Stability of stochastic di erential equations in

Stability of stochastic dierential equations in manifolds Marc Arnaudon

Institut de Recherche Mathematique Avancee Universite Louis Pasteur et CNRS 7, rue Rene Descartes F{67084 Strasbourg Cedex France [email protected]

Anton Thalmaier

Naturwissenschaftliche Fakultat I { Mathematik Universitat Regensburg D{93040 Regensburg Germany

[email protected]

Abstract. | We extend the so-called topology of semimartingales to continuous semimartingales with values in a manifold and with lifetime, and prove that if the manifold is endowed with a connection r then this topology and the topology of compact convergence in probability coincide on the set of continuous r-martingales. For the topology of manifold-valued semimartingales, we give results on dierentiation with respect to a parameter for second order, Stratonovich and Itô stochastic dierential equations and identify the equation solved by the derivative processes. In particular, we prove that both Stratonovich and Itô equations dierentiate like equations involving smooth paths (for the Itô equation the tangent bundles must be endowed with the complete lifts of the connections on the manifolds). As applications, we prove that dierentiation and antidevelopment of C 1 families of semimartingales commute, and that a semimartingale with values in a tangent bundle is a martingale for the complete lift of a connection if and only if it is the derivative of a family of martingales in the manifold. 1. Introduction

Let ( ; (Ft)0t 0g [0; 1) of an R d -valued continuous adapted process (resp. semi ? martingale) de ned on the probability space f > 0g; FA?s s0 ; P( j > 0) . They can be endowed with a complete metric space structure, as in the case = 1, which gives respectively the topology of compact convergence in probability and the topology of semimartingales (see [E1]). Let T denote the set of predictable stopping times and let S S D^c (R d ) = Dc (R d ; ) ; S^ (Rd ) = S (Rd ; ) : 1

2T

2T

The sum X + Y , dierence X ? Y , product (X; Y ) of two processes with lifetime is a process with lifetime the in mum of the lifetimes of the two processes. The lifetime of a process X will be denoted by X . If T is a predictable stopping time, we can de ne the operations of stopping at T and killing at T on the sets D^c (R d ) and S^ (R d ): let X be an element of D^c (R d ) or S^ (R d ). Then the process X T stopped at time T is the continuous process with lifetime +1 1fT 0g and T = 0 on f = 0g. Let us de ne a topology on the sets D^ c (R d ) and S^ (Rd ). If X 2 D^c (R d ) with lifetime X , T a predictable stopping time such that T < X and " > 0, one de nes neighbourhoods of X with the sets

Vcp (X; T; ") = Y 2 D^c (R d );

E

h

i

1 ^ sup kYt ? Xt k < " 0 X + " \ t! lim Xt exists < " X (the second condition will insure that the topology is separated). Analogously, one de nes neighbourhoods of X 2 S^(R d ) by setting n o d ^ V (X; T; ") = Y 2 S (R ); E 1 ^ v(Y ? X )T < " Wcp (X; ") = Y

where if Z = Z0 + M + A is the canonical decomposition of Z 2 S^ (Rd ),

v(Z )t =

d X i=1

jZ0i j + 1t =2 +

t

Z

0

jdAij

(with the convention that v(Z )t = +1 if t Z ) and

W (X; ") = Y

2 S^ (R d ); P

Y > X + " \ t! lim Xt exists X

i 0 and a predictable stopping time T with T < X ^ Y and E 1 ^ sup kYt ? Xt k > 2" in which case Vcp (X; T; ") \ Vcp (Y; T; ") = , or 0 0 and stopping nthere exists " > 0 and a predictable o time T satisfying T < X such that P Y + 2" < T; t! lim Yt exists > 2"; in this Y case, one veri es that Vcp (X; T; ") \ Wcp (Y; ") = . Remark. | Convergence for the topology of semimartingales implies compact

convergence in probability. For 1 p 1 and 2 T , let S p (R d ; ) denote the Banach space of processes X 2 Dc (R d ; ) such that kX kSp(Rd; ) = kXkLp < 1, where Xt = sups 0, and this gives the result.

3. Regularity of solutions of stochastic dierential equations

Let M and N be connected smooth manifolds. In this section, we will study stability of second order stochastic dierential equations of the type

DZ = f (X; Z ) DX

(3:1)

where f 2 ?( (M ) (N )) is a Schwartz morphism, X belongs to S^ (M ) and Z to S^ (N ). Remark. | If P is a submanifold of M N such that the canonical projection P ! M is a surjective submersion, and if f is only de ned on P and constrained to P (see [E3]), then one can extend f in a smooth way to M N , and one knows that a solution (X; Z ) of (3.1) with (X0; Z0) 2 P will stay on P . PROPOSITION 3.1. | Let (X n)n2N be a sequence of elements in S^(M ) converging to X in S^ (M ), let (Z0n )n2N be a sequence of N -valued random variables converging to Z0 in probability, and let (f n)n2N be a sequence of locally Lipschitz Schwartz morphisms in ?( (M ) (N )) with uniform Lipschitz constant on compact sets, converging to a Schwartz morphism f 2 ?( (M ) (N )). If Z n is the maximal solution starting from Z0n to DZ n = f n (X n; Z n ) DX n, then (X n; Z n ) converges in S^(M N ) to (X; Z ) where Z is the solution to DZ = f (X; Z ) DX starting from Z0. Moreover, if X n converges in probability to X then Z n converges to Z in S^(N ).

