Directory UMM :Data Elmu:jurnal:S:Stochastic Processes And Their Applications:Vol92.Issue2.2001:
Stochastic Processes and their Applications 92 (2001) 219–236
www.elsevier.com/locate/spa
Stochastic
ows of di eomorphisms on manifolds driven
by in nite-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 of stochastic di erential
equations on manifolds which are written in Marcus canonical form and driven by in nitec 2001
dimensional semimartingales with jumps give rise to stochastic
ows of di eomorphisms.
Elsevier Science B.V. All rights reserved.
Keywords: Semimartingale; Manifold; Stochastic
ow; Interlacing construction
1. Introduction
A number of authors 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
of Levy type, i.e. having independent increments. In this paper our main contributions
are as follows:
1. We work on a quite general class of nite-dimensional smooth manifolds.
2. We study general stochastic
ows of di eomorphisms arising as solutions of SDEs
driven by a wide class of vector- eld-valued semimartingales with jumps.
3. We establish the solution of the
ow as the almost sure limit of an interlacing
sequence so that each term of the sequence consists of stochastic
ows with continuous sample paths broken up by a nite number of random jumps in each nite
time interval.
We remark that interlacing of stochastic
ows on compact manifolds was also considered by Fujiwara (1991) but only at the level of the L2 -limit.
One of the main aims of this 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.
E-mail addresses: dba@maths.ntu.ac.uk (D. Applebaum), fuchang.tang@yahoo.ca (F. Tang).
c 2001 Elsevier Science B.V. All rights reserved.
0304-4149/01/$ - see front matter
PII: S 0 3 0 4 - 4 1 4 9 ( 0 0 ) 0 0 0 9 1 - 0
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D. Applebaum, F. Tang / Stochastic Processes and their Applications 92 (2001) 219–236
of embedding into Euclidean space and utilizing the tubular neighborhood theorem.
This speci c approach used herein was rst carried out by Elworthy for
ows with
continuous sample paths in Elworthy (1988) and was extended to a class of Levy
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
Let M be a connected, paracompact C ∞ -manifold of dimension d and denote as
C k (M ) = C k (M; R);
k = 0; 1; : : : ; +∞
the collection of all k-times continuously di erentiable real-valued functions on M .
Suppose 0 6 m 6 + ∞ and de ne Xm (M ) to be the real linear space of all C m -vector
elds ( rst-order linear partial di erential operators with C m -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 di eomorphism
between M and its range ‘(M ) which is a closed subset of R2d+1 .
For f ∈ C ∞ (M ) and X ∈ Xm (M ), we de ne a smooth function f and vector eld
X on ‘(M ) as follows:
= f ◦ ‘−1 (x);
f(x)
@
X (x) = D‘(‘−1 (x))(X (‘−1 (x))) = X i (x) i ;
@x
(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 di erential 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 (‘−1 (x))f(‘−1 (x));
X (x)f(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 for X0 (M ) as follows:
2d+1
∞
j
j
1 supx ∈ (Gi ) j = 1 |X (x) − Y (x)|
;
% (X; Y ) =
j
j
2i 1 + supx ∈ (G ) 2d+1
j = 1 |X (x) − Y (x)|
i=1
i
where X; Y ∈ X0 (M ) are represented as in (2.1) and (Gi , 1 6 i¡ + ∞) 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
of the choice of embedding . So we will write % simply as %.
We say that v ∈ Xm (M ) is (deterministically) complete if the partial di erential equation given below has a unique solution for all initial conditions p ∈ M and
D. Applebaum, F. Tang / Stochastic Processes and their Applications 92 (2001) 219–236
221
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 the exponential map.
De nition 2.2 (cf. Kunita, 1990, pp. 186). Suppose = {t ; t ¿ 0} is an Xm (M )valued process on M . If for any f ∈ C ∞ (M ), t f is a C m (M )-valued local martingale,
then is called a Xm (M )-valued local martingale. We say that is continuous if t f
is continuous for each f ∈ C ∞ (M ) with respect to t.
We can similarly de ne an Xm (M )-valued semimartingale.
Suppose we are given an X0 (M )-valued semimartingale which has the following
decomposition:
t+
t+
vNM (dr dv);
(2.4)
vN˜ M (dr dv) +
Xt = Ht + Jt +
s
s
U
Uc
0
where U is a Borel subset of X (M ), and
1. Ht is a continuous X0 (M )-valued stochastic process and Jt is a continuous X0 (M )valued local martingale. Furthermore, for any f; g ∈ C ∞ (M ), p1 ; p2 ∈ M and
0 6 s 6 t, the following holds (cf. Fujiwara, 1991; Kunita, 1990, pp. 186 –187):
t
b(f; r)(p1 ) dAr ;
Ht (s)f(p1 ) =
s
Jf(p1 ); Jg(p2 ) (s; t] =
t
a(f; g; r)(p1 ; p2 ) dAr ;
(2.5)
s
where At is a continuous increasing process which is independent of f and g.
2. NM (dr dv) is a random measure on R+ ×(X0 (M )\{0}) with a predictable compensator (intensity measure) dAr r; M (dv) and
N˜ M (dr dv) = NM (dr dv) − dAr r; 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 ) → R are linear
and
a(· ; · ; t)(p1 ; p2 ) : C ∞ (M )×C ∞ (M ) → R are 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 ) → C m (M ).
2. a(· ; · ; t) : C ∞ (M )×C ∞ (M ) → C m (M ×M ).
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D. Applebaum, F. Tang / Stochastic Processes and their Applications 92 (2001) 219–236
3. The map t → At is continuous.
4. The measure t; M satis es
(a) supp t; M ⊆ Xm+1 (M ).
(b) every member of supp t; M is a complete vector eld,
(c) t; M (U c ) 6 ft ,
(d) supv ∈ U {v
m+ : ‘() } 6 Q()¡ + ∞,
(e)
2
(v
0+1 : ‘() ∨ ∇v · v
Lip : ‘() )t; M (dv) 6 Kt (); m = 0;
U
U
2
(v
m+ : ‘() ∨ ∇v · v
m+ : ‘() )t; M (dv) 6 Kt ();
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+
t+
fr dAr ¡ + ∞
Kr () dAr ¡ + ∞;
s
s
a.s. for any compact subset ⊂ M and Q : M → R+ .
These conditions are natural generalizations of those 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 of di eomorphisms whenever
all driving vector elds have bounded derivatives to all orders in every co-ordinate
system.
Some examples of semimartingales satisfying these conditions are given at the end of
this paper. For convenience, we will below often identify a vector eld X ∈ Xm (R2d+1 )
(Y ∈ Xm (‘(M )), respectively) with its coordinate representation in R2d+1 , thus we will
have X ∈ C m (R2d+1 ; R2d+1 ) (Y ∈ C m (‘(M ); R2d+1 ), respectively).
For the rest of this paper, we will x our embedding to be ‘ given by Whitney’s
embedding theorem as above.
3. Stochastic di erential equations on manifolds
Consider the canonical SDE driven by X = (Xt ; t ¿ s ¿ 0) on the manifold M which
is given as follows, where denotes the exponential map de ned in (2.3)
t+
(s; t)(p) = p +
X ( (s; r−)(p); ♦dr);
s
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
t+
t+
(s; t)(p) = p +
J ( (s; r−)(p); ◦ dr)
H ( (s; r−)(p); dr) +
s
s
D. Applebaum, F. Tang / Stochastic Processes and their Applications 92 (2001) 219–236
+
t+
t+
t+
s
+
[(v)( (s; r−)(p)) − (s; r−)(p)]NM (dr dv)
[(v)( (s; r−)(p)) − (s; r−)(p)]N˜ M (dr dv)
[(v)( (s; r−)(p)) − (s; r−)(p)
Uc
s
+
223
U
s
U
− v( (s; r−)(p))]r; M (dv) dAr :
(3.6)
∞
Using Itˆo’s formula in (3.6), we have for any f ∈ C (M ) :
t+
f( (s; t)(p)) = f(p) +
Hf( (s; r−)(p); dr) +
+
t+
t+
t+
s
+
s
[f((v)( (s; r−)(p))) − f( (s; r−)(p))]NM (dr dv)
[f((v)( (s; r−)(p))) − f( (s; r−)(p))]N˜ M (dr dv)
[f((v)( (s; r−)(p))) − f( (s; r−)(p))
Uc
s
+
Jf( (s; r−)(p); ◦ dr)
s
s
t+
U
U
− 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 modi cation which is a stochastic
ow of local C m -di eomorphisms.
