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Stochastic ows ofdieomorphisms on manifolds driven
by innite-dimensional semimartingales with jumps
David Applebaum
∗, Fuchang Tang
Department of Mathematics, Statistics & Operational Research, Nottingham Trent University, Burton Street, Nottingham NG1 4BU, UK
Received 7 March 2000; received in revised form 3 October 2000; accepted 31 October 2000
Abstract
We employ the interlacing construction to show that the solutions ofstochastic dierential equations on manifolds which are written in Marcus canonical form and driven by innite-dimensional semimartingales with jumps give rise to stochastic ows ofdieomorphisms. c2001 Elsevier Science B.V. All rights reserved.
Keywords:Semimartingale; Manifold; Stochastic ow; Interlacing construction
1. Introduction
A number ofauthors have studied SDEs with jumps on manifolds – see in particular Cohen (1996), Applebaum (1995), Fujiwara (1991) and Kurtz et al. (1995) but apart from Fujiwara (1991), they were all restricted to nite-dimensional noise while in Fujiwara (1991) the manifold was taken to be compact and the ow was required to be ofLevy type, i.e. having independent increments. In this paper our main contributions are as follows:
1. We work on a quite general class ofnite-dimensional smooth manifolds.
2. We study general stochastic ows ofdieomorphisms arising as solutions ofSDEs driven by a wide class ofvector-eld-valued semimartingales with jumps.
3. We establish the solution ofthe ow as the almost sure limit ofan interlacing sequence so that each term ofthe sequence consists ofstochastic ows with con-tinuous sample paths broken up by a nite number ofrandom jumps in each nite time interval.
We remark that interlacing ofstochastic ows on compact manifolds was also consid-ered by Fujiwara (1991) but only at the level oftheL2-limit.
One ofthe main aims ofthis paper is to give a complete and systematic exposition of the technique for constructing solutions of SDEs with jumps on manifolds by means
∗Corresponding author.
ofembedding into Euclidean space and utilizing the tubular neighborhood theorem. This specic approach used herein was rst carried out by Elworthy for ows with continuous sample paths in Elworthy (1988) and was extended to a class ofLevy ows in Applebaum (1995) however the argument given there was incomplete and has been extended herein to a far more general set-up.
A full list of notations for semi-norms and function spaces used in this paper can be found in the appendix.
2. Preliminaries
LetM be a connected, paracompact C∞-manifold of dimensiond and denote as
Ck(M) =Ck(M;R); k= 0;1; : : : ;+∞
the collection ofall k-times continuously dierentiable real-valued functions on M. Suppose 06m6+∞and dene Xm(M) to be the real linear space ofallCm-vector elds (rst-order linear partial dierential operators with Cm-coecients) on M. By Whitney’s embedding theorem (see Hirsh, 1976), there exists a smooth embedding ‘
of M in R2d+1. We will always nd it convenient to regard ‘ as a dieomorphism betweenM and its range‘(M) which is a closed subset of R2d+1.
For f∈C∞(M) and X∈Xm(M), we dene a smooth function f and vector eld
X on ‘(M) as follows:
f(x) =f◦‘−1(x); X(x) =D‘(‘−1(x))(X(‘−1(x))) =Xi(x) @
@xi; (2.1) where x= (x1; x2; : : : ; x2d+1)∈‘(M) is the standard coordinate of R2d+1, and if p=
‘−1(x)∈M, D‘(p) is the dierential which is a linear map from the tangent space at
p to the tangent space at x. We call f and X the push forwards of f and X by ‘, respectively (see Abraham et al., 1988, pp. 265 –266). One can see that
X(x) f(x) =X(‘−1(x))f(‘−1(x)); ∀x∈‘(M): (2.2)
Suppose in general that we are given an embedding of M into an arbitrary Euclidean space. We introduce the topology of uniform convergence forX0(M) as follows:
%(X; Y) =
∞
i= 1
1 2i
supx∈(Gi) 2d+1
j= 1 |Xj(x)−Yj(x)|
1 + supx∈(Gi) 2d+1
j= 1 |Xj(x)−Yj(x)|
;
where X; Y∈X0(M) are represented as in (2.1) and (Gi, 16i¡+∞) are compact
subsets of M with Gi⊆Gi+1 and i Gi=M. X0(M) is a complete separable metric space under %.