Proof. | Let Z be the lifetime of Z . We will show that (Z n )Z - converges to Z and that limt!Z Zt does not exist on fZ < X g, which is stronger than the results of Proposition 3.1. The second point is known, let us prove the rst one. We have to show that there exists a stopping time T as close to Z as we want and a subsequence Z nk converging to Z . Hence we can assume that X n, X take their values in a compact subset KM , Z0n in a compact subset KN and that X n converge in H 1 (KM ; 1) and in S 1 (KM ; 1) to X . We can also assume that Z takes its values in KN and has lifetime 1. Consider Schwartz morphisms fKn , fK satisfying the same convergence assumptions as f n and f , with compact support K containing a neighbourhood of the product KM KN . Using the continuity results of Proposition 2.6 and [E2] theorem 0, we obtain that the solution ZKn of DZKn = fKn (X n; ZKn ) DX n with (ZKn )0 = Z0n converge in S (N; 1) to the solution ZK of DZK = fK (X; ZK ) DX with (ZK )0 = Z0. This implies that a subsequence converges locally in H^ 1 (N ) and in S^1 (N ), but then locally, for indices suciently large, the solutions to the truncated equation coincide with the solutions to the original equation. This gives the claim. Immediate consequences of Proposition 3.1 are the following results. COROLLARY 3.2. | 1) Let ?1 ( (M ) (N )) be the set of C 1 Schwartz morphisms endowed with the topology of uniform convergence on compact sets of the maps and their derivatives, and let L0 (N ) be the set of N -valued random variables endowed with the topology of convergence in probability. Then the map

S^(M ) ?1( (M ) (N )) L0(N ) ?! S^(M N ); de ned by (X; f; Z0) 7! (X; Z ) with Z the maximal solution of DZ = f (X; Z ) DX , is continuous. 2) Let ?1 ( (M )) be the set of C 1 forms of order 2 endowed with the topology of uniform convergence on compact sets of the maps and their derivatives. Then the map ?1 ( (M )) S^(M ) ?! S^ (R ) Z

(; X ) 7?! h(X ); DX i 0

is lower semicontinuous. Example. | Here we give an example of a sequence of deterministic paths converging uniformly to a constant path, but such that parallel transports above the elements of this sequence do not converge. This shows in particular that in 1) we cannot replace the topology of semimartingales in S^(M ) by the topology of compact convergence in probability, unless we restrict for instance to the sets of martingales with respect to a given connection. Let M be a simply connected surface endowed with a rotationally invariant metric about o 2 M , represented in polar coordinates as ds2 = dr2 + g2(r) d#2 for some smooth function g. Let t 7! x(t) 2 M be a path in M , de ned on the unit

interval [0; 1], with polar coordinates r(t) " and #(t) = t for some > 0. A straightforward calculation shows that the rotational speed of a parallel transport above x in polar coordinates is ?g0 ("). Hence the rotational speed in an exponential chart with centre o which realizes an isometry at o is (1 ? g0(")) (note that this gives 0 if the metric is at). In the following, M is taken to be an open subset of the sphere S 2. Thus we have g(r) = sin r, and (1 ? g0(")) = (1 ? cos "). Consider the sequence of paths (xn )n2N de ned in polar coordinates by #n (t) = 2nt and rn (t) "n = arccos(1 ? 21n ) (hence 2n(1 ? cos "n ) = ). Since "n ! 0, we get uniform convergence of (xn )n2N to the constant path o. But for all n, the rotation at time 1 of a parallel transport above xn is . Hence parallel transports above xn do not converge to a parallel transport above o. In the sequel we are seeking dierentiability results. This requires some geometric preliminaries. We will use the maps : M N M ! N de ned by Cohen [C1] and [C2] to describe stochastic dierential equations in manifolds with cadlag semimartingales. DEFINITION 3.3. | Let k 2 N . A Schwartz morphism f 2 ?( (M ) (N )) k if f (resp. e) is C k (resp. a section e 2 ?(TM TN )) is said to be of class CLip with locally Lipschitz derivatives of order k. k;1 if there We say that a measurable map : M N M ! N is of class CLip exists a neighbourhood of the submanifold f(x; z; x); (x; z) 2 M N g on which is C 1 with respect to the third variable and all the derivatives with respect to this variable are C k with locally Lipschitz derivatives of order k (with respect to the three variables). LEMMA (AND DEFINITION) 3.4. | Let k 2 N . For every Schwartz morphism k , there exists a map : M N M ! N of class f 2 ?( (M ) (N )) of class CLip k;1 such that for all (x; z ) 2 M N CLip

f (x; z) = 3 (x; z; x) where 3 denotes the second order derivative of with respect to the third variable. Such a map will be called a Cohen map associated to f . In particular, a Cohen map satis es (x; z; x) = z for all (x; z) 2 M N . Proof. | First, we remark that it is sucient to construct in a neighbourhood of the submanifold f(x; z; x); (x; z) 2 M N g and to extend it then in a measurable way to M N M . Let rM (resp. rN ) be a connection on M (resp. N ). There exists a neighbourhood of the diagonal of M M on which the maps (x; z) 7! v(x; z) = _ (0) and (x; z) 7! u(x; z) = (0) are smooth, where is the geodesic such that (0) = x and (1) = z. There exists a neighbourhood of the null section in TN on which the exponential map, denoted by expN , is smooth. If u 2 N is a second order vector, denote by F (u) 2 TN its rst order part with respect to the connection rN (see [E4] for the de nition).

