# ROUGH VOLTERRA EQUATIONS THE ALGEBRAIC

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### pefav

ROUGH VOLTERRA EQUATIONS 1: THE ALGEBRAIC INTEGRATION SETTING. AURÉLIEN DEYA AND SAMY TINDEL Abstract. We deﬁne and solve Volterra equations driven by an irregular signal, by means of a variant of the rough path theory called algebraic integration. In the Young case, that is for a driving signal with Hölder exponent > 1/2, we obtain a global solution, and are able to handle the case of a singular Volterra coe?cient. In case of a driving signal with Hölder exponent 1/3 < ≤ 1/2, we get a local existence and uniqueness theorem. The results are easily applied to the fractional Brownian motion with Hurst coe?cient H > 1/3. 1. Introduction This article is the ﬁrst of a series of two papers dealing with Volterra equations driven by rough paths. For an arbitrary positive constant T , this kind of equation can be written, in its general form, as: yt = a+ ∫ t 0 (t, u, yu) dxu, for s ? [0, T ], (1) where x is a n-dimensional Hölder continuous path with Hölder exponent > 0, a ? ?d stands for an initial condition, and : ?+??+??d ? ?d,n is a smooth enough function. Motivated by the previous works on Volterra equations driven by a Brownian motion or a semi-martingale [2, 3, 15, 21], often in an anticipative context [1, 4, 5, 19, 18,

• ?su ?

• unique ?? ?

• then

• dimensional hölder

• algebraic integration

• rough volterra

• hölder exponent

• smooth functions

• called algebraic

Publié le : mardi 19 juin 2012
Lecture(s) : 52
Tags :
##### Smooth function
Source : iecn.u-nancy.fr
Nombre de pages : 33
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1 N [z^;C ([T ;T +"])]c 1 +" N [z;C ([0;T +"])] :0 0 ;x 01 1
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z^ [0;T +"])]0

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1 1=(1 ) 1 (1)(1)" = (2c ) " y N = 2c 1 +N [y ;C ([0;T ])] ;1 0;x ;x 1
N N1 1N [z;C ([0;T +"])]N N [z^;C ([0;T +"])] + =N0 1 0 11 1 2 2

N (1)1B = z2C ([0;T +"]) : z =y ;N [z;C ([0;T +"])]N(1) 0 j[0;T ] 0 11 0 1y ;T ;"0
2 (0;"]
[0;T +]0
N(1) (2) 1 (1) (1) (2) (2) (1)z ;z 2B z^ = ( z ) z^ = ( z ) z^(1)y ;T ; 0
(2) (1) (2)z^ [0;T ] N [z^ z^ ;C ([0;T +])] =0 01
(1) (2)N [z^ z^ ;C ([T ;T +])] T s < t T +0 0 0 01
1;1 1;2(1) (2) 2(z^ z^ ) =J +J +Jst st st st
1;1 1;2t (1) t (2) t (1) t (2)J = ( (Z ) (Z )) (x) ; J = (( (Z ) (Z ))x);st stst sts s
2 t s (1) t s (2)J = (([ ](Z ) [ ](Z ))x):0sst
1;1J
1;1 (1) (2)kJ kkDk kz z kN [x;C ]jt sj :st 1 s s 1
(1) (2) (1) (2) (1) (2) (1) (2) kz z k =k[z z ] [z z ]kN [z z ;C ([0;T +])] ;0s s s s T T 10 0
1;1 (1) (2)N [J ;C ([T ;T +])]c N [z z ;C ([0;T +])]:0 0 x; 02 1
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