THÈSE Koléhè Abdoulaye COULIBALY-PASQUIER

Define the Itô stochastic equation in the sense of [13]:d ˜∇Yt (x) = P ˜∇Yt(x 0 ),Y t(x)d ˜∇Yt (x 0 ) (4.2)Remark : The above equation is well defined, for x sufficiently close to x 0 ,because d T −t (X t (x), X t (x 0 )) is a finite variation process, with bounded derivative(by a short computation and [18], [6]).Let ˜// 0,tbe the parallel transport, associated to the connection ˜∇, over the semimartingale Y t (x 0 ).In the next lemma, we will explain the relationship between the two paralleltransport ˜// 0,tand // T 0,t.Lemma 4.1 Let (e i ) i=1..n be a orthonormale of (T M x0 , g(T )) thend((// T 0,t) −1 dπ ˜// 0,t)(0, e i ) = 1 2 (//T 0,t) −1 ( ∂ ∂ tg(T − t)) #T −t (dπ ˜// 0,t(0, e i )) dt.Proof :The parallel transport ˜// 0,tdoes not modify the time vector, i.e.,˜// −1(t,X t)(0, ...) = (0, ...),as can be shown for every curves, and hence for the semi-martingale Y t by thetransfer principle.We can identify T (0,x0 )I × M and T x0 M with the help of (0, v) ↦−→ v. Hence(// T 0,t) −1 dπ ˜// 0,t: T (0,x0 )I × M → T x0 Mbecomes an element in M n,n (R).Recall that // T 0,t = Ut T U T,−10 . By definition of D S,t given in (1.2). We get usingthe shorthand e i = U0 T ẽ i , with (ẽ i ) i=1..n an orthonormal frame of R n ,∗d((// T 0,t) −1 dπ ˜// 0,t) = ∗d(〈(// T 0,t) −1 dπ ˜// 0,te i , e j 〉 T ) i,j= ∗d(〈dπ ˜// 0,te i , // T 0,te j 〉 T −t ) i,j(= 〈D S,T −t dπ ˜// 0,te i , Ut T ẽ j 〉 T −t+ ∂ (g(T − t))(dπ ˜// ∂ 0,te i , Ut T ẽ j ) dtt+〈dπ ˜//)0,te i , D S,T −t Ut T ẽ j 〉 T −t .i,j5621