Remark 2.1. It is not dicult to verify that the topology induced by% is independent ofthe choice ofembedding . So we will write % simply as %.
all u∈R
(d=du)(v)(u; p) =v((v)(u; p));
(v)(0; p) =p: (2.3)
In the sequel, we will nd it convenient to use the notation (p) =(1; p). is often called theexponential map.
Denition 2.2(cf. Kunita, 1990, pp. 186). Suppose ={t; t¿0} is an Xm(M )-valued process onM. Iffor anyf∈C∞(M),tf is aCm(M)-valued local martingale, then is called a Xm(M)-valued local martingale. We say that iscontinuous iftf is continuous for each f∈C∞(M) with respect tot.
We can similarly dene an Xm(M)-valued semimartingale.
Suppose we are given an X0(M)-valued semimartingale which has the following decomposition:
Xt=Ht+Jt+ t+
s
U
vN˜M(drdv) + t+
s
Uc
vNM(drdv); (2.4)
whereU is a Borel subset ofX0(M), and
1. Ht is a continuous X0(M)-valued stochastic process and Jt is a continuousX0(M )-valued local martingale. Furthermore, for any f; g∈C∞(M), p1; p2∈M and
06s6t, the following holds (cf. Fujiwara, 1991; Kunita, 1990, pp. 186 –187):
Ht(s)f(p1) =
t s
b(f; r)(p1) dAr;
Jf(p1); Jg(p2)(s; t]=
t s
a(f; g; r)(p1; p2) dAr; (2.5)
where At is a continuous increasing process which is independent of f and g. 2. NM(drdv) is a random measure on R+×(X0(M)\{0}) with a predictable
compen-sator (intensity measure) dArr; M(dv) and ˜
NM(drdv) =NM(drdv)−dArr; M(dv)
is an X0(M)-valued local martingale.
From (2.5), one can see immediately that for any t¿s¿0 and p1; p2∈M, the maps
b(·; t)(p1) :C∞(M)→Rare linear
and
a(·;·; t)(p1; p2) :C∞(M)×C∞(M)→Rare bi-linear:
Condition 2.3. Throughout this paper, we assume for any t¿s¿0 and for every embedding ‘ of M into a Euclidean space that
1. b(·; t) :C∞(M)→Cm(M).
3. The map t→At is continuous. 4. The measure t; M satises
(a) suppt; M⊆Xm+1(M).
(b) every member of suppt; M is a complete vector eld, (c) t; M(Uc)6ft,
(d) supv∈U{vm+:‘()}6Q()¡+∞, (e)
U
(v20+1 :‘()∨ ∇v·vLip :‘())t; M(dv)6Kt(); m= 0;
U
(v2m+:‘()∨ ∇v·vm+:‘())t; M(dv)6Kt(); m¿0;
where U is a Borel subset of X0(M), Kt() = (Kt(); t¿s¿0) and f= (ft;
t¿s¿0)are nonnegative predictable processes with
t+ s
Kr() dAr¡+∞; t+
s
frdAr¡+∞
a.s. for any compact subset ⊂M and Q:M→R+.
These conditions are natural generalizations ofthose given by Fujiwara and Kunita (see below and Fujiwara and Kunita, 1999) in the Euclidean space case. Note in particular that they ensure that we have a stochastic ow ofdieomorphisms whenever all driving vector elds have bounded derivatives to all orders in every co-ordinate system.
Some examples ofsemimartingales satisfying these conditions are given at the end of this paper. For convenience, we will below often identify a vector eldX∈Xm(R2d+1) (Y∈Xm(‘(M)), respectively) with its coordinate representation in R2d+1, thus we will
have X∈Cm(R2d+1;R2d+1) (Y∈Cm(‘(M);R2d+1), respectively).
For the rest ofthis paper, we will x our embedding to be ‘ given by Whitney’s embedding theorem as above.