Thus there exists a neighbourhood V of (x; y; x); (x; y) 2 M N such that the map : V ! N ? (x; y; z) 7! expN f (x; y) v(x; z) + 21 F f (x; y) u(x; z) is de ned and satis es the regularity assumptions. We have to verify the equation 3(x; y; x) = f (x; y). For this, it is sucient to check that these maps coincide on elements of xM of the form _ (0) and (0) where is a geodesic with (0) = x and

(1) = z. A change of time gives 2 ? (x; y; (t)) = expN t f (x; y) v(x; z) + t2 F f (x; y) u(x; z) Taking successively rst and second order derivatives with respect to t at time 0 gives the result. THEOREM 3.5. | Let a 7! X (a) be C 1 from I to S^ (M ), let f 2 ?( (M ) (N )) 1 , and a 7! Z (a) the maximal solution of be a Schwartz morphism of class CLip ? DZ (a) = f X (a); Z (a) DX (a) (3:2) where a 7! Z0 (a) is C 1 in probability. Then the map a 7! (X (a); Z (a)) de ned on I and with values in S^(M N ) is C 1 , and the process @a Z (a) is the maximal solution of ? D@a Z (a) = f 0 @a X (a); @aZ (a) D@a X (a) (3:3) 0 de ned with initial condition @a Z0 (a) where f 0 is the Schwartz morphism of class CLip 1;1 Cohen map associated to f , then as follows : if f (x; z) = 3 (x; z; x) with a CLip 0;1 Cohen map f 0(u; v) = 3 T(u; v; u) for (u; v) 2 TM TN , i.e., T is a CLip associated to f 0. If moreover a 7! X (a) is continuous in probability, then a 7! Z (a) is C 1 in S^ (N ). Remark. | If P is a submanifold of M N such that the canonical projection P ! M is a surjective submersion, and if f is only de ned on P and is constrained to P , then one can show that f 0 is constrained to TP . As a consequence, by the ? remark at the beginning of this section, if @a X0 (a); @aZ0 (a) belongs to TP , then ? @a X (a); @aZ (a) takes its values in TP . LEMMA 3.6. | Let P; Q; R; S be manifolds, ': Q ! P and : R ! S maps, and let : Q R Q ! R and 0 : P S P ! S be Cohen maps such that 0 ('; ; ') = . Then, for all (x; y) 2 Q R, we have ? 30 '(x); (y); '(x) '(x) = (y) 3(x; y; x): If semimartingales X; Z take values in Q, resp. R, and satisfy the equation DZ = 3 (X; Z; X ) DX , then U = '(X ) and V = (Z ) satisfy DV = 3 0 (U; V; U ) DU: Proof. | It is sucient and easy to prove the rst equality with second order derivatives of curves. The second equality is a consequence of the rst one.

Proof of Theorem 3.5. | Assume that 0 2 I . Using Proposition 3.1, it is sucient to prove that a 7! (X (a); Z (a)) is dierentiable at a = 0 and that the derivative of a 7! Z (a) is the maximal solution of (3.3). Let rM be a connection on M . There exists an open neighbourhood M 1 of the M diagonal in M M such that for a 6= 0 the function M M M 'M a : 1 ?! Ua := 'a (1 ) (x; y) 7?! 1 (expN )?1 y

a x is well-de ned and a dieomorphism. The same objects on N are denoted with the 1;1 Cohen map associated to f . It is easy to see that superscript N . Let be a CLip ? ? D(Z (0); Z (a)) = 3 (; ) (X (0); X (a)); (Z (0); Z (a)); (X (0); X (a)) D X (0); X (a) : e (a) = (X (0); X (a))T M (a)- and Let T M (a) be the exit time of (X (0); X (a)) of M 1 ,X Ze(a) be the maximal solution to ? DZe(a) = 3 (; ) Xe (a); Ze(a); Xe (a)) DXe (a) with initial condition (Z0(0); Z0(a)). Let then T N (a) be the exit time of Ze(a) of N1 . Using Proposition 2.6, it is easy to see that T M (a) ^ X (0) converges in probability to X (0) as a tends to 0, and then that T N (?a) ^ Z (0) converges in probability to Z (0). ? N (a)- N T M e e e e By Lemma 3.6, de ning V (a) = 'a Z (a) and Y (a) = 'a X (a)T N (a)for a 6= 0, we have that Ve (a) is the maximal solution in S^ (TM ) of DVe (a) = 3?'Na (; ) ?('Ma )?1 ; ('Na )?1 ; ('Ma )?1?Ye (a); Ve (a); Ye (a) DYe (a) ? with initial condition Ve0 (a) = 'Na Z0 (0); Z0(a) on f(Z0(0); Z0(a)) 2 N1 g. For N u 2 UaM (resp. u 2 UaN ), denote by `M a (u) (resp. à (u)) the second coordinate of ?1 N ?1 ('M a ) (u) (resp. ('a ) (u)). Then the mapping N? ? ? M N (v ); `M (w) if a 6= 0, ' ( u ) ; ( v ) ; ( w ) ; ` ( u ) ; ` a a a a (a; u; v; w) 7! T(u; v; w) if a = 0, de ned on an open subset of (?1; 1) TM TN TM containing the elements of the form (0; u; v; u) with (u; v) 2 TM TN , depends C 1 on the last variable and its derivatives with respect to this variable are locally ?Lipschitz (as functions of all four variables). This implies the convergence of 3 'Na (; ) ? M ?1 ?1 to 3 T as a ! 0, and the existence of uniform ('a ) ; ('Na )?1 ; ('M a) Lipschitz constants on compact sets. Since a 7! X (a) is dierentiable at a = 0, T M (a) ^ X (0) converges in probability to X (0) and T N (a) ^ Z (0) converges in probability to Z (0) , we have that Ye (a) converges to Y (0)Z with Y (0) := @a X (0); on the other side, Ve0 (a) converges in probability to @a Z0 (0) = V0 (0) on fZ (0) > 0g; hence we get by Proposition 3.1 that (Ye (a); Ve (a)) converges to (Y (0); V (0)) where V (0) is the maximal solution of ? DV (0) = 3 T Y (0); V (0); Y (0) DY (0) with initial condition V0 (0) = @a Z0(0). This implies that a 7! (X (a); Z (a)) is dierentiable at a = 0 and that its derivative is (Y (0); V (0)). (0)