Our main aim in this paper is to give a complete proof of this 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). Let X be a C(Rd ; Rd )-valued semimartingale
with characteristics (a; b; ; A) (associated with U ).
Condition 3.2 (cf. Fujiwara and Kunita, 1999; Kunita, 1996). Let m be a non-negative
integer, q¿0 and ∈ (0; 1], suppose for any t ¿ s ¿ 0
1.
=
1;
;
m = 0 or + ∞:
0¡m¡ + ∞:
2. The map t → At is continuous.
∼
3. a(t)(m∨2)+ 6 Lt .
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D. Applebaum, F. Tang / Stochastic Processes and their Applications 92 (2001) 219–236
4. c(t)m+ 6 Lt , where
d
ij
@a
1
i
c (x; t) =
(x; y; t)
j
2
@x
j=1
:
y=x
5. b(t)m+ 6 Lt .
6. The measures t satis es
(a) (supp t ) ⊆(Cbm; ∩ C m+1 ).
(b) t (U c ) 6 Lt .
(c) supv ∈ U vm+ 6 q.
(d)
2
U
(v0+1 ∨ ∇v · vLip )t (dv) 6 Lt ;
m = 0;
2
U
(vm+ ∨ ∇v · vm+ )t (dv) 6 Lt ;
m¿0:
where Lt is nonnegative predictable processes with
t+
Lr dAr ¡ + ∞ a:s:
s
Theorem 3.3. Let X be a C(Rd ; Rd )-valued semimartingale with characteristics satisfying condition (3:2); then there exists a unique global solution (s; t) to the following equation. Furthermore; has a modi cation which is a stochastic
ow of
C m -di eomorphisms.
(s; t)(x) = x +
t+
X ( (s; r−)(x); ♦dr):
s
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 construction for was given.
First we consider the equation without big jumps
t+
t+
[(v)((s; r−)(p)) − (s; r−)(p)]N˜ Rd (dr dv)
t+
[(v)((s; r−)(p)) − (s; r−)(p)
H ((s; r−)(p); dr) +
s
+
s
J ((s; r−)(p); ◦ dr)
s
s
+
t+
(s; t)(p) = p +
U
U
− v((s; r−)(p))]r; Rd (dv) dAr :
(3.8)
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D. Applebaum, F. Tang / Stochastic Processes and their Applications 92 (2001) 219–236
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 and vm+ ¿ n }
such that
Un+1 \Un
2
(vm+ ∨ ∇v · vm+ )t (dv) 6
1
nL
U
(3.9)
2
(vm+ ∨ ∇v · vm+ )t (dv)
(3.10)
holds for any n ∈ N.
Consider the following approximating equation (cf. Applebaum, 2000).
t
t
n (s; t)(x) = x +
{c( n (s; r)(x); r)
J ( n (s; r)(x); dr) +
s
s
+ b( n (s; r)(x); r) − &n ( n (s; r)(x))} dA(r);
(3.11)
v(x)t (dv):
(3.12)
where
&n (x; t) =
Un
then it follows from Kunita (1990) that each n is a stochastic
ow of di eomorphism
of Rd .
We now de ne
t+
i
i
n (x; t) =
v (x)N (dr dv)
s
Un
n (t) = (1n (x; t); 2n (x; t); : : : ; dn (x; t)).
and
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 of n (x; t) by ( 1n ; 2n ; : : :). Following (cf. Applebaum, 2000),
we construct an interlacing sequence as follows:
n (s; t)(x) = n (s; t)(x);
0 6 s 6 t¡ 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 each x ∈ Rd ,
lim n (s; t)(x) = (s; t)(x) a:s:
n→∞
and the convergence is uniform on compacta.
The stochastic
ow is then obtained from by a single interlacing with big
jumps.
We see below that a similar construction holds on manifolds.
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D. Applebaum, F. Tang / Stochastic Processes and their Applications 92 (2001) 219–236
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
t+
t+
J ((s; r−)(p); ◦ dr)
H ((s; r−)(p); dr) +
(s; t)(p) = p +
s
s
+
t+
t+
s
+
[(v)((s; r−)(p)) − (s; r−)(p)]N˜ M (dr dv)
U
s
[(v)((s; r−)(p)) − (s; r−)(p)
U
− 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 C m -di eomorphisms.
Proof. Fix 0 6 m¡ + ∞ in the sequel.
Step 1: Euclidean version of (3.14). Let H t ; Jt and v be the push forwards of Ht ,
Jt and v, respectively. Moreover, de ne U‘(M ) = {v : v ∈ U }.
For clarity, we denote the random measure on X0 (‘(M )) by N‘(M ) (dr d v),
its predictable compensator by dAr r; ‘(M ) (d v)
and
N˜ ‘(M ) (dr d v)
= N‘(M ) (dr d v)
− dAr r; ‘(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:
t+
t+
r−)(x); ◦ dr)
J((s;
H ((s; r−)(x); dr) +
(s; t)(x) = x +
s
s
+
t+
t+
s
+
s
r−)(x)) − (s;
r−)(x)]N˜ ‘(M ) (dr d v)
[(v)(
(s;
U‘(M )
r−)(x)) − (s;
r−)(x)
[(v)(
(s;
U‘(M )
r−)(x))]r; ‘(M ) (d v)
− v(
(s;
dAr :
(3.15)
t)(x) = ‘((s; t)(p)) with p ∈ M and x = ‘(p).
where (s;
Step 2: Extend (3.15) to the whole of R2d+1 . Let G ∈ N and B(0; G) be an open
ball in R2d+1 whose centre is the origin and radius is G. By the tubular neighbourhood
theorem (see Hirsh (1976)), there exists a positive smooth function
: ‘(M ) → R+
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D. Applebaum, F. Tang / Stochastic Processes and their Applications 92 (2001) 219–236
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)
2d+1
where d is the usual Euclidean metric in R
and
I‘(M
{y ∈ R2d+1 : |y − x|¡(x)}:
)=
x ∈ ‘(M )
De ne
WG =
LG = ‘(M ) ∩ B(0; G);
inf
x ∈ LG+1
(3.17)
(x):
Let >G ∈ C ∞ (R2d+1 ; R+ ) and G ∈ C ∞ (R+ ; R+ ) be given by (cf. Elworthy, 1988,
p. 8)
G 2
1
= 1; x ∈ B(0; G);
= 1; x¡ 2 (W ) ;
G
G
c
G 2
> (x) = 0; x ∈ B(0; G + 1) ;
(x) = 0; x ¿ (W ) ;
6 1; x ∈ B(0; G + 1)\B(0; G);
6 1; 12 (WG )2 6 x¡(WG )2 :
Next, we extend the vector elds H t ; Jt and v (see step 1), and smooth function
f smoothly from ‘(M ) to the whole of R2d+1 as follows (cf. Elworthy, 1988, p. 8;
Applebaum, 1995; Fujiwara, 1991):
G
f˜ (x) = >G (x) G (d(x; ‘(M ))2 )f((x));
G
H˜ t (x) = >G (x) G (d(x; ‘(M ))2 )H t ((x));
G
J˜t (x) = >G (x) G (d(x; ‘(M ))2 ) Jt ((x));
v˜ G (x) = >G (x) G (d(x; ‘(M ))2 ) v((x)):
(3.18)
Then, we obtain the following SDE on R2d+1 :
t+
G
G
G
˜
(s; t)(x) = x +
H ( (s; r−)(x); dr) +
+
t+
t+
+
G
G
G
G
[(v˜ G )( (s; r−)(x)) − (s; r−)(x)]N˜ G (dr d v˜ G )
UG
s
[(v˜ G )( (s; r−)(x)) − (s; r−)(x)
UG
G
− v˜ G ( (s; r−)(x))]r; G (d v˜ G ) dAr
which can be represented by
t+
G
G
G
(s; t)(x) = x +
X˜ ( (s; r−)(x); ♦dr);
s
where
G
G
G
X˜ t = H˜ t + J˜t +
G
G
J˜ ( (s; r−)(x); ◦ dr)
s
s
s
t+
s
t+
UG
v˜ G N˜ G (dr d v˜ G )
(3.19)
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D. Applebaum, F. Tang / Stochastic Processes and their Applications 92 (2001) 219–236
and
UG = {v˜ G : v ∈ U‘(M ) };
c
UGc = {v˜ G : v ∈ U‘(M
)}
and the random measure on KG = UG ∪ UGc (⊂ X0 (R2d+1 )) is NG (dr d v˜ G ), its predictable compensator is dAr r; G (d v˜ G ) and (cf. Step 1)
N˜ G (dr d v˜ G ) = NG (dr d v˜ G ) − dAr r; G (d v˜ G )
which satisfy for any A ∈ B(X0 (R2d+1 )) ∩ KG
NG (t; A) = N‘(M ) (t; B)
where B = {v ∈ B(X0 (‘(M ))) : v˜ G ∈ A}:
One can see that KG is a Borel subset of X0 (R2d+1 ) by (3.18). Applying (2.2) and
G
(3.18), we next verify that the characteristics of X˜ t satisfy condition (3:2).