3. Stochastic dierential equations on manifolds
Consider the canonical SDE driven byX= (Xt; t¿s¿0) on the manifoldM which is given as follows, where denotes the exponential map dened in (2.3)
(s; t)(p) =p+ t+
s
X( (s; r−)(p);♦dr);
where p∈M and t¿s¿0. The ♦ notation means that our stochastic integral is in Marcus canonical form which generalises the Stratonovitch integral to the extent that its form is invariant under local co-ordinate changes and so it is a natural geometric object. Indeed its coordinate form is written as
(s; t)(p) = p+ t+
s
H( (s; r−)(p);dr) + t+
s
+ t+
s
Uc
[(v)( (s; r−)(p))− (s; r−)(p)]NM(drdv)
+ t+
s
U
[(v)( (s; r−)(p))− (s; r−)(p)] ˜NM(drdv)
+ t+
s
U
[(v)( (s; r−)(p))− (s; r−)(p)
−v( (s; r−)(p))]r; M(dv) dAr: (3.6)
Using Itˆo’s formula in (3.6), we have for anyf∈C∞(M) :
f( (s; t)(p)) =f(p) + t+
s
Hf( (s; r−)(p);dr) + t+
s
Jf( (s; r−)(p);◦dr)
+ t+
s
Uc
[f((v)( (s; r−)(p)))−f( (s; r−)(p))]NM(drdv)
+ t+
s
U
[f((v)( (s; r−)(p)))−f( (s; r−)(p))] ˜NM(drdv)
+ t+
s
U
[f((v)( (s; r−)(p)))−f( (s; r−)(p))
−vf( (s; r−)(p))]r; M(dv) dAr (3.7)
from which we easily deduce the required invariance under co-ordinate changes. For further discussion see the original article by Marcus (1981) and the account in relation to stochastic ows given in Applebaum and Kunita (1993).
Theorem 3.1. There exists a unique maximal solution to(3:7)which has a modica-tion which is a stochastic ow of local Cm-dieomorphisms.
Our main aim in this paper is to give a complete proofofthis result.
3.1. Stochastic ows in Euclidean space
Here we recall some results from Euclidean space (see Fujiwara and Kunita, 1999; Applebaum and Tang, 2000; Tang, 2000). LetX be aC(Rd;Rd)-valued semimartingale with characteristics (a; b; ;A) (associated with U).
Condition 3.2(cf. Fujiwara and Kunita, 1999; Kunita, 1996). Letmbe a non-negative integer, q¿0 and ∈(0;1], suppose for anyt¿s¿0
1.
=
1; m= 0or +∞: ; 0¡m¡+∞:
4. c(t)m+6Lt, where
ci(x; t) =1 2
d j= 1
@aij
@xj(x; y; t)
y=x
:
5. b(t)m+6Lt.
6. The measures t satises (a) (suppt)⊆(Cbm; ∩Cm+1). (b) t(Uc)6Lt.
(c) supv∈Uvm+6q. (d)
U
(v20+1∨ ∇v·vLip)t(dv)6Lt; m= 0;
U
(v2m+∨ ∇v·vm+)t(dv)6Lt; m¿0:
where Lt is nonnegative predictable processes with
t+
s
LrdAr¡+∞a:s:
Theorem 3.3. Let X be a C(Rd;Rd)-valued semimartingale with characteristics
sat-isfying condition (3:2); then there exists a unique global solution (s; t) to the fol-lowing equation. Furthermore; has a modication which is a stochastic ow of
Cm-dieomorphisms.
(s; t)(x) =x+ t+
s
X( (s; r−)(x);♦dr):
Proof. See Fujiwara and Kunita (1999) or Corollary 5:3:9 in Tang (2000, p. 110).
In Applebaum and Tang (2000) and Tang (2000) the following interlacing construc-tion for was given.
First we consider the equation without big jumps
(s; t)(p) =p+ t+
s
H((s; r−)(p);dr) + t+
s
J((s; r−)(p); ◦dr)
+ t+
s
U
[(v)((s; r−)(p))−(s; r−)(p)] ˜NRd(drdv)
+ t+
s
U
[(v)((s; r−)(p))−(s; r−)(p)
For any L∈N\{1}, it is shown that there exists a monotonic decreasing sequence
(n; n∈N) with 1= 1=2 and limn→ ∞n= 0 with
Un={v∈U:v∈Cbm; ∩C
m+1 andv
m+¿n} (3.9) such that
Un+1\Un
(v2m+∨ ∇v·vm+)t(dv)6 1
nL
U
(v2m+∨ ∇v·vm+)t(dv)
(3.10)
holds for any n∈N.
