Random Oper. Stoch. Equ. 18 (2010), 267–284
DOI 10.1515/ROSE.2010.015 © de Gruyter 2010
Existence and uniqueness of solutions
of stochastic functional differential equations
Max-K. von Renesse and Michael Scheutzow
Communicated by S. Molchanov
Abstract. Using a variant of the Euler–Maruyama scheme for stochastic functional dif-
ferential equations with bounded memory driven by Brownian motion we show that only
weak one-sided local Lipschitz (or “monotonicity”) conditions are sufficient for local ex-
istence and uniqueness of strong solutions. In case of explosion the method yields the
maximal solution up to the explosion time. We also provide a weak growth condition
which prevents explosions to occur. In an appendix we formulate and prove four lem-
mas which may be of independent interest: three of them can be viewed as rather general
stochastic versions of Gronwall’s Lemma, the final one provides tail bounds for Hölder
norms of stochastic integrals.
Keywords. Stochastic functional differential equation, existence of solution, maximal so-
lution, uniqueness of solution, Dereich lemma, stochastic Gronwall lemma.
2000 Mathematics Subject Classification. 34K50, 34K05, 60H10, 60H20.
1 Introduction
There is by now a rather comprehensive mathematical literature on the mathe-
matical theory and on applications of stochastic functional (or delay) differential
equations driven by Brownian motion. Existence and uniqueness of global solu-
tions have been established under global Lipschitz conditions on the coefficients
(e.g. [10]) or under local Lipschitz and linear growth conditions (e.g. [9, 12]).
On the other hand it is common knowledge for non-delay (stochastic) differential
equations that only one-sided Lipschitz conditions are sufficient for local existence
of solutions. This distinction becomes particularly relevant in infinite dimensions
where the drift in (stochastic) evolution equations is unbounded and discontinuous
in almost all interesting cases but nevertheless satisfies a one-sided Lipschitz. i.e.
“monotonicity/dissipativity” condition, cf. e.g. [11]. In this paper we show that
monotonicity of the coefficients guarantees local existence of solutions to delay
equations with bounded memory, thereby closing a systematic gap in the existing
literature.
Bereitgestellt von | Technische Universität Berlin
Angemeldet
Heruntergeladen am | 27.11.17 15:58
268 M.-K. von Renesse and M. Scheutzow
We choose the classical framework of the space of continuous functions as a nat-
ural state space of the equation. Note that, due to the absence of an inner product
on this space, the right formulation of monotonicity is not obvious in this case. The
proposed condition (M) below fits well to our needs, since it recovers the classical
monotonicity condition for the non-delay case as a limit and yet is weak enough
to cover a rather big set of equations.
In our proof we define a specific Euler–Maruyama scheme, which is generally
a very powerful tool in the Markovian case [1, 6, 7]. Other variants have been
treated for the numerical simulation of stochastic delay equations under Lipschitz
conditions in e.g. [4, 5, 8] and most recently [3]. We point out that our method
yields an approximation in the strong sense even in the case of an explosion. In
particular our proof below shows how the explosion time can be recovered numer-
ically, which seems to be a question typically neglected in the literature.
As for the proofs, note that the left hand side of condition (M) is quite weak
w.r.t. the C0-norm. As a consequence the standard two-step Burkholder–Davis–
Gundy and Gronwall argument cannot be applied to obtain the crucial contraction
estimates. We overcome this difficulty by what we call stochastic Gronwall lem-
mas and which are presented in the appendix. We think that they may be of inde-
pendent interest. These lemmas are also crucial for the global existence assertion
which holds under a rather familiar growth (or “coercivity”, [11]) condition (C),
which is again weak in the C0-topology.
2 Set up and main results
For r > 0, let Cdenote the space of continuous Rd-valued functions on Œr; 0
endowed with the sup-norm kk. For a function or a process Xdefined on Œt r; t
we write Xt.s/ WD X.t Cs/,s2Œr; 0. Consider the stochastic functional
differential equation
´dX.t/ Df .Xt/dtCg.Xt/dW.t/;
X0D'; (2.1)
where Wis an Rm-valued Brownian motion defined on a complete probability
space .; F;P/with the augmented Brownian filtration
FW
tD.W.u/; 0 ut/_NF;
where Ndenotes the null-sets in F,'is an .FW
t/-independent C-valued random
variable and fWC!Rd,gWC!Rdmare continuous maps.
Bereitgestellt von | Technische Universität Berlin
Angemeldet
Heruntergeladen am | 27.11.17 15:58
Well-posedness of stochastic functional differential equations 269
We will suppose throughout this work the following monotonicity assumption
on fand g:
For each compact subset CC, there exists a number KCand some rC2
.0; r such that for all x; y 2Cwith x.s/ Dy.s/ 8s2Œr; rC
2hf .x/ f .y/; x.0/ y.0/iCjjjg.x/ g.y/jjj2KCkxyk2;
(M)
where h;i denotes the standard inner product on Rdand jjjMjjj2Dtr.MM /
for M2Rdm.
As an example in dD1take f .x/ D'.PN
iD1wix.ti//, where ti2Œr; 0,
wi0,iD1; : : : ; N , and '2C.R/is a non-increasing continuous (not
necessarily Lipschitz) function, e.g. '.s/ D sign.s/pjsjand glocally Lips-
chitz on C. Another example is fDf1Cf2Cf3with f1locally Lipschitz
on C,f2.x/ DRr0
r .x.s//k.s/ dsfor some 0 < r0< r,k; 2C.R/and
f3.x/ D'.x.0// with '2C.R/non-increasing as above.