First, note that for f; g ∈ C ∞ (R) and x; y ∈ R2d+1 , we have
t
G
G
G
G
˜
˜
b˜ (f˜ ; r)(x) dAr ;
H t (s)f (x) =
s
G G
G
J˜ f˜ (x); J˜ g˜ G (y) (s; t] =
t
G
a˜G (f˜ ; g˜ G; r)(x; y) dAr ;
s
where
G
G
b˜ (f˜ ; t)(x) = >G (x) G (d(x; ‘(M ))2 )b(f; t)(‘−1 ◦ (x));
G
a˜G (f˜ ; 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
v˜ G (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) satis es
the following conditions:
1. Applying condition 2.3(4a) and (3.21), we have
supp t; G ⊆(Cbm; ∩ C m+1 ):
2. Applying condition 2.3(4d), the de nition of · m+ and v˜ G (x), we obtain
m+ : LG+1 }
sup {v˜ G m+ } 6 C˜ 1 (m; G) sup {v
v ∈ U‘(M )
v˜ G ∈ UG
6 C˜ 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
2
{v˜ G m+ ∨ ∇v˜ G · v˜ G m+ }t; G (d v˜ G )
UG
6 C˜ 2 (m; G)
U‘(M )
2
{v
m+ : LG+1 ∨ ∇v · v
m+ : LG+1 }t; ‘(M ) (d v)
D. Applebaum, F. Tang / Stochastic Processes and their Applications 92 (2001) 219–236
= C˜ 2 (m; G)
U
229
2
{v
m+ : LG+1 ∨ ∇v · v
m+ : LG+1 }t; M (dv)
6 C˜ 2 (m; G) Kt (‘−1 (LG+1 ))
and
UG
2
{v˜ G 0+1 ∨ ∇v˜ G · v˜ G Lip }t; G (d v˜ G ) 6 C˜ 2 (m; G)Kt (‘−1 (LG+1 ))
holds as above for m = 0, where C˜ j (m; G) are positive nite constants for j = 1; 2.
From (3.20) and 1–3, one can see that for any G ∈ N, there exists a unique global
solution to (3.19). Furthermore, it has a modi cation which is a stochastic
ow of
C m -di eomorphisms by Theorem 3.3. From now on, we identify the solution with this
very modi cation.
G
Now we are going to investigate the behaviour of the solution . 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). De ne
G
G
sG (x; w) = inf {t¿0 : (| (s; t)(x; w)| ∨ | (s; t−)(x; w)|) ¿ G}
(3.23)
and set
s (x; w) = sup sG (x; w);
G
t)(x; w) = G (s; t)(x; w)
(s;
when s 6 t¡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 s 6 t¡s (x) is the unique maximal solution to
(3.15).
Step 4: Flow properties. For each 0 6 s 6 t and G ∈ N, de ne as in Kunita (1990,
p. 178):
d
G
DG
s; t (w) = {x ∈ R : s (x; w)¿t};
Ds; t (w) = {x ∈ Rd : s (x; w)¿t}:
One can see that both DG
s; t (w) and Ds; t (w) are open sets and
Ds; t (w) =
+∞
DG
s; t (w):
G=0
Since for
t¡sG (x):
d
t) = G (s; t) : DG
(s;
s; t (w) → R
is a stochastic
ow of local C m -di eomorphisms by Theorem 3.3, then we have that
is a stochastic
ow of local C m -di eomorphisms on [s; s (x)).
In Lemma 3.6, we will show that for almost all w ∈
, the following holds:
G
(s; t)(x) ∈ LG
(3.25)
230
D. Applebaum, F. Tang / Stochastic Processes and their Applications 92 (2001) 219–236
for any x ∈ LG and s 6 t¡sG (x; w). Therefore,
t)(x) ∈ ‘(M ) for s 6 t¡s (x)
(s;
(3.26)
t)(x) onto the
when x ∈ ‘(M ) by (3.24). We will then be able to “pull back” (s;
manifold M itself to 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 de ne for any p ∈ M ,
t)(‘(p));
(s; t)(p) = ‘−1 ◦ (s;
s 6 t 6 s (‘(p))
then (s; t)(p) is the unique solution of (3.14) which has a modi cation which is a
ow of C m -di eomorphisms.
Step 6: m = + ∞. From Steps 1– 4, we obtain that Theorem 3.4 holds for any
0 6 m¡ + ∞. 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 following equation in R2d+1 (cf. (2:3)) :
(d=du)(v˜ G )(u; x) = v˜ G ((v˜ G )(u; x));
(v˜ G )(0; x) = x
then for each x ∈ LG ; the following holds:
(v˜ G )(1; x) ∈ ‘(M ):
Proof. Let UG ; >G and G be as in Theorem 3.4. Now we follow Applebaum (1995,
pp. 173, 174). De ne
sup
¿
x
∈ LG
sup
sup |(v˜ G )(u; x)|
∨G
0 6 u 6 1 v˜ G ∈ UG
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 0 6 u 6 1 and v˜ G ∈ UG , de ne IuG : C ∞ (R2d+1 ; R) → C m (R2d+1 ; R) by
IuG (f)(·) = f ◦ (v˜ G )(u; ·)
then we have that
dIuG (f)
= IuG (v˜ G f):
du
Take f = , then by (3.16) and (3.18), we have for any x ∈ LG (also cf. Elworthy,
1988, p. 8),
IuG (v˜ G
)(x) = v˜ G ((v˜ G )(u; x))
with the help of the de nition of
((v˜ G )(u; x)) = 0
in (3.27).
D. Applebaum, F. Tang / Stochastic Processes and their Applications 92 (2001) 219–236
231
Hence,
((v˜ G )(u; x)) =
((v˜ G )(0; x)) =
(x) = 0;
∀x ∈ LG and 0 6 u 6 1: (3.28)
Suppose there exist x0 ∈ LG such that (v˜ G )(1; x0 ) ∈= ‘(M ), we will show this is a
contradiction. De ne
ux0 = inf {u¿0 : (v˜ G )(u; x0 ) ∈= ‘(M )}:
Since ‘(M ) is a closed subset of R2d+1 and (v˜ G )(0; x0 ) = x0 ∈ LG , we have
(v˜ G )(ux0 ; x0 ) ∈ ‘(M ). Hence, ux0 ¡1.
Note that since u → (v˜ G )(u; x0 ) is continuous, there exists ux0 ¡u0 6 1 such that
0¡d((v˜ G )(u0 ; x0 ); ‘(M ))¡
W
2
hence, by (3.27),
((v˜ G )(u0 ; x0 )) = d((v˜ G )(u0 ; x0 ); ‘(M ))2 ¿0
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
G
(s; t)(x) ∈ LG
for any x ∈ LG and 0 6 s 6 t¡sG (x).
Proof. De ne
UG; = {v˜ G ∈ UG : v˜ G m+ ¿}:
G
G
Let H˜ ; J˜ ; v˜ G be as in (3.18), and 0¡n ¡1 satisfy n ↓ 0 and
2
{v˜ G m+ ∨ ∇v˜ G · v˜ G m+ }t; G (d v˜ G )
UG; n+1 \UG; n
6
1
n5
2
UG
{v˜ G m+ ∨ ∇v˜ G · v˜ G m+ }t; G (d v˜ G ):
De ne
n (t) =
t+
s
G
G
v˜ N (dr d v˜ )
UG; n
and let ( 1n ; 2n ; : : :) be the jump times of the process n .