We now want to investigate Stratonovich and Itô equations. In the following, if (t; a) 7! x(t; a) is a map de ned on an open subset of R 2 and with values in a manifold M , x_ (t; a) will denote its derivative with respect to t, and x(t; a) will denote the second order tangent vector such that for all smooth function g on M , x(t; a)(g) = @t2 (g x)(t; a). For a smooth function g on M , d2 g will denote the second order form de ned by hd2g; x(t; a)i = x(t; a)(g) (see [E4]). LEMMA 3.7. | Let J; I be two intervals in R . Suppose that (t; a) 7! x(t; a) 2 M and (t; a) 7! z(t; a) 2 N are C 2;1 maps de ned on J I , and satisfy for each a ?

z(0; a) = 3 x(0; a); z(0; a); x(0; a) x(0; a) (3:4) 1;1 Cohen map. Then where : M N M ! N is a CLip ? (@a z)(0; a) = 3 T @a x(0; a); @az(0; a); @ax(0; a) (@ax)(0; a): Proof. | It is sucient to prove

2

? d `; (@az)(0; a) = d2 `; 3 T @a x(0; a); @az(0; a); @ax(0; a) (@a x)(0; a) (3:5) and

?

d`; (@az)_(0; a) 2 = d`; T3 T @a x(0; a); @az(0; a); @ax(0; a) (@a x)_(0; a) 2 (3:6) for ` = g : TN ! R and ` = dg: TN ! R where g: N ! R is smooth. Equations (3.5) and (3.6) for ` = g : TN ! R , g 2 C 1 (N; R) are direct consequences of? assumption (3.4). To establish (3.5) for ` = dg: TN ! R , we de ne z0 (t; a) = x(0; a); z(0; a); x(t; a) . Then

hd2`; (@az)(0; a)i = @t2 @a (g z)(0? ; a) = @a @t2(g z)(0; a) = @a d2 g; z(0; a) = @a d2g; 3 x(0; a); z(0; a); x(0; a) x(0; a)

= @a d2g; (z0)(0; a) = @a @t2(g z0 )(0; a) = @t2@a (g z0 )(0; a) ? = @t2 dg T @a x(0; a); @az(0; a); @ax(t; a)

? = d2`; 3 T @a x(0; a); @az(0; a); @ax(0; a) (@a x)(0; a) : Finally, to verify (3.6) for ` = dg: TN ! R , we have to prove that ? ? @t @a (g z)(0; a) 2 = @t @a (g z0 )(0; a) 2 : We rst derive from (3.4) that ? ? @t (g z)(0; a) 2 = @t (g z0 )(0; a) 2 ; and by taking the square of the derivative with respect to a, ? ? ? ? @t (g z)(0; a) 2 @t@a (g z)(0; a) 2 = @t (g z0 )(0; a) 2 @t @a (g z0 )(0; a) 2 : ? Let a0 2?I . If @t (gz)(0; a0) 2 6= ?0, equality (3.6) is satis ed?for a = a0. Now consider the case @t (g ?z)(0; a0) 2 = 0. If @t @a (g z)(0; a0) 2 6= 0 or @t @a (g z0)(0; a0) 2 6= 0, then we have @t (g z)(0; a) 2 6= 0 in a neighbourhood of a0 (a0 excepted) and (3.6) is satis ed for a = a0 by continuity.

DEFINITION 3.8. | A Cohen map : M N M ! N is said to be a Cohen map of Stratonovich type if in addition it has? the following property : if a C 2 curve ( ; ) in M N satis es _ (t) = T3 (t); (t); (t) _ (t) then ? (t) = 3 (t); (t); (t) (t). k section of the vector bundle PROPOSITION 3.9. | Let k 1 and e be a CLip k?1;1 Cohen map of Stratonovich T M TN over M N . Then there exists a CLip k;1 Cohen map type such that e(x; z) = T3 (x; z; x) for all (x; z) 2 M N . If is a CLip k?1;1 Cohen map of of Stratonovich type, then T: TM TN TM ! TN is a CLip Stratonovich type. k?1;1 is a consequence of [E3 Theorem 8], Proof. | The existence of of class CLip which gives the existence of a unique Schwartz morphism of Stratonovich type f of k?1 associated to e, together with Lemma 3.4. class CLip k;1 Cohen map of Stratonovich type; we want to show that T is Let be a CLip also a Cohen map of Stratonovich type. Let t 7! (t) be a smooth curve with values in TN and t 7! (t) a smooth curve with values in TM such that ? _ (t) = T3 T (t); (t); (t) _ (t):

(3:7)

We have to prove that ? (t) = 3 T (t); (t); (t) (t):

This will be done by means of Lemma 3.7. More precisely, let (t; a) 7! x(t; a) satisfy @a x(t; 0) = (t), and let (t; a) 7! z(t; a) 2 M be a solution of ?

z_ (t; a) = T3 x(t; a); z(t; a); x(t; a) x_ (t; a)

(3:8)

with the property @a z(0; 0) = (0). It is easy to verify that (t) = @a z(t; 0) then already for all t, by exploiting uniqueness of solutions to (3.7) with given initial conditions and by calculating hdh; (@a z)_(t; 0)i for h = dg and h = g where g: N ! R is smooth. Since is a Cohen map of Stratonovich type, together with equation (3.8), we get from Lemma 3.7 ?

