ROUGH VOLTERRA EQUATIONS THE ALGEBRAIC

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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 :
Source : iecn.u-nancy.fr
Nombre de pages : 33
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AT 0B =fy2C ([0;T ]) : y =a;N [y;C ([0;T ])]A g0 0 0 T1 1 0T ;a0
A TT 00
AT0T 2 (0;T ] y;y~2 B z =0 1 T ;a0
1;1 1;2 2 ( y);z~ = (~y) (z z~) (z z~) =J +J +J

1;1 1;2t t t t~ ~J = ( (Y ) (Y )) (x) ; J = ( (Y) (Y))x ;s s st stst st

2 t s t s ~J = ([ ](Y) [ ](Y))x :0sst

1;1 1;1 0J N [J ;C ]kDk N [y y~;C ]N [x;C ] y =1 02 1 1
0y~ =a y y~ =y y~ (y y~ ) N [y y~;C ]N [y y~;C ]T0 s s s s 0 0 1 1 0
1;1N [J ;C ]c N [y y~;C ]T :x; 2 1 0
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1;2 2t t ~kJ k cN [ (Y) (Y);C ]N [x;C ]jt sjst 1 1
c jt sj (1 +N [y;C ] +N [y~;C ])N [y y~;C ]T ;x; 1 1 1 0
1;2N [J ;C ]c (1 +N [y;C ] +N [y~;C ])N [y y~;C ]Tx; 2 1 1 1 0
2J
(1+)2 t s t s ~kJ k cN [[ ](Y) [ ](Y);C ]N [x;C ]Tst 1 1 0
1+ c jt sj T N [y y~;C ]f1 +N [y;C ] +N [y~;C ]g;;x 0 1 1 1
1+ 2 N [J ;C ]c T N [y y~;C ]f1 +N [y;C ] +N [y~;C ]g;x2 0 1 1 1

N [z z~;C ]c T N [y y~;C ]f1 +A g A;x T T1 0 1 0 0
T 0 T 2 (0;T ]0 0 1
AT0BT ;a0

(1)y
dC ([0;T ];R )1
" > 0
(1) (1)y y [0;T +"]0

z2C ([0;T +"])01
(1)z =yj[0;T ]0
(
(1)
y t2 [0;T ]t 0
z^ = ( z) = :t t ta +J ( (Z)dx) t2 [T ;T +"]0t 0 0

N [z^;C ([0;T +"])]01
s;t2 [0;T ] s;t2 [T ;T +"] sT tT +"0 0 0 0 0
(1)N [z^;C ([0;T ])]N [y ;C ([0;T ])]0 01 1
1;1 1;2
s;t2 [T ;T +"] (z^) (z^) =I +I +0 0 st st st st
2;1 2;2
I +Ist st
1;1 1;2t tI = (Z ) (x) ; I = (( (Z))x);st s st st st
2;1 t s 2;2 t sI = [ ](Z ) (x) ; I = (([ ](Z))x):0 0s 0sst st
1;2Ist
1;2 2tkI k cN [ (Z);C ([0;T +"])]N [x;C ]jt sj0st 1 1
2
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2;2
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2;2 t s 2kI k cN [[ ](Z);C ([0;T +"])]N [x;C ]T0st 1 1

c jt sjf1 +N [z;C ([0;T +"])]g;;x 01
2;2 1 N [I ;C ([T ;T +"])] c " f1 +N [z;C ([0;T +"])]g0 0 ;x 02 1
i;1N [I ;C ([T ;T +"])]c i = 1; 20 0 ;x2

1 N [z^;C ([T ;T +"])]c 1 +" N [z;C ([0;T +"])] :0 0 ;x 01 1
0sT tT +"0 0
k(z^) k =k(z^) + (z^) kst sT T t0 0
(1) N [y ;C ([0;T ])]jT sj +N [z^;C ([T ;T +"])]jt Tj0 0 0 0 01 1
(1) N [y ;C ([0;T ])] +N [z^;C ([T ;T +"])] jt sj :0 0 01 1
z^ [0;T +"])]0

1 (1) 1 N [z^;C ([0;T +"])]c 1 +N [y ;C ([0;T ])] +" N [z;C ([0;T +"])] :0 0 01 ;x 1 1

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
1;2
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1;2 2t (1) t (2)kJ k cN [ (Z ) (Z );C ([0;T +])]N [x;C ]jt sj0st 1 1
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