Consider the following approximating equation (cf. Applebaum, 2000).
n(s; t)(x) = x+ t
s
J(n(s; r)(x);dr) + t
s
{c(n(s; r)(x); r)
+b(n(s; r)(x); r)−&n(n(s; r)(x))}dA(r); (3.11)
where
&n(x; t) =
Un
v(x)t(dv): (3.12)
then it follows from Kunita (1990) that eachn is a stochastic ow ofdieomorphism of Rd.
We now dene
in(x; t) =
t+
s
Un
vi(x)N(drdv)
andn(t) = (1
n(x; t); 2n(x; t); : : : ; dn(x; t)).
It is then not dicult to verify that n(t) = (n(x; t); t¿0) is a Cbm; -valued point process and has nitely many jump times on each interval (0; t].
Denote the jump times ofn(x; t) by (1n; 2n; : : :). Following (cf. Applebaum, 2000), we construct an interlacing sequence as follows:
n(s; t)(x) =n(s; t)(x); 06s6t¡1n;
n(s; 1n)(x) =(n(1n))◦n(s; 1n−)(x); t=1n;
n(s; t)(x) =n(1n; t)◦n(s; 1n)(x); 1n¡t¡2n;
n(s; 2n)(x) =(n(2n))◦n(s; 2n−)(x); t=2n ..
. (3.13)
and so on inductively, where n is the solution to (3.11). We then nd that for eachx∈Rd,
lim
n→ ∞n(s; t)(x) =(s; t)(x) a:s:
and the convergence is uniform on compacta.
The stochastic ow is then obtained from by a single interlacing with big jumps.
3.2. Stochastic ows on manifolds
We begin, as above by discussing the SDE on the manifold without big jumps and then apply the interlacing technique to obtain its solution, so our aim is to solve
(s; t)(p) =p+ t+
s
H((s; r−)(p);dr) + t+
s
J((s; r−)(p); ◦dr)
+ t+
s
U
[(v)((s; r−)(p))−(s; r−)(p)] ˜NM(drdv)
+ t+
s
U
[(v)((s; r−)(p))−(s; r−)(p)
−v((s; r−)(p))]r; M(dv) dAr: (3.14)
Theorem 3.4. There exists a unique maximal solution to (3:14) which has a modi-cation which is a stochastic ow of local Cm-dieomorphisms.
Proof. Fix 06m¡+∞ in the sequel.
Step 1: Euclidean version of (3.14). Let Ht; Jt and v be the push forwards of Ht,
Jt and v, respectively. Moreover, deneU‘(M)={v:v∈U}.
For clarity, we denote the random measure on X0(‘(M)) by N‘(M)(drd v), its
pre-dictable compensator by dArr; ‘(M)(d v) and
˜
N‘(M)(drd v) =N‘(M)(drd v)−dArr; ‘(M)(d v);
where for any A∈B(X0(‘(M))), we have
N‘(M)(t; A) =NM(t; D‘−1(A)):
Thus, we obtain an SDE on‘(M)⊂R2d+1 as follows:
(s; t)(x) =x+ t+
s
H( (s; r−)(x);dr) + t+
s
J( (s; r−)(x);◦dr)
+ t+
s
U‘(M)
[( v)( (s; r−)(x))−(s; r−)(x)] ˜N‘(M)(drd v)
+ t+
s
U‘(M)
[( v)( (s; r−)(x))−(s; r−)(x)
−v( (s; r−)(x))]r; ‘(M)(d v) dAr: (3.15)
where (s; t)(x) =‘((s; t)(p)) with p∈M and x=‘(p).
Step 2: Extend (3.15) to the whole of R2d+1. Let G∈N and B(0; G) be an open ball inR2d+1 whose centre is the origin and radius isG. By the tubular neighbourhood theorem (see Hirsh (1976)), there exists a positive smooth function
and a smooth projection (also cf. Applebaum, 1995)
:I
‘(M)→‘(M); |‘(M)= Identity map;
d(y; ‘(M)) =|y−(y)| when y∈I
‘(M); (3.16)
whered is the usual Euclidean metric in R2d+1 and
and
UG={v˜G: v∈U‘(M)}; UGc={v˜ G
: v∈U‘c(M)}
and the random measure on KG=UG ∪UGc (⊂X0(R2d+1)) is NG(drd ˜vG), its pre-dictable compensator is dArr; G(d ˜vG) and (cf. Step 1)
˜
NG(drd ˜vG) =NG(drd ˜vG)−dArr; G(d ˜vG)
which satisfy for any A∈B(X0(R2d+1))∩KG
NG(t; A) =N‘(M)(t; B) whereB={v∈B(X0(‘(M))) : ˜vG∈A}:
One can see that KG is a Borel subset of X0(R2d+1) by (3.18). Applying (2.2) and (3.18), we next verify that the characteristics of ˜XGt satisfy condition (3:2).