First, note that, by arguing similarly to Elworthy (1988, p. 8), we obtain that the
following SDE
t
t
G
G G
G
˜
n (s; t)(x) = x +
J˜ ( nG (s; r)(x); ◦ dr)
H ( n (s; r)(x); dr) +
s
s
−
s
t
UG; n
v˜ G ( nG (s; r)(x))r; G (d v˜ G ) dAr
(3.29)
232
D. Applebaum, F. Tang / Stochastic Processes and their Applications 92 (2001) 219–236
has a global solution. Furthermore, for almost all w ∈
nG (s; t)(x) ∈ LG
(3.30)
holds for any x ∈ LG and t¡G
s; n (x; w) where
G
G
s; n (x; w) = inf {t¿s : | n (s; t)(x; w)| ¿ G}:
(3.31)
G
Secondly, let be the interlacing with jumps in UG; n from nG in (3.29) de ned as
follows:
G
n (s; t)(x) = nG (s; t)(x);
0 6 s 6 t¡ 1n ;
G
G
n (s; 1n )(x) = (n ( 1n )) ◦ n (s; 1n −)(x); t = 1n ;
G
G
n (s; t)(x) = nG ( 1n ; t) ◦ n (s; 1n )(x);
1n ¡t¡ 2n ;
G
G
n (s; 2n )(x) = (n ( 2n )) ◦ 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 all w ∈
,
G
G
lim n (s; t)(x) = (s; t)(x)
(3.33)
n→∞
uniformly on compact sets containing x ∈ R2d+1 and t ∈ [s; +∞).
Thirdly, x w ∈
which satis es (3.33), x ∈ LG and t0 ¡sG (x; w) in the sequel. By
(3.23), we have
G
G
(| (s; t)(x; w)| ∨ | (s; t−)(x; w)|)¡G
for any 0 6 s 6 t 6 t0 .
Hence, there exists ′ ¿0 such that
G
G
(| (s; t)(x; w)| ∨ | (s; t−)(x; w)|) 6 G − ′
holds for any 0 6 s 6 t 6 t0 .
Now apply (3.33), we have G0 (′ ) ∈ N such that
G
G
(| n (s; t)(x; w)| ∨ | n (s; t−)(x; w)|) 6 G −
′
2
holds for any 0 6 s 6 t 6 t0 and n¿G0 (′ ).
G
Finally, x n¿G0 (′ ). We will show n (s; t0 )(x) ∈ ‘(M ).
When 0 6 s 6 t0 ¡ 1n : By (3.32) and (3.34), we have
G
| n (s; t)(x)| = | nG (s; t)(x)| 6 G −
′
2
for all s 6 t 6 t0 . Hence t0 ¡G
s; n (x), i.e.
G
n (s; t)(x) = nG (s; t)(x) ∈ LG ⊂ ‘(M ) by (3:30):
(3.34)
D. Applebaum, F. Tang / Stochastic Processes and their Applications 92 (2001) 219–236
233
When t0 = 1n : Similarly as above, we have
′
2
for any 0 6 s 6 t 6 t0 , i.e.
| nG (s; t)(x)| 6 G −
nG (s; t0 )(x) ∈ LG ⊂ ‘(M ):
Note that
G
n (s; 1n )(x) = (n ( 1n )) ◦ nG (s; 1n )(x)
G
then by Lemma 3.5, we have n (s; t0 )(x) ∈ ‘(M ).
G
Other values of t0 : Similarly, we can show n (s; t0 )(x) ∈ ‘(M ).
G
Now apply (3.33). As ‘(M ) is a closed subset of R2d+1 , we have (s; t0 )(x) ∈ ‘(M ).
One can see that
G
(s; t0 )(x) ∈ LG
G
since | (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 e ectively 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 proof in Lemma 4:8:2 in Kunita (1990,
p. 190), hence sG (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. De ne
Xt = Vt j Yj
for t ¿ 0;
where Y1 ; Y2 ; : : : ; Yl and all their nite linear combinations are complete vector elds
in Xm (M ) (cf. Applebaum, 1995, p. 172) and each Yj has bounded derivatives to all
orders in every co-ordinate system. Then X satis es Condition 2.3. Moreover, one can
see that (3.6) can be written as follows :
t+
t+
J ( (s; r−)(p); ◦ dr)
H ( (s; r−)(p); r) dr +
(s; t)(p) = p +
s
s
+
t+
t+
s
[(z j Yj )( (s; r−)(p)) − (s; r−)(p)]N (dr d z)
[(z j Yj )( (s; r−)(p)) − (s; r−)(p)]N˜ (dr d z)
|z| ¿ 1
s
+
|z|¡1
234
D. Applebaum, F. Tang / Stochastic Processes and their Applications 92 (2001) 219–236
+
t+
s
[(z j Yj )( (s; r−)(p)) − (s; r−)(p)
|z|¡1
− z j Yj ( (s; r−)(p))](d z) dr;
(3.35)
where
Ht (x) = t j Yj (x);
Jt (x) = kj Btk Yj (x):
The above can be easily extended to the case where the Levy process is replaced by an
Rd -valued semimartingale of similar 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
C m = C m (D; Rd ) = {v : v : D → Rd is m-times continuously di erentiable};
2
m
m
C˜ = C˜ (D×D; Rd )
2
= {f : f : D×D → Rd is m-times continuously di erentiable};
where D is a region in Rd and m is a non-negative integer. De ne
D ≡ Dx =
@| |
(@x1 ) · · · (@xd ) d
1
as a di erential operator where x ∈ Rd and
; = ( 1 ; 2 ; : : : ; d ) are multi-indexes of
i
non-negative integers with | | =
.
m
m
For v ∈ C m ; f ∈ C˜ , de ne the following semi-norms on C m and C˜ (see Kunita,
1990, pp. 72,73):
vLip; = sup
x;y ∈
x
=y
vm : = sup
x∈
|v(x) − v(y)|
;
|x − y|
|v(x)|
+
1 + |x|
vLip = sup
x
=y
1 6 | |¡(m+1)
vm+ : = vm : + 1(m¡+∞)
|v(x) − v(y)|
;
|x − y|
sup |Dx v(x)|;
x∈
sup
| | = m x;y ∈
x
=y
|Dx v(x) − Dx v(y)|
;
|x − y|
D. Applebaum, F. Tang / Stochastic Processes and their Applications 92 (2001) 219–236
∼
fm : = sup
x;y ∈
|f(x; y)|
+
(1 + |x|)(1 + |y|)
1 6 | | = |
|¡(m+1)
235
sup |Dx Dy
f(x; y)|;
x;y ∈
∼
fm+ :
= 1(m¡+∞)
sup
| | = m x;y ∈
x
=y
|Dx Dy f(x; x) − 2Dx Dy f(x; y) + Dx Dy f(y; y)|
|x − y|2
∼
+ fm :
and
Dx Dy f(x0 ; y0 ) = Dx Dy f(x; y)|x = x0 ; = y0 ;
d
@v j
(x) vi (x);
(∇v · v) =
@xi
j
i=1
@v
=
@x j
@v1 @v2
@vd
;
;
:
:
:
;
@x j @x j
@x j
;
where is a compact subset of D; ∈ (0; 1] and 1 6 j 6 d.
And
Cbm; ≡ Cbm; (D; Rd ) = {v ∈ C m : vm+ ¡∞};
where · m+ : D = · m+ .
References
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Mathematical Sciences, Vol. 75, 2nd Edition, Springer, Berlin.
Applebaum, D., 1995. A horizontal Levy processes on the bundle of orthonormal frames over a complete
Riemannian manifold. In: Azema, J., Emery, M., Meyer, P.A., Yor, M (Eds.), Seminaire de Probabilites
XXIX, Lecture Notes in Mathematics, Vol. 1613. Springer, Berlin, pp. 166–181.
Applebaum, D., 2000. Compound Poisson processes and Levy processes in groups and symmetric spaces.
J. Theoret. Probab. 13 (2), 383–425.
Applebaum, D., Kunita, H., 1993. Levy
ows on manifolds and Levy processes on Lie groups. J. Math.
Kyoto Univ. 33, 1103–1123.
Applebaum, D., Tang, F., 2000. The interlacing construction for stochastic
ows of di eomorphisms on
euclidean spaces. Preprint.
Cohen, S., 1996. Geometrie di erentielle stochastique avec sauts I, II. Stochastics and Stochastic Reports
56, 179 –203, 205 –225.
Elworthy, K.D., 1988. Geometric Aspects of Di usion on Manifolds. Lecture Notes in Mathematics, Vol.
1362. Springer, Berlin.
Fujiwara, T., 1991. Stochastic di erential equations of jump type on manifolds and Levy processes in
di eomorphisms group. J. Math. Kyoto Univ. 31, 99–119.