(@a z)(t; a) = 3 T @a x(t; a); @az(t; a); @ax(t; a) (@a x)(t; a) which can be rewritten for a = 0 as ? (t) = 3 T (t); (t); (t) (t):

This proves that T is indeed a Cohen map of Stratonovich type. Rephrased in terms of Cohen maps of Stratonovich type, the following result is a consequence of [E3 Theorem 8].

k section of T M TN PROPOSITION 3.10. | Let k 1 and let e be a CLip k?1;1 Cohen map satisfying e(x; z ) = T (x; z; x). The over M N . Let be a CLip 3 equations Z = T3 (X; Z; X ) X and DZ = 3 (X; Z; X ) DX are equivalent if and only if is a Cohen map of Stratonovich type. For the rest of this section we assume that both M and N are endowed with connections rM and rN . On the tangent bundles TM and TN we consider the corresponding complete lifts (rM )0 and (rN )0 of these connections (see [Y-I] for a de nition). We will say that a Schwartz morphism f 2 ?( (M ) (N )) is semi-ane if for every rM -geodesic with values in M and de ned at time 0, for every y 2 N , f ( (0); y) (0) is the second derivative of a rN -geodesic in N (see [E3] for details). In fact f ( (0); y) (0) is the second order derivative (0) of the geodesic which satis es (0) = y and _ (0) = f ( (0); (0)) _ (0). DEFINITION 3.11. | We say that a Cohen map is a Cohen map of Itô type (with respect to the connections rM and rN ) if 3 (x; z; x): x M ! z N is a semi-ane morphism. k section of T M TN over PROPOSITION 3.12. | Let k 0 and let e be a CLip k;1 Cohen map of Itô type such that e(x; z ) = T (x; z; x) M N . There exists a CLip 3 k; 1 for all (x; z) 2 M N . If k 1 and is a CLip Cohen map of Itô type, then T is k?1;1 Cohen map of Itô type (with respect to the connections (rM )0 and (rN )0 ). a CLip Proof. | The existence of is a consequence of [E3 Lemma 11] which gives the existence of a unique Schwartz morphism of Itô type associated to e, together with Lemma 3.4. Let be a Cohen map of Itô type; we want to show that T is also a Cohen map of Itô type. We have to prove that for all (y0; v0 ) 2 TM TN , 3 T(y0 ; v0; y0) is semi-ane, i.e., if t 7! y(t) is a (rM )0-geodesic in TM with y(0) = y0 , then the (rN )0-geodesic t 7! v(t) in TN with v_ (0) = T3 T(y0 ; v0; y0) y_ (0) satis es v(0) = 3 T(y0; v0; y0) y(0). Let (t; a) 7! x(t; a) 2 M satisfy @a ja=0 x(t; a) = y(t) and such that t 7! x(t; a) is a rM -geodesic for all a. Note that this is possible because y is a Jacobi eld. Let (t; a) 7! z(t; a) 2 N be such that for all a, t 7! z(t; a) is a rN -geodesic with ?

z_ (0; a) = T3 x(0; a); z(0; a); x(0; a) x_ (0; a) and @a ja=0 z(0; a) = v(0). Since t 7! x(t; a) and t 7! z(t; a) are geodesics and T3 (x; z; x) is semi-ane, we deduce that ?

z(0; a) = 3 x(0; a); z(0; a); x(0; a) x(0; a): Now we can apply Lemma 3.7 to obtain (@a z)(0; a) = 3 T(@a x(0; a); @az(0; a); @ax(0; a)) (@ax)(0; a):

(3:9)

It remains to prove that (@a z)(0; 0) = v(0) and (@ax)(0; 0) = y(0). But t 7! @a z(t; a) and t 7! @a x(t; a) are geodesics for (rN )0 and (rM )0 , hence it is sucient to know that (@a z)_(0; 0) = v_ (0) and (@a x)_(0; 0) = y_ (0) (the last equality is already known). For this, we want to calculate hdh; (@az)_(0; a)i for h = dg and h = g with g: N ! R smooth (we will do the veri cation only for h = dg). Let h = dg, then

hdh; (@az)_(0; a)i = @t jt=0 dg; @az(t; a) = @t jt=0@a (g z)(t; a) = @a @t jt=0 (g z)(t; a) ? = @a @t jt=0(g ) x(0; a); z(0; a); x(t; a) ? = @t jt=0@a (g ) x(0; a); z(0; a); x(t; a)

? = @t jt=0 dg; T @a x(0; a); @az(0; a); @ax(t; a)

? = dh; T3T @a x(0; a); @az(0; a); @ax(0; a) (@a x)_(0; a) : In particular, for a = 0, this gives

