 0 t X u ⊗dX u  it can be easily checked,. Then