Fujiwara, T., Kunita, H., 1999. Canonical SDE’s based on semimartingales with spatial parameters–part I.
stochastics
ows of di eomorphisms. Kyushu J. Math. 53, 265–300.
Hirsh, M., 1976. Di erential Topology, GTM 33. Springer, Berlin.
Ikeda, N., Watanabe, S., 1981. Stochastic Di erential Equations and Di usion Processes, 2nd Edition.
North-Holland=Kodansha, Amsterdam=Tokyo.
Kunita, H., 1990. Stochastic Flows and Stochastic Di erential Equations. Cambridge University Press,
Cambridge.
Kunita, H., 1996. Stochastic di erential equations with jumps and stochastic
ows of di eomorphisms. In:
Ikeda, N., et al. (Eds.), Itˆo’s Stochastic Calculus and Probability Theory. Springer, Tokyo, pp. 197–212.
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Kurtz, T.G., Pardoux, E., Protter, P., 1995. Stratonovich stochastic di erential equations driven by general
semimartingales. Ann. Inst. H. Poincare Probab. Statist. 31, 77–351.
Marcus, S.I., 1981. Modeling and approximation of stochastic di erential equations driven by
semimartingales. Stochastics 4, 223–245.
Protter, P.E., 1990. Stochastic Integration and Di erential Equations – A New Approach. Appl. Math., Vol.
21 (Second printing). Springer, Berlin.
Tang, F., 2000. The interlacing construction for stochastic
ows of di eomorphisms. Ph.D. Thesis,
Nottingham Trent University.
www.elsevier.com/locate/spa
Stochastic
ows of di eomorphisms on manifolds driven
by in nite-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 of stochastic di erential
equations on manifolds which are written in Marcus canonical form and driven by in nitec 2001
dimensional semimartingales with jumps give rise to stochastic
ows of di eomorphisms.
Elsevier Science B.V. All rights reserved.
Keywords: Semimartingale; Manifold; Stochastic
ow; Interlacing construction
1. Introduction
A number of authors 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
of Levy type, i.e. having independent increments. In this paper our main contributions
are as follows:
1. We work on a quite general class of nite-dimensional smooth manifolds.
2. We study general stochastic
ows of di eomorphisms arising as solutions of SDEs
driven by a wide class of vector- eld-valued semimartingales with jumps.
3. We establish the solution of the
ow as the almost sure limit of an interlacing
sequence so that each term of the sequence consists of stochastic
ows with continuous sample paths broken up by a nite number of random jumps in each nite
time interval.
We remark that interlacing of stochastic
ows on compact manifolds was also considered by Fujiwara (1991) but only at the level of the L2 -limit.
One of the main aims of this 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.
E-mail addresses: dba@maths.ntu.ac.uk (D. Applebaum), fuchang.tang@yahoo.ca (F. Tang).
c 2001 Elsevier Science B.V. All rights reserved.
0304-4149/01/$ - see front matter
PII: S 0 3 0 4 - 4 1 4 9 ( 0 0 ) 0 0 0 9 1 - 0
220
D. Applebaum, F. Tang / Stochastic Processes and their Applications 92 (2001) 219–236
of embedding into Euclidean space and utilizing the tubular neighborhood theorem.
This speci c approach used herein was rst carried out by Elworthy for
ows with
continuous sample paths in Elworthy (1988) and was extended to a class of Levy
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
Let M be a connected, paracompact C ∞ -manifold of dimension d and denote as
C k (M ) = C k (M; R);
k = 0; 1; : : : ; +∞
the collection of all k-times continuously di erentiable real-valued functions on M .
Suppose 0 6 m 6 + ∞ and de ne Xm (M ) to be the real linear space of all C m -vector
elds ( rst-order linear partial di erential operators with C m -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 di eomorphism
between M and its range ‘(M ) which is a closed subset of R2d+1 .
For f ∈ C ∞ (M ) and X ∈ Xm (M ), we de ne a smooth function f and vector eld
X on ‘(M ) as follows:
= f ◦ ‘−1 (x);
f(x)
@
X (x) = D‘(‘−1 (x))(X (‘−1 (x))) = X i (x) i ;
@x
(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 di erential 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 (‘−1 (x))f(‘−1 (x));
X (x)f(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 for X0 (M ) as follows:
2d+1
∞
j
j
1 supx ∈ (Gi ) j = 1 |X (x) − Y (x)|
;
% (X; Y ) =
j
j
2i 1 + supx ∈ (G ) 2d+1
j = 1 |X (x) − Y (x)|
i=1
i
where X; Y ∈ X0 (M ) are represented as in (2.1) and (Gi , 1 6 i¡ + ∞) 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
of the choice of embedding . So we will write % simply as %.
We say that v ∈ Xm (M ) is (deterministically) complete if the partial di erential equation given below has a unique solution for all initial conditions p ∈ M and
D. Applebaum, F. Tang / Stochastic Processes and their Applications 92 (2001) 219–236
221
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 the exponential map.
De nition 2.2 (cf. Kunita, 1990, pp. 186). Suppose = {t ; t ¿ 0} is an Xm (M )valued process on M . If for any f ∈ C ∞ (M ), t f is a C m (M )-valued local martingale,
then is called a Xm (M )-valued local martingale. We say that is continuous if t f
is continuous for each f ∈ C ∞ (M ) with respect to t.
We can similarly de ne an Xm (M )-valued semimartingale.
Suppose we are given an X0 (M )-valued semimartingale which has the following
decomposition:
t+
t+
vNM (dr dv);
(2.4)
vN˜ M (dr dv) +
Xt = Ht + Jt +
s
s
U
Uc
0
where U is a Borel subset of X (M ), and
1. Ht is a continuous X0 (M )-valued stochastic process and Jt is a continuous X0 (M )valued local martingale. Furthermore, for any f; g ∈ C ∞ (M ), p1 ; p2 ∈ M and
0 6 s 6 t, the following holds (cf. Fujiwara, 1991; Kunita, 1990, pp. 186 –187):
t
b(f; r)(p1 ) dAr ;
Ht (s)f(p1 ) =
s
Jf(p1 ); Jg(p2 ) (s; t] =
t
a(f; g; r)(p1 ; p2 ) dAr ;
(2.5)
s
where At is a continuous increasing process which is independent of f and g.
2. NM (dr dv) is a random measure on R+ ×(X0 (M )\{0}) with a predictable compensator (intensity measure) dAr r; M (dv) and
N˜ M (dr dv) = NM (dr dv) − dAr r; 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 ) → R are linear
and
a(· ; · ; t)(p1 ; p2 ) : C ∞ (M )×C ∞ (M ) → R are 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 ) → C m (M ).
2. a(· ; · ; t) : C ∞ (M )×C ∞ (M ) → C m (M ×M ).
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D. Applebaum, F. Tang / Stochastic Processes and their Applications 92 (2001) 219–236
3. The map t → At is continuous.
4. The measure t; M satis es
(a) supp t; M ⊆ Xm+1 (M ).
(b) every member of supp t; M is a complete vector eld,
(c) t; M (U c ) 6 ft ,
(d) supv ∈ U {v
m+ : ‘() } 6 Q()¡ + ∞,
(e)
2
(v
0+1 : ‘() ∨ ∇v · v
Lip : ‘() )t; M (dv) 6 Kt (); m = 0;
U
U
2
(v
m+ : ‘() ∨ ∇v · v
m+ : ‘() )t; M (dv) 6 Kt ();
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+
t+
fr dAr ¡ + ∞
Kr () dAr ¡ + ∞;
s
s
a.s. for any compact subset ⊂ M and Q : M → R+ .
These conditions are natural generalizations of those 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 of di eomorphisms whenever
all driving vector elds have bounded derivatives to all orders in every co-ordinate
system.
Some examples of semimartingales satisfying these conditions are given at the end of
this paper. For convenience, we will below often identify a vector eld X ∈ Xm (R2d+1 )
(Y ∈ Xm (‘(M )), respectively) with its coordinate representation in R2d+1 , thus we will
have X ∈ C m (R2d+1 ; R2d+1 ) (Y ∈ C m (‘(M ); R2d+1 ), respectively).
For the rest of this paper, we will x our embedding to be ‘ given by Whitney’s
embedding theorem as above.