First, note that forf; g∈C∞(R) and x; y∈R2d+1, we have
˜
HGt(s) ˜fG(x) = t
s ˜
bG( ˜fG; r)(x) dAr;
J˜Gf˜G(x);J˜Gg˜G
(y)(s; t]=
t s
˜
aG( ˜fG;g˜G; r)(x; y) dAr;
where
˜
bG( ˜fG; t)(x) =>G(x)G(d(x; ‘(M))2)b(f; t)(‘−1◦(x));
˜
aG( ˜fG;g˜G; t)(x; y) =>G(x)>G(y)G(d(x; ‘(M))2)G(d(y; ‘(M))2)
×a(f; g; t)(‘−1◦(x); ‘−1◦(y)): (3.20)
Secondly, since
˜
vG(x) =>G(x)G(d(x; ‘(M))2) v((x)) (3.21)
which is 0 whenever |x|¿G+ 1, one can see the measure (t; G; t¿s¿0) satises the following conditions:
1. Applying condition 2.3(4a) and (3.21), we have
suppt; G⊆(Cbm; ∩C m+1):
2. Applying condition 2.3(4d), the denition of · m+ and ˜vG(x), we obtain
sup
˜
vG∈U G
{v˜Gm+} 6C˜1(m; G) sup
v∈U‘(M)
{vm+:LG+1}
6C˜1(m; G)Q(‘−1(LG+1)): (3.22)
3. Similarly as immediately above, applying condition 2.3(4e), one can see that for
m¿0
UG
{v˜G2m+∨ ∇v˜G·v˜Gm+}t; G(d ˜vG)
6C˜2(m; G)
U‘(M)
= ˜C2(m; G)
U
{v2m+:LG+1∨ ∇v·vm+:LG+1}t; M(dv)
6C˜2(m; G)Kt(‘−1(LG+1))
and
UG
{v˜G20+1∨ ∇v˜G·v˜GLip}t; G(d ˜vG)6C˜2(m; G)Kt(‘−1(LG+1))
holds as above form= 0, where ˜Cj(m; G) are positive nite constants for j= 1;2. From (3.20) and 1–3, one can see that for anyG∈N, there exists a unique global
solution to (3.19). Furthermore, it has a modication which is a stochastic ow of
Cm-dieomorphisms by Theorem 3.3. From now on, we identify the solution with this very modication.
Now we are going to investigate the behaviour ofthe solution G. We expect (cf. Elworthy (1988); Applebaum (1995)) that the solution remains in ‘(M) until some explosion time whenever the initial point is in ‘(M).
Step 3: Maximal solution to (3.15). Dene
Gs(x; w) = inf{t¿0 : (|G(s; t)(x; w)| ∨ |G(s; t−)(x; w)|)¿G} (3.23)
and set
s(x; w) = sup G
Gs(x; w);
(s; t)(x; w) = G(s; t)(x; w) when s6t¡G(s; t)(x; w): (3.24)
Then by the arbitrariness of G and the argument as in Theorem 38 in Protter (1990, pp. 247–249), we have that with s6t¡s(x) is the unique maximal solution to (3.15).
Step 4: Flow properties. For each 06s6t andG∈N, dene as in Kunita (1990,
p. 178):
DGs; t(w) = {x∈Rd:Gs(x; w)¿t};
Ds; t(w) = {x∈Rd:s(x; w)¿t}:
One can see that bothDGs; t(w) and Ds; t(w) are open sets and
Ds; t(w) =
+∞
G= 0
DGs; t(w):
Since fort¡G s(x):
(s; t) = G(s; t) :DG
s; t(w)→Rd
is a stochastic ow oflocal Cm-dieomorphisms by Theorem 3.3, then we have that
is a stochastic ow oflocal Cm-dieomorphisms on [s; s(x)).