dh; (@a z)_(0; 0) = dh; T3T (y0; v0 ; y0) y_ (0) : Since v_ (0) = T3 T(y0; v0 ; y0) y_ (0) we obtain v_ (0) = (@a z)_(0; 0) which nally gives with (3:9) v(0) = (@a z)(0; 0) = 3 T(y0; v0 ; y0) y(0): This proves that T is a Cohen map of Itô type. Rewritten with Cohen maps of Itô type, we get the following result as a consequence of [E3 Theorem 12]. k section of T M TN over M N . PROPOSITION 3.13. | Let k 0 and e be a CLip k;1 Cohen map satisfying e(x; z ) = T (x; z; x) for all (x; z ) 2 M N . Let be a CLip 3 N M r r Then the equations d Z = T3 (X; Z; X ) d X and DZ = 3 (X; Z; X ) DX are equivalent if and only if is a Cohen map of Itô type. The main motivation in our study of Cohen maps of Stratonovich and Itô type is the following result. k+1 section of the vector bundle COROLLARY 3.14. | 1) Let k 0 and e be a CLip T M TN over M N . Assume that a 7! X (a) is C k in S^ (M ), and a 7! Z (a) is the maximal solution of ? Z (a) = e X (a); Z (a) X (a) (3:10) where a 7! Z0 (a) is C k in probability. Then a 7! (X (a); Z (a)) is C k in S^ (M N ), and if k 1, the derivative @a Z (a) is the maximal solution of ? @a Z (a) = e0 @a X (a); @aZ (a) @a X (a) (3:11) k section of T TM TTN over with initial condition @a Z0 (0) where e0 is the CLip k+1;1 Cohen map TM TN de ned as follows : if e(x; z) = T3 (x; z; x) with a CLip then e0 (u; v) = T3 T(u; v; u) for (u; v) 2 TM TN . If moreover a 7! X (a) is continuous in probability, then a 7! Z (a) is C k in S^ (N ).

k section of the vector bundle T M TN over M N . 2) Let k 0 and e be a CLip Assume that M (resp. N ) is endowed with a connection rM (resp. rN ), and denote by (rM )0 (resp. (rN )0 ) the complete lift of rM (resp. rN ) in TM (resp. TN ). Assume that a 7! X (a) is C k in S^ (M ), a 7! Z (a) is the maximal solution of ? drNZ (a) = e X (a); Z (a) drMX (a)

(3:12)

where a 7! Z0 (a) is C k in probability. Then a 7! (X (a); Z (a)) is C k in S^ (M N ), and the derivative @a Z (a) is the maximal solution of

d(rN )0@a Z (a) = e0 @a X (a); @aZ (a) d(rM )0@a X (a) (3:13) k?1 section of T TM TTN over with initial condition @a Z0(0) where e0 is the CLip TM TN de ned in 1). If moreover a 7! X (a) is continuous in probability, then a 7! Z (a) is C k in S^ (N ). Remark. | We like to stress the pleasant point that both Stratonovich and Itô equations dierentiate like equations involving smooth paths. Proof of Corollary 3.14. | 1) We only have to consider the case k 1. Let k;1 Cohen map of Stratonovich type such that T (x; z; x) = e(x; z ) for all be a CLip 3 (x; z) 2 M N . By Proposition 3.10, equation (3.10) is equivalent to ?

DZ (a) = 3 ?X (a); Z (a); X (a) DX (a): Applying Theorem 3.5, we can dierentiate with respect to a and we get ?

D@aZ (a) = 3 T @a X (a); @aZ (a); @aX (a) D@a X (a):

(3:14)

k?1;1 Cohen map of Stratonovich type, and again But by Proposition 3.9, T is a CLip by Proposition 3.10, equation (3.14) is equivalent to ?

@a Z (a) = T3 T @a X (a); @aZ (a); @aX (a) @a X (a) which is precisely equation (3.11). 2) The proof of 1) carries over verbatim, replacing Stratonovich by Itô, Proposition 3.10 by Proposition 3.13, and Proposition 3.9 by Proposition 3.12. We want to rephrase equation (3.13) in terms of covariant derivatives. For this we need some de nitions and lemmas. Let RM denote the curvature tensor of the connection rM on M , which is assumed here to be torsion-free. If J is a semimartingale with values in TM endowed with the horizontal lift (rM )h of rM (see [Y-I] for a de nition),Mleth DJ denote its covariant derivative, i.e. the projection M )h v ? 1 ( r ( r ) J ) with vj : Tx M ! Tj TM of the vertical part of d J , thus DJ = vJ (d denoting the vertical lift for j 2 Tx M . We observe that also DJ = ==0;. d(==0?;.1J ) where ==0;t means parallel translation along (J ). Indeed, this equality is veri ed if J is a smooth curve, and since by [Y-I] (9.2) p. 114, J is a geodesic if and only if ((J ); ==0?;.1J ) is a geodesic in M T(J ) M for the product connection, using [E3] corollary 16, it extends to semimartingales as an Itô equation. 0

LEMMA 3.15. | Let J be a semimartingale with values in TM , and X = (J ) its projection to M . Then ? (3:15) d(rM )0 J = d(rM )h J + 21 vJ RM (J; dX )dX where vj (u) is the vertical lift above j 2 Tx M of a vector u 2 Tx M . e is a connection on TM , the It^ Proof. | Following [E3], if r o dierential e e e r r r d J may be written as F (DJ ) where F : TM ! TTM is the projection de ned as follows: if A and B are vector elds on TM , then F (A) = A and ? 1 F (AB ) = 2 re A B + re B A + [A; B ] . The result is a direct consequence of the following Lemma. For ` 2 M , let b(`) 2 TM TM denote its symmetric bilinear part, i.e., 1 hdf dg; b(`)i = 2 `(fg) ? f `(g) ? g `(f ) for f; g smooth functions on M . LEMMA 3.16. | Let L be an element of u TM with u 2 Tx M . Then

F (rM )0 ? F (rM )h

?