3. Stochastic di erential equations on manifolds
Consider the canonical SDE driven by X = (Xt ; t ¿ s ¿ 0) on the manifold M which
is given as follows, where denotes the exponential map de ned in (2.3)
t+
(s; t)(p) = p +
X ( (s; r−)(p); ♦dr);
s
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
t+
t+
(s; t)(p) = p +
J ( (s; r−)(p); ◦ dr)
H ( (s; r−)(p); dr) +
s
s
D. Applebaum, F. Tang / Stochastic Processes and their Applications 92 (2001) 219–236
+
t+
t+
t+
s
+
[(v)( (s; r−)(p)) − (s; r−)(p)]NM (dr dv)
[(v)( (s; r−)(p)) − (s; r−)(p)]N˜ M (dr dv)
[(v)( (s; r−)(p)) − (s; r−)(p)
Uc
s
+
223
U
s
U
− v( (s; r−)(p))]r; M (dv) dAr :
(3.6)
∞
Using Itˆo’s formula in (3.6), we have for any f ∈ C (M ) :
t+
f( (s; t)(p)) = f(p) +
Hf( (s; r−)(p); dr) +
+
t+
t+
t+
s
+
s
[f((v)( (s; r−)(p))) − f( (s; r−)(p))]NM (dr dv)
[f((v)( (s; r−)(p))) − f( (s; r−)(p))]N˜ M (dr dv)
[f((v)( (s; r−)(p))) − f( (s; r−)(p))
Uc
s
+
Jf( (s; r−)(p); ◦ dr)
s
s
t+
U
U
− 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 modi cation which is a stochastic
ow of local C m -di eomorphisms.
Our main aim in this paper is to give a complete proof of this 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). Let X be a C(Rd ; Rd )-valued semimartingale
with characteristics (a; b; ; A) (associated with U ).
Condition 3.2 (cf. Fujiwara and Kunita, 1999; Kunita, 1996). Let m be a non-negative
integer, q¿0 and ∈ (0; 1], suppose for any t ¿ s ¿ 0
1.
=
1;
;
m = 0 or + ∞:
0¡m¡ + ∞:
2. The map t → At is continuous.
∼
3. a(t)(m∨2)+ 6 Lt .
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D. Applebaum, F. Tang / Stochastic Processes and their Applications 92 (2001) 219–236
4. c(t)m+ 6 Lt , where
d
ij
@a
1
i
c (x; t) =
(x; y; t)
j
2
@x
j=1
:
y=x
5. b(t)m+ 6 Lt .
6. The measures t satis es
(a) (supp t ) ⊆(Cbm; ∩ C m+1 ).
(b) t (U c ) 6 Lt .
(c) supv ∈ U vm+ 6 q.
(d)
2
U
(v0+1 ∨ ∇v · vLip )t (dv) 6 Lt ;
m = 0;
2
U
(vm+ ∨ ∇v · vm+ )t (dv) 6 Lt ;
m¿0:
where Lt is nonnegative predictable processes with
t+
Lr dAr ¡ + ∞ a:s:
s
Theorem 3.3. Let X be a C(Rd ; Rd )-valued semimartingale with characteristics satisfying condition (3:2); then there exists a unique global solution (s; t) to the following equation. Furthermore; has a modi cation which is a stochastic
ow of
C m -di eomorphisms.
(s; t)(x) = x +
t+
X ( (s; r−)(x); ♦dr):
s
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 construction for was given.
First we consider the equation without big jumps
t+
t+
[(v)((s; r−)(p)) − (s; r−)(p)]N˜ Rd (dr dv)
t+
[(v)((s; r−)(p)) − (s; r−)(p)
H ((s; r−)(p); dr) +
s
+
s
J ((s; r−)(p); ◦ dr)
s
s
+
t+
(s; t)(p) = p +
U
U
− v((s; r−)(p))]r; Rd (dv) dAr :
(3.8)
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D. Applebaum, F. Tang / Stochastic Processes and their Applications 92 (2001) 219–236
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 and vm+ ¿ n }
such that
Un+1 \Un
2
(vm+ ∨ ∇v · vm+ )t (dv) 6
1
nL
U
(3.9)
2
(vm+ ∨ ∇v · vm+ )t (dv)
(3.10)
holds for any n ∈ N.
Consider the following approximating equation (cf. Applebaum, 2000).
t
t
n (s; t)(x) = x +
{c( n (s; r)(x); r)
J ( n (s; r)(x); dr) +
s
s
+ b( n (s; r)(x); r) − &n ( n (s; r)(x))} dA(r);
(3.11)
v(x)t (dv):
(3.12)
where
&n (x; t) =
Un
then it follows from Kunita (1990) that each n is a stochastic
ow of di eomorphism
of Rd .
We now de ne
t+
i
i
n (x; t) =
v (x)N (dr dv)
s
Un
n (t) = (1n (x; t); 2n (x; t); : : : ; dn (x; t)).
and
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 of n (x; t) by ( 1n ; 2n ; : : :). Following (cf. Applebaum, 2000),
we construct an interlacing sequence as follows:
n (s; t)(x) = n (s; t)(x);
0 6 s 6 t¡ 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 each x ∈ Rd ,
lim n (s; t)(x) = (s; t)(x) a:s:
n→∞
and the convergence is uniform on compacta.
The stochastic
ow is then obtained from by a single interlacing with big
jumps.
We see below that a similar construction holds on manifolds.
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D. Applebaum, F. Tang / Stochastic Processes and their Applications 92 (2001) 219–236
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
t+
t+
J ((s; r−)(p); ◦ dr)
H ((s; r−)(p); dr) +
(s; t)(p) = p +
s
s
+
t+
t+
s
+
[(v)((s; r−)(p)) − (s; r−)(p)]N˜ M (dr dv)
U
s
[(v)((s; r−)(p)) − (s; r−)(p)
U
− 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 C m -di eomorphisms.
Proof. Fix 0 6 m¡ + ∞ in the sequel.
Step 1: Euclidean version of (3.14). Let H t ; Jt and v be the push forwards of Ht ,
Jt and v, respectively. Moreover, de ne U‘(M ) = {v : v ∈ U }.
For clarity, we denote the random measure on X0 (‘(M )) by N‘(M ) (dr d v),
its predictable compensator by dAr r; ‘(M ) (d v)
and
N˜ ‘(M ) (dr d v)
= N‘(M ) (dr d v)
− dAr r; ‘(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:
t+
t+
r−)(x); ◦ dr)
J((s;
H ((s; r−)(x); dr) +
(s; t)(x) = x +
s
s
+
t+
t+
s
+
s
r−)(x)) − (s;
r−)(x)]N˜ ‘(M ) (dr d v)
[(v)(
(s;
U‘(M )
r−)(x)) − (s;
r−)(x)
[(v)(
(s;
U‘(M )
r−)(x))]r; ‘(M ) (d v)
− v(
(s;
dAr :
(3.15)
t)(x) = ‘((s; t)(p)) with p ∈ M and x = ‘(p).