In Lemma 3.6, we will show that for almost all w∈, the following holds:
for any x∈LG and s6t¡G
s(x; w). Therefore,
(s; t)(x)∈‘(M) f or s6t¡s(x) (3.26)
when x∈‘(M) by (3.24). We will then be able to “pull back” (s; t)(x) onto the manifold M itselfto get our required solution.
Assume for now that (3.26) holds.
Step 5: Maximal solution to (3.14). Apply Itˆo’s formula (2.2) and (3.26), then argue as in Lemma 3.5 in Fujiwara (1991), so dene for any p∈M,
(s; t)(p) =‘−1◦(s; t)(‘(p)); s6t6s(‘(p))
then (s; t)(p) is the unique solution of(3.14) which has a modication which is a ow of Cm-dieomorphisms.
Step 6: m= +∞. From Steps 1– 4, we obtain that Theorem 3.4 holds for any 06m¡+∞. By the arbitrariness of m, one can see that it also holds for m= +∞.
To show (3.25), we rst need the following technical lemma.
Lemma 3.5. Fix G∈N and let v˜G∈UG as in (3:18). Suppose we are given the fol-lowing equation inR2d+1 (cf. (2:3)) :
(d=du)( ˜vG)(u; x) = ˜vG(( ˜vG)(u; x)); ( ˜vG)(0; x) =x
then for eachx∈LG; the following holds:
( ˜vG)(1; x)∈‘(M):
Proof. LetUG; >G andG be as in Theorem 3.4. Now we follow Applebaum (1995, pp. 173, 174). Dene
¿
sup x∈LG
sup
06u61
sup
˜
vG∈U G
|( ˜vG)(u; x)|
∨G
and
(x) =(d(x; ‘(M))2)d(x; ‘(M))2: (3.27)
Applying (3.22), we have¡+∞. Hence, W in (3.17) is a positive constant. For 06u61 and ˜vG∈UG, dene IG
u :C∞(R2d+1;R)→Cm(R2d+1;R) by
IuG(f)(·) =f◦( ˜vG)(u;·)
then we have that dIuG(f)
du =I
G u ( ˜v
G
f):
Take f= , then by (3.16) and (3.18), we have for any x∈LG (also cf. Elworthy, 1988, p. 8),
IuG( ˜vG )(x) = ˜vG(( ˜vG)(u; x)) (( ˜vG)(u; x)) = 0
Hence,
which contradicts (3.28), therefore the required result holds.
In the following lemma, we use the interlacing technique to establish (3.25).
Lemma 3.6 (cf. Fujiwara, 1991). For almost all w∈; we have
has a global solution. Furthermore, for almost all w∈
Gn(s; t)(x)∈LG (3.30)
holds for any x∈LG and t¡G
s; n(x; w) where
Gs; n(x; w) = inf{t¿s:|nG(s; t)(x; w)|¿G}: (3.31)
Secondly, let G be the interlacing with jumps in UG; n from
G
n in (3.29) dened as follows:
Gn(s; t)(x) =Gn(s; t)(x); 06s6t¡1n;
Gn(s; 1n)(x) =(n(1n))◦ G
n(s; 1n−)(x); t=1n;
Gn(s; t)(x) =nG(1n; t)◦G
n(s; 1n)(x); 1n¡t¡2n;
Gn(s; 2n)(x) =(n(2n))◦ G
n(s; 2n−)(x); t=2n; ..
.
(3.32)
By Theorems 4:3:1 in Tang (2000, p. 63) (also cf. Fujiwara, 1991), we nd that for almost allw∈,
lim n→ ∞
Gn(s; t)(x) = G(s; t)(x) (3.33)
uniformly on compact sets containingx∈R2d+1 and t∈[s;+∞).
Thirdly, xw∈which satises (3.33),x∈LG andt0¡Gs(x; w) in the sequel. By (3.23), we have
(|G(s; t)(x; w)| ∨ |G(s; t−)(x; w)|)¡G
for any 06s6t6t0.
Hence, there exists′¿0 such that
(|G(s; t)(x; w)| ∨ |G(s; t−)(x; w)|)6G−′
holds for any 06s6t6t0.
Now apply (3.33), we have G0(′)∈N such that
(|Gn(s; t)(x; w)| ∨ |Gn(s; t−)(x; w)|)6G−
′
2 (3.34)
holds for any 06s6t6t0 and n¿G0(′).