(L) = vu RM (u; . ) . b(L)

where : TM ! M is induced by : TM ! M . Proof. | It is sucient to prove this for Lu = (AB )u with A and B horizontal or vertical vector elds. But since among these possibilities (rM )0AB and (rM )hAB coincide except if both A and B are horizontal, we can restrict to this case. Let A (resp. B ) be the horizontal lift of A (resp. B ). Then by [Y-I], ? (rM )0Au B ? (rM )hAu B = vu RM (u; Ax)Bx ? where x = (u), and this gives the result, since b(Lu ) = 21 Ax Bx + Bx Ax .

k section of the vector bundle COROLLARY 3.17. | Let k 0 and e be a CLip T M TN over M N . Assume that M (resp. N ) is endowed with a torsion-free connection rM (resp. rN ). Assume that a 7! X (a) is C k in S^(M ), a 7! Z (a) the maximal solution of ? drNZ (a) = e X (a); Z (a) drMX (a)

(3:16)

? where a 7! Z0(a) is C k in probability. Then a 7! X (a); Z (a) is C k in S^ (M N ), and the derivative @a Z (a) is the maximal solution of the covariant stochastic differential equation

D@a Z = e(X; Z ) D@aX + re(@a X; @aZ ) drMX (3:17) ? ? + 21 e(X; Z )RM @a X; dX dX ? RN @a Z; e(X; Z )dX e(X; Z )dX

with initial condition @a Z0 (0). If moreover a 7! X (a) is continuous in probability, then a 7! Z (a) is C k in S^ (N ).

Remarks. | 1) If rM and rN are allowed to have torsion, one can write rst a

covariant equation of the form (3.17) with respect to the symmetrized connections a X , D@ a X , R M and R N r M and r N . With the obvious notations, expressing D@ in terms of D@a X , D@a X , RM , RN and the torsion tensors, one obtains then a covariant equation with respect to rM and rN . 2) Starting with (3.11) in Corollary 3.14, one can also easily determine a Stratonovich covariant equation, identical to the equation for smooth processes. Proof of Corollary 3.17. | Applying Lemma 3.15 to part 2) of Corollary 3.14 gives the following equation for @a Z : ? ? d(rN )h@a Z = e0 @a X; @aZ d(rM )h@a X + 21 v@a X RM (@aX; dX )dX ? ? 12 v@aZ RN (@a Z; dZ )dZ : ? ? But, if u; w 2 Tx M , z 2 TN , we have e0 (u; z)vu (w) = vz e (u); (?z) w , and by de nition, if hru M (w) 2 Tu TM is the horizontal lift of w, then vz re(u; z)w is the vertical part of e0 (u; z)hru M (w). TheseM equalities applied to u = @a X , z = @a Z , and successively to w = D@a X , w = dr X and w = 21 RM (@aX; dX )dX , give the desired equation. As an application of Corollary 3.14, we get dierentiability results for stochastic integrals, considered as particular instances of stochastic dierential equations: k+1 section of the vector bundle COROLLARY 3.18. | 1) Let k 0 and be a CLip T M . Assume that aZ7!. X (a) is C k in S^ (M ).

Then a 7! X (a); (X (a)); X (a) is C k in S^(M R ). 0 k section of the vector bundle T M TN over M N . 2) Let k 0 and be a CLip Assume that M (resp. N ) is endowed with a connection rM . Assume that a 7! X (a) is C k in S^ (M). Z .

M r Then a 7! X (a); (X (a)); d X (a) is C k in S^ (M R ). 0

4. Application to antidevelopment

If M is a manifold, we will denote by s: TTM ! TTM the following canonical isomorphism: if (t; a) 7! x(t; a) is a smooth M -valued map de ned on some open subset of R 2 , then @t @a x(t; a) = s (@a @t x(t; a)). THEOREM 4.1. | Let M be a manifold endowed with a connection r. Denote by r0 the complete lift of r on TM . Let A0 denote the antidevelopment with respect to r0 . Let a? 7! X (a) 2 S^ (M ) be a map of class C 1 de ned on some interval I of R . Then a 7! X (a); A(X (a)) 2 S^ (TM TM ) is of class C 1 and ? ? s @a A(X (a)) = A0 @a X (a) : Moreover, if a 7! X (a) is continuous in probability, then a 7! A(X (a)) is C 1 in S^ (TN ).

Before proving this result we introduce some de nitions and lemmas. Let M be a manifold of dimension m. If r is a connection on M , we consider the complete lift r0 of r on TM , which is characterized by the relation r0X c Y c = (rX Y )c valid for all vector elds X; Y 2 ?(TM ). Here X c ?denotesthe complete lift of X , i.e. the vector eld in ?(TTM ) de ned by Xuc = s TX (u) (see [Y-I] for details). Recall that the geodesics for r0 are the Jacobi elds for r. Let L(M ) be the principal bundle of linear frames on TM : thus Lx (M ) is the set of linear isomorphisms R m ! Tx M for each x 2 M . There is a canonical embedding |: TL(M ) ! L(TM ) de ned as follows: if W 2 TL(M ) is equal to (@a U )(0) where a 7! U (a) is a smooth path in L(M ), and if v 2 T R m = R 2m is equal to (@a z)(0) where a 7! z(a) is a smooth path in R m , then one has |(W )v = s ((@a (Uz))(0)). Let (e1 ; : : :; em ; e1; : : :; en ) be the standard basis of T R n . Then |((@a U )(0))e = s ((@a (Ue))(0)) and |((@a U )(0))e is the vertical lift (Ue )v (0) of (Ue )(0) above @a ( U )(0) where : L(M ) ! M is the canonical projection. LEMMA 4.2. | If a 7! X (a) 2 S^(M ) is C 1 and U (a) 2 S^ (L(M )) is a horizontal lift of X (a) such that a 7! U0(a) is C 1 in probability, then a 7! U (a) is C 1 in S^(L(M )) and | (@a U (a)) is a horizontal lift of @a X (a) with respect to the connection r0 . Proof. | The fact that a 7! U (a) is C 1 is a direct consequence of Corollary 3.14 and the fact that limt!U a Ut (a) exists is included in limt!X a Xt(a) exists . Another consequence of Corollary 3.14 is that it suces to prove the assertion with both Xt(a) = x(t; a) and a 7! U0(a) = u(0; a) deterministic and smooth. Write u(t; a) = Ut (a). It is sucient to verify that for all i 2 f1; : : :; mg, t 7! |(@a u(t; a)) ei and t 7! |(@a u(t; a)) e{ are parallel transports. But by [Y-I chapt. I, prop. 6.3], we have ( )

( )

?