where (s;
Step 2: Extend (3.15) to the whole of R2d+1 . Let G ∈ N and B(0; G) be an open
ball in R2d+1 whose centre is the origin and radius is G. By the tubular neighbourhood
theorem (see Hirsh (1976)), there exists a positive smooth function
: ‘(M ) → R+
227
D. Applebaum, F. Tang / Stochastic Processes and their Applications 92 (2001) 219–236
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)
2d+1
where d is the usual Euclidean metric in R
and
I‘(M
{y ∈ R2d+1 : |y − x|¡(x)}:
)=
x ∈ ‘(M )
De ne
WG =
LG = ‘(M ) ∩ B(0; G);
inf
x ∈ LG+1
(3.17)
(x):
Let >G ∈ C ∞ (R2d+1 ; R+ ) and G ∈ C ∞ (R+ ; R+ ) be given by (cf. Elworthy, 1988,
p. 8)
G 2
1
= 1; x ∈ B(0; G);
= 1; x¡ 2 (W ) ;
G
G
c
G 2
> (x) = 0; x ∈ B(0; G + 1) ;
(x) = 0; x ¿ (W ) ;
6 1; x ∈ B(0; G + 1)\B(0; G);
6 1; 12 (WG )2 6 x¡(WG )2 :
Next, we extend the vector elds H t ; Jt and v (see step 1), and smooth function
f smoothly from ‘(M ) to the whole of R2d+1 as follows (cf. Elworthy, 1988, p. 8;
Applebaum, 1995; Fujiwara, 1991):
G
f˜ (x) = >G (x) G (d(x; ‘(M ))2 )f((x));
G
H˜ t (x) = >G (x) G (d(x; ‘(M ))2 )H t ((x));
G
J˜t (x) = >G (x) G (d(x; ‘(M ))2 ) Jt ((x));
v˜ G (x) = >G (x) G (d(x; ‘(M ))2 ) v((x)):
(3.18)
Then, we obtain the following SDE on R2d+1 :
t+
G
G
G
˜
(s; t)(x) = x +
H ( (s; r−)(x); dr) +
+
t+
t+
+
G
G
G
G
[(v˜ G )( (s; r−)(x)) − (s; r−)(x)]N˜ G (dr d v˜ G )
UG
s
[(v˜ G )( (s; r−)(x)) − (s; r−)(x)
UG
G
− v˜ G ( (s; r−)(x))]r; G (d v˜ G ) dAr
which can be represented by
t+
G
G
G
(s; t)(x) = x +
X˜ ( (s; r−)(x); ♦dr);
s
where
G
G
G
X˜ t = H˜ t + J˜t +
G
G
J˜ ( (s; r−)(x); ◦ dr)
s
s
s
t+
s
t+
UG
v˜ G N˜ G (dr d v˜ G )
(3.19)
228
D. Applebaum, F. Tang / Stochastic Processes and their Applications 92 (2001) 219–236
and
UG = {v˜ G : v ∈ U‘(M ) };
c
UGc = {v˜ G : v ∈ U‘(M
)}
and the random measure on KG = UG ∪ UGc (⊂ X0 (R2d+1 )) is NG (dr d v˜ G ), its predictable compensator is dAr r; G (d v˜ G ) and (cf. Step 1)
N˜ G (dr d v˜ G ) = NG (dr d v˜ G ) − dAr r; G (d v˜ G )
which satisfy for any A ∈ B(X0 (R2d+1 )) ∩ KG
NG (t; A) = N‘(M ) (t; B)
where B = {v ∈ B(X0 (‘(M ))) : v˜ G ∈ A}:
One can see that KG is a Borel subset of X0 (R2d+1 ) by (3.18). Applying (2.2) and
G
(3.18), we next verify that the characteristics of X˜ t satisfy condition (3:2).
First, note that for f; g ∈ C ∞ (R) and x; y ∈ R2d+1 , we have
t
G
G
G
G
˜
˜
b˜ (f˜ ; r)(x) dAr ;
H t (s)f (x) =
s
G G
G
J˜ f˜ (x); J˜ g˜ G (y) (s; t] =
t
G
a˜G (f˜ ; g˜ G; r)(x; y) dAr ;
s
where
G
G
b˜ (f˜ ; t)(x) = >G (x) G (d(x; ‘(M ))2 )b(f; t)(‘−1 ◦ (x));
G
a˜G (f˜ ; 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
v˜ G (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) satis es
the following conditions:
1. Applying condition 2.3(4a) and (3.21), we have
supp t; G ⊆(Cbm; ∩ C m+1 ):
2. Applying condition 2.3(4d), the de nition of · m+ and v˜ G (x), we obtain
m+ : LG+1 }
sup {v˜ G m+ } 6 C˜ 1 (m; G) sup {v
v ∈ U‘(M )
v˜ G ∈ UG
6 C˜ 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
2
{v˜ G m+ ∨ ∇v˜ G · v˜ G m+ }t; G (d v˜ G )
UG
6 C˜ 2 (m; G)
U‘(M )
2
{v
m+ : LG+1 ∨ ∇v · v
m+ : LG+1 }t; ‘(M ) (d v)
D. Applebaum, F. Tang / Stochastic Processes and their Applications 92 (2001) 219–236
= C˜ 2 (m; G)
U
229
2
{v
m+ : LG+1 ∨ ∇v · v
m+ : LG+1 }t; M (dv)
6 C˜ 2 (m; G) Kt (‘−1 (LG+1 ))
and
UG
2
{v˜ G 0+1 ∨ ∇v˜ G · v˜ G Lip }t; G (d v˜ G ) 6 C˜ 2 (m; G)Kt (‘−1 (LG+1 ))
holds as above for m = 0, where C˜ j (m; G) are positive nite constants for j = 1; 2.
From (3.20) and 1–3, one can see that for any G ∈ N, there exists a unique global
solution to (3.19). Furthermore, it has a modi cation which is a stochastic
ow of
C m -di eomorphisms by Theorem 3.3. From now on, we identify the solution with this
very modi cation.
G
Now we are going to investigate the behaviour of the solution . 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). De ne
G
G
sG (x; w) = inf {t¿0 : (| (s; t)(x; w)| ∨ | (s; t−)(x; w)|) ¿ G}
(3.23)
and set
s (x; w) = sup sG (x; w);
G
t)(x; w) = G (s; t)(x; w)
(s;
when s 6 t¡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 s 6 t¡s (x) is the unique maximal solution to
(3.15).
Step 4: Flow properties. For each 0 6 s 6 t and G ∈ N, de ne as in Kunita (1990,
p. 178):
d
G
DG
s; t (w) = {x ∈ R : s (x; w)¿t};
Ds; t (w) = {x ∈ Rd : s (x; w)¿t}:
One can see that both DG
s; t (w) and Ds; t (w) are open sets and
Ds; t (w) =
+∞
DG
s; t (w):
G=0
Since for
t¡sG (x):
d
t) = G (s; t) : DG
(s;
s; t (w) → R
is a stochastic
ow of local C m -di eomorphisms by Theorem 3.3, then we have that
is a stochastic
ow of local C m -di eomorphisms on [s; s (x)).
In Lemma 3.6, we will show that for almost all w ∈
, the following holds:
G
(s; t)(x) ∈ LG
(3.25)
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D. Applebaum, F. Tang / Stochastic Processes and their Applications 92 (2001) 219–236
for any x ∈ LG and s 6 t¡sG (x; w). Therefore,
t)(x) ∈ ‘(M ) for s 6 t¡s (x)
(s;
(3.26)
t)(x) onto the
when x ∈ ‘(M ) by (3.24). We will then be able to “pull back” (s;
manifold M itself to 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 de ne for any p ∈ M ,
t)(‘(p));
(s; t)(p) = ‘−1 ◦ (s;
s 6 t 6 s (‘(p))
then (s; t)(p) is the unique solution of (3.14) which has a modi cation which is a
ow of C m -di eomorphisms.
Step 6: m = + ∞. From Steps 1– 4, we obtain that Theorem 3.4 holds for any
0 6 m¡ + ∞. 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 following equation in R2d+1 (cf. (2:3)) :
(d=du)(v˜ G )(u; x) = v˜ G ((v˜ G )(u; x));
(v˜ G )(0; x) = x
then for each x ∈ LG ; the following holds:
(v˜ G )(1; x) ∈ ‘(M ):
Proof. Let UG ; >G and G be as in Theorem 3.4. Now we follow Applebaum (1995,
pp. 173, 174). De ne
sup
¿
x
∈ LG
sup
sup |(v˜ G )(u; x)|
∨G
0 6 u 6 1 v˜ G ∈ UG
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 0 6 u 6 1 and v˜ G ∈ UG , de ne IuG : C ∞ (R2d+1 ; R) → C m (R2d+1 ; R) by
IuG (f)(·) = f ◦ (v˜ G )(u; ·)
then we have that
dIuG (f)
= IuG (v˜ G f):
du
Take f = , then by (3.16) and (3.18), we have for any x ∈ LG (also cf. Elworthy,
1988, p. 8),
IuG (v˜ G
)(x) = v˜ G ((v˜ G )(u; x))
with the help of the de nition of
((v˜ G )(u; x)) = 0
in (3.27).
D. Applebaum, F. Tang / Stochastic Processes and their Applications 92 (2001) 219–236
231
Hence,
((v˜ G )(u; x)) =
((v˜ G )(0; x)) =
(x) = 0;
∀x ∈ LG and 0 6 u 6 1: (3.28)
Suppose there exist x0 ∈ LG such that (v˜ G )(1; x0 ) ∈= ‘(M ), we will show this is a
contradiction. De ne
ux0 = inf {u¿0 : (v˜ G )(u; x0 ) ∈= ‘(M )}:
Since ‘(M ) is a closed subset of R2d+1 and (v˜ G )(0; x0 ) = x0 ∈ LG , we have
(v˜ G )(ux0 ; x0 ) ∈ ‘(M ). Hence, ux0 ¡1.