Finally, xn¿G0(′). We will show
G
n(s; t0)(x)∈‘(M).
When 06s6t0¡1n: By (3.32) and (3.34), we have
|Gn(s; t)(x)|=|Gn(s; t)(x)|6G−
′
2
for all s6t6t0. Hence t0¡Gs; n(x), i.e.
When t0=1n: Similarly as above, we have
|nG(s; t)(x)|6G−
′
2 for any 06s6t6t0, i.e.
Gn(s; t0)(x)∈LG⊂‘(M):
Note that
Gn(s; 1n)(x) =(n(1n))◦Gn(s; 1n)(x)
then by Lemma 3.5, we have Gn(s; t0)(x)∈‘(M).
Other values oft0: Similarly, we can show
G
n(s; t0)(x)∈‘(M).
Now apply (3.33). As‘(M) is a closed subset ofR2d+1, we have G(s; t0)(x)∈‘(M).
One can see that
G(s; t0)(x)∈LG since|
G
(s; t0)(x)|¡G:
The result now follows immediately from the arbitrariness of t0.
Proof of Theorem 3.1 (cf. Applebaum, 1995, p. 174).
This is eectively applying Theorem 3.4 and the interlacing technique.
Remark 3.7. When the manifold M is compact, we can take G big enough to satisfy
LG=‘(M), then G
s; n= +∞ in (3.31) by the proofin Lemma 4:8:2 in Kunita (1990, p. 190), hence Gs(x) = +∞in (3.23).
Note. By the above construction, it is clear that when the starting point x is on the manifold then each term in the interlacing sequence of (3.33) also lies on the manifold and so we have the required interlacing construction for manifold-valued stochastic ows.
Example. Let V be a k-dimensional Levy process in a Euclidean space with Levy–Itˆo decomposition. Dene
Xt=VtjYj for t¿0;
where Y1; Y2; : : : ; Yl and all their nite linear combinations are complete vector elds inXm(M) (cf. Applebaum, 1995, p. 172) and each Y
j has bounded derivatives to all orders in every co-ordinate system. ThenX satises Condition 2.3. Moreover, one can see that (3.6) can be written as follows :
(s; t)(p) = p+ t+
s
H( (s; r−)(p); r) dr+ t+
s
J( (s; r−)(p);◦dr)
+ t+
s
|z|¿1
[(zjYj)( (s; r−)(p))− (s; r−)(p)]N(drdz)
+ t+
s
|z|¡1
+ t+
s
|z|¡1
[(zjYj)( (s; r−)(p))− (s; r−)(p)
−zjYj( (s; r−)(p))](dz) dr; (3.35)
where
Ht(x) =tjYj(x); Jt(x) =kjBkt Yj(x):
The above can be easily extended to the case where the Levy process is replaced by an Rd-valued semimartingale ofsimilar type (see e.g. Ikeda and Watanabe, 1981, p. 64).
Acknowledgements
Both authors thank H. Kunita for helpful discussions during his visit to Nottingham in July 1999. F. Tang would like to thank Nottingham Trent University for nancial support from its Research Enhancement Fund during the period of his graduate studies. We would also like to thank the referee for several helpful comments.
Appendix. Notations and terminology
Cm=Cm(D;Rd) ={v:v:D→Rd ism-times continuously dierentiable};
˜
Cm = ˜Cm(D×D;Rd2)
={f:f:D×D→Rd2 ism-times continuously dierentiable};
whereD is a region in Rd and m is a non-negative integer. Dene
D≡Dx=
@||
(@x1)1
· · ·(@xd)d
as a dierential operator where x∈Rd and ; = (1; 2; : : : ; d) are multi-indexes of
non-negative integers with||=
i.
For v∈Cm; f∈C˜m, dene the following semi-norms on Cm and ˜Cm (see Kunita, 1990, pp. 72,73):
vLip; = sup x;y∈
x=y
|v(x)−v(y)|
|x−y| ; vLip= supx=y
|v(x)−v(y)| |x−y| ;
vm:= sup x∈
|v(x)|
1 +|x|+
16||¡(m+1)
sup x∈
|Dxv(x)|;
vm+:=vm:+ 1(m¡+∞)
||=m sup x;y∈
x=y
|D
xv(x)−Dxv(y)|
f∼m:= sup
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