?

?

r0(@ax)_(t;a) s @a (u(t; a)ei) = s @a rx_ (t;a) u(t; a)ei = 0 and

r0(@a x)_(t;a) u(t; a)ei v = rx_ (t;a) u(t; a)ei v = 0: ?

?

This proves Lemma 4.2. ? Proof of Theorem 4.1. | The fact that a 7! X (a); A(X (a)) is C 1 is a consequence of Corollary 3.14. We can calculate as if dealing with smooth deterministic paths. Let a 7! U0(a) 2 LX (a) (M ) be C 1 in probability and denote by U (a) the parallel transport of U0 (a) along X (a). Write Z (a) = A(X (a)). Then we have the equation 0

?

U (a)U0?1(a) p Z (a) = X (a) ?

where if z 2 Tx M and v 2 Tz TM is a vertical vector, p v denotes its canonical projection onto Tx M .

Denoting by m the dimension of M , we de ne a family of R m -valued processes R(a) by R(a) = U0?1 (a)Z (a), hence ?

R(a) = U0?1 (a) p Z (a) :

(4:1)

We have U (a)R(a) = X (a) and by dierentiation with respect to a, using the de nition of |, ?

?

| @a U (a) s @a R(a) = @a X (a):

(4:2)

On the other hand, dierentiation of (4.1) gives ?

?

?

?

| @a U0(a) s @a R(a) = p0 s @a Z (a)

(4:3)

where p0 on the vertical vectors to TTM is de ned like p. Putting together (4.2) and (4.3) gives ? ? ? ? ? | @a U (a) | @a U0(a) ?1 p0 s @a Z (a) = @a X (a): ?

But | @?a U (a) is the parallel transport of |(U0(a)) above @a X (a) by Lemma 4.2, hence s @a Z (a) = A0 (@a X (a)). COROLLARY 4.3. | Let J be a TM -valued semimartingale with lifetime = (0). There exists a C 1 family (X (a))a2R of elements in S^ (M ) such that the equality J = @a X (0) is satis ed. In particular, if (a) is the lifetime of X (a), then (a) ^ (0) converges in probability to (0) as a tends to 0. The semimartingale J is a r0 martingale if and only if one can choose (X (a))a2R such that X (a) is a r-martingale for each a 2 R . Proof. | With the notations of Theorem 4.1 de ne V = A0 (J ), and for a 2 R , ? ?

Z (a) = T exp s as(V ) : Note that the lifetime of ?Z (a) can be 0 if exp aJ0 is not de ned. A straightforward calculation shows that s @a Z (0) = V . De ne now? X (a) as the stochastic development of Z (a). We have the relation Z (a) = A X (a) ; by Corollary 3.14, the map a 7! (Z (a); X (a)) is C 1 in S^(M ) and in particular, (a) ^ (0) converges in probability to (0) as a tends to 0.? By Theorem 4.1, the antidevelopment of the derivative of a 7! X (a) at a = 0 is s @a Z (0) = V . This implies that @a X (0) = J . If J is a martingale, then V is a local martingale (with possibly nite lifetime). It is easy to see that for each a 2 R , Z (a) is also a local martingale, hence its development X (a) is a martingale.

REFERENCES [C1] Cohen (S.) | Geometrie dierentielle stochastique avec sauts 1, Stochastics and Stochastics Reports, t. 56, 1996, p. 179{203. [C2] Cohen (S.) | Geometrie dierentielle stochastique avec sauts 2: discretisation et applications des EDS avec sauts, Stochastics and Stochastics Reports, t. 56, 1996, p. 205{225. [E1] Emery (M.) | Une topologie sur l'espace des semimartingales, Seminaire de Probabilites XIII, Lecture Notes in Mathematics, Vol 721, Springer, 1979, p. 260{280. [E2] Emery (M.) | Equations dierentielles stochastiques lipschitziennes: etude de la stabilite, Seminaire de Probabilites XIII, Lecture Notes in Mathematics, Vol 721, Springer, 1979, p. 281{293. [E3] Emery (M.) | On two transfer principles in stochastic dierential geometry, Seminaire de Probabilites XXIV, Lecture Notes in Mathematics, Vol 1426, Springer, 1989, p. 407{441. [E4] Emery (M.) | Stochastic calculus in manifolds. | Springer, 1989. [K] Kendall (W.S.) | Convex geometry and noncon uent ?-martingales II: Wellposedness and ?-martingale convergence, Stochastics, t. 38, 1992, p. 135{147. [Pi] Picard (J.) | Convergence in probability for perturbed stochastic integral equations, Probability Theory and Related Fields, t. 81, 1989, p. 383{451. [Pr] Protter (P.) | Stochastic Integration and Dierential Equations. A New Approach. | Springer, 1990. [Y-I] Yano (K.), Ishihara (S.) | Tangent and Cotangent Bundles. | Marcel Dekker, Inc., New York, 1973.