Note that since u → (v˜ G )(u; x0 ) is continuous, there exists ux0 ¡u0 6 1 such that
0¡d((v˜ G )(u0 ; x0 ); ‘(M ))¡
W
2
hence, by (3.27),
((v˜ G )(u0 ; x0 )) = d((v˜ G )(u0 ; x0 ); ‘(M ))2 ¿0
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
G
(s; t)(x) ∈ LG
for any x ∈ LG and 0 6 s 6 t¡sG (x).
Proof. De ne
UG; = {v˜ G ∈ UG : v˜ G m+ ¿}:
G
G
Let H˜ ; J˜ ; v˜ G be as in (3.18), and 0¡n ¡1 satisfy n ↓ 0 and
2
{v˜ G m+ ∨ ∇v˜ G · v˜ G m+ }t; G (d v˜ G )
UG; n+1 \UG; n
6
1
n5
2
UG
{v˜ G m+ ∨ ∇v˜ G · v˜ G m+ }t; G (d v˜ G ):
De ne
n (t) =
t+
s
G
G
v˜ N (dr d v˜ )
UG; n
and let ( 1n ; 2n ; : : :) be the jump times of the process n .
First, note that, by arguing similarly to Elworthy (1988, p. 8), we obtain that the
following SDE
t
t
G
G G
G
˜
n (s; t)(x) = x +
J˜ ( nG (s; r)(x); ◦ dr)
H ( n (s; r)(x); dr) +
s
s
−
s
t
UG; n
v˜ G ( nG (s; r)(x))r; G (d v˜ G ) dAr
(3.29)
232
D. Applebaum, F. Tang / Stochastic Processes and their Applications 92 (2001) 219–236
has a global solution. Furthermore, for almost all w ∈
nG (s; t)(x) ∈ LG
(3.30)
holds for any x ∈ LG and t¡G
s; n (x; w) where
G
G
s; n (x; w) = inf {t¿s : | n (s; t)(x; w)| ¿ G}:
(3.31)
G
Secondly, let be the interlacing with jumps in UG; n from nG in (3.29) de ned as
follows:
G
n (s; t)(x) = nG (s; t)(x);
0 6 s 6 t¡ 1n ;
G
G
n (s; 1n )(x) = (n ( 1n )) ◦ n (s; 1n −)(x); t = 1n ;
G
G
n (s; t)(x) = nG ( 1n ; t) ◦ n (s; 1n )(x);
1n ¡t¡ 2n ;
G
G
n (s; 2n )(x) = (n ( 2n )) ◦ 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 all w ∈
,
G
G
lim n (s; t)(x) = (s; t)(x)
(3.33)
n→∞
uniformly on compact sets containing x ∈ R2d+1 and t ∈ [s; +∞).
Thirdly, x w ∈
which satis es (3.33), x ∈ LG and t0 ¡sG (x; w) in the sequel. By
(3.23), we have
G
G
(| (s; t)(x; w)| ∨ | (s; t−)(x; w)|)¡G
for any 0 6 s 6 t 6 t0 .
Hence, there exists ′ ¿0 such that
G
G
(| (s; t)(x; w)| ∨ | (s; t−)(x; w)|) 6 G − ′
holds for any 0 6 s 6 t 6 t0 .
Now apply (3.33), we have G0 (′ ) ∈ N such that
G
G
(| n (s; t)(x; w)| ∨ | n (s; t−)(x; w)|) 6 G −
′
2
holds for any 0 6 s 6 t 6 t0 and n¿G0 (′ ).
G
Finally, x n¿G0 (′ ). We will show n (s; t0 )(x) ∈ ‘(M ).
When 0 6 s 6 t0 ¡ 1n : By (3.32) and (3.34), we have
G
| n (s; t)(x)| = | nG (s; t)(x)| 6 G −
′
2
for all s 6 t 6 t0 . Hence t0 ¡G
s; n (x), i.e.
G
n (s; t)(x) = nG (s; t)(x) ∈ LG ⊂ ‘(M ) by (3:30):
(3.34)
D. Applebaum, F. Tang / Stochastic Processes and their Applications 92 (2001) 219–236
233
When t0 = 1n : Similarly as above, we have
′
2
for any 0 6 s 6 t 6 t0 , i.e.
| nG (s; t)(x)| 6 G −
nG (s; t0 )(x) ∈ LG ⊂ ‘(M ):
Note that
G
n (s; 1n )(x) = (n ( 1n )) ◦ nG (s; 1n )(x)
G
then by Lemma 3.5, we have n (s; t0 )(x) ∈ ‘(M ).
G
Other values of t0 : Similarly, we can show n (s; t0 )(x) ∈ ‘(M ).
G
Now apply (3.33). As ‘(M ) is a closed subset of R2d+1 , we have (s; t0 )(x) ∈ ‘(M ).
One can see that
G
(s; t0 )(x) ∈ LG
G
since | (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 e ectively 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 proof in Lemma 4:8:2 in Kunita (1990,
p. 190), hence sG (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. De ne
Xt = Vt j Yj
for t ¿ 0;
where Y1 ; Y2 ; : : : ; Yl and all their nite linear combinations are complete vector elds
in Xm (M ) (cf. Applebaum, 1995, p. 172) and each Yj has bounded derivatives to all
orders in every co-ordinate system. Then X satis es Condition 2.3. Moreover, one can
see that (3.6) can be written as follows :
t+
t+
J ( (s; r−)(p); ◦ dr)
H ( (s; r−)(p); r) dr +
(s; t)(p) = p +
s
s
+
t+
t+
s
[(z j Yj )( (s; r−)(p)) − (s; r−)(p)]N (dr d z)
[(z j Yj )( (s; r−)(p)) − (s; r−)(p)]N˜ (dr d z)
|z| ¿ 1
s
+
|z|¡1
234
D. Applebaum, F. Tang / Stochastic Processes and their Applications 92 (2001) 219–236
+
t+
s
[(z j Yj )( (s; r−)(p)) − (s; r−)(p)
|z|¡1
− z j Yj ( (s; r−)(p))](d z) dr;
(3.35)
where
Ht (x) = t j Yj (x);
Jt (x) = kj Btk Yj (x):
The above can be easily extended to the case where the Levy process is replaced by an
Rd -valued semimartingale of similar 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
C m = C m (D; Rd ) = {v : v : D → Rd is m-times continuously di erentiable};
2
m
m
C˜ = C˜ (D×D; Rd )
2
= {f : f : D×D → Rd is m-times continuously di erentiable};
where D is a region in Rd and m is a non-negative integer. De ne
D ≡ Dx =
@| |
(@x1 ) · · · (@xd ) d
1
as a di erential operator where x ∈ Rd and
; = ( 1 ; 2 ; : : : ; d ) are multi-indexes of
i
non-negative integers with | | =
.
m
m
For v ∈ C m ; f ∈ C˜ , de ne the following semi-norms on C m and C˜ (see Kunita,
1990, pp. 72,73):
vLip; = sup
x;y ∈
x
=y
vm : = sup
x∈
|v(x) − v(y)|
;
|x − y|
|v(x)|
+
1 + |x|
vLip = sup
x
=y
1 6 | |¡(m+1)
vm+ : = vm : + 1(m¡+∞)
|v(x) − v(y)|
;
|x − y|
sup |Dx v(x)|;
x∈
sup
| | = m x;y ∈
x
=y
|Dx v(x) − Dx v(y)|
;
|x − y|
D. Applebaum, F. Tang / Stochastic Processes and their Applications 92 (2001) 219–236
∼
fm : = sup
x;y ∈
|f(x; y)|
+
(1 + |x|)(1 + |y|)
1 6 | | = |
|¡(m+1)
235
sup |Dx Dy
f(x; y)|;
x;y ∈
∼
fm+ :
= 1(m¡+∞)
sup
| | = m x;y ∈
x
=y
|Dx Dy f(x; x) − 2Dx Dy f(x; y) + Dx Dy f(y; y)|
|x − y|2
∼
+ fm :
and
Dx Dy f(x0 ; y0 ) = Dx Dy f(x; y)|x = x0 ; = y0 ;
d
@v j
(x) vi (x);
(∇v · v) =
@xi
j
i=1
@v
=
@x j
@v1 @v2
@vd
;
;
:
:
:
;
@x j @x j
@x j
;
where is a compact subset of D; ∈ (0; 1] and 1 6 j 6 d.
And
Cbm; ≡ Cbm; (D; Rd ) = {v ∈ C m : vm+ ¡∞};
where · m+ : D = · m+ .
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