Our first result is a local existence and uniqueness statement for solutions to
(2.1) for which we recall some basic notions. Given any filtration .Ft/on , an
.Ft/-stopping time W!R0is called predictable if there exists a sequence of
(“announcing”) stopping times nsuch that n< and n%P-almost surely.
A tuple XD.X; / of a predictable stopping time and a map XW.Œr; 0[
Œ0; // !Rdis called a local .Ft/-semimartingale up to time starting from
'2C, if X0D'holds P-almost surely and for any (announcing) stopping time
n< , the process .Xn.t//t0with Xn.t/ DX.t ^n/is an Rd-valued
.Ft/-adapted semimartingale.
Definition 2.1 (Local Solution). Let FtDFW
t_.'/. A local .Ft/-semimartin-
gale .X; / up to a predictable stopping time is called a local strong solution to
equation (2.1) if X0D'and for any stopping time n< and any t0
X.t ^n/DX.0/ CZt^n
0
f .Xu/duCZt^n
0
g.Xu/dW.u/ P-a.s.
The pair .X; / is called maximal strong solution if in addition .Xt/eventually
leaves any compact set KCfor t!,P-almost surely on ¹ < 1º; i.e.
P¹9a compact set KCand ti%s.t. Xti2Kº\¹ < 1ºD0:
Theorem 2.2. Equation (2.1) admits a unique maximal strong solution .X; / pro-
vided (M) holds.
Bereitgestellt von | Technische Universität Berlin
Angemeldet
Heruntergeladen am | 27.11.17 15:58
270 M.-K. von Renesse and M. Scheutzow
Theorem 2.3. In addition to the assumptions of Theorem 2.2 let fand gbe
bounded on bounded subsets of Cand let the pair .f; g/ be weakly coercive in
the sense that there exists a non-decreasing function WŒ0; 1/!.0; 1/such that
R1
01=.u/ duD 1and for all x2C
2hf .x/; x.0/iCjjjg.x/jjj2.kxk2/: (C)
Then Xis globally defined, i.e. D 1P-almost surely.
3 Proof of Theorem 2.2
The proof of Theorem 2.2 is based on an iteration of Lemma 3.1 below, which
requires some auxiliary notation. For ˆCand R > 0 let
Cˆ;R D®2Cˇˇ9'2ˆ; r02Œ0; r W.u/ D'.u Cr0/; u 2.r; r0;
k'.0/k1=4IŒr0;0 R¯C;
where
kk˛IŒa;b Dsup
au<vbj.v/ .u/j=.v u/˛Csup
aubj.u/j
denotes the Hölder-˛-norm on C.Œa; b; Rd/,˛2.0; 1/.
Note that Cˆ;R is compact in Cprovided ˆis.
Below we drop the subscript ˆwhenever this causes no confusion.
Lemma 3.1. In addition to the conditions of Theorem 2.2 assume there is a com-
pact subset ˆCsuch that '2ˆP-almost surely. For R > 0, let rRDrCbe
the constant appearing in (M) for choosing CDCˆ;R. Then there exists a stop-
ping time 0 < RrRand a unique (up to indistinguishability) .Ft/-adapted
process X.t/,t2Œ0; Rsuch that Xt2CRfor all t2Œ0; Rwhich solves (2.1)
up to time R. Moreover,
kX./'.0/k1=4IŒ0;RR
2P-a.s. on ¹R< rRº:(3.1)
Proof. The proof is inspired by the arguments for finite dimensional monotone
SDEs in [7], cf. e.g. [11]. For n2N, we define an Euler-like approximation to
(2.1) with step size 1
nby
´dXn.t/ Df .Xn
t/dtCg.Xn
t/dW.t/;
Xn
0D'; (3.2)
Bereitgestellt von | Technische Universität Berlin
Angemeldet
Heruntergeladen am | 27.11.17 15:58
Well-posedness of stochastic functional differential equations 271
where we define Xn
s./2C,s0, by
Xn
s.u/ DXn.s Cu/ ^bnsc
n; u 2Œr; 0:
Equation (3.2) admits a global in time solution via the recursion Xn
0D'and
Xn.t/ DXnbntc
nCZt
bntc=n
fXn
sdsCZt
bntc=n
gXn
sdW.s/:
The process t7! Xn.t/ is adapted and continuous, hence
t7! pn
t./WD Xn
t./Xn
t./; t 0;
defines an adapted C-valued process (which is càdlàg). With this, (3.2) is equiva-
lent to Xn
0D'and
Xn.t/ D'.0/ CZt
0
f .Xn
sCpn
s/dsCZt
0
g.Xn
sCpn
s/dW.s/:
Without loss of generality, we may assume that the set ˆhas the property that
02ˆand 2ˆ; s 2Œr; 0/ implies that the function u7! .u ^s/; u 2Œr; 0
also belongs to ˆ. Then, Xn
t2CRimplies Xn
t2CR, hence pn
t2e
CRD
¹12ji2CRºprovided
tn
RWD inf¹t > 0 jXn
t…CRº:
Since e
CRCis again compact,
e.R/ Dsup
x2e
CRkxk<1(3.3)
and the continuity of fand gensures that
C1.R/ WD sup
x2e
CR¹jf .x/jCjjjg.x/jjjº <1:(3.4)
Fix n; m 2Nand let 0be a finite stopping time. Then, by Itô’s formula,
jXn./ Xm./j2
D2Z
0hXn.u/ Xm.u/; g.Xn
uCpn
u/g.Xm
uCpm
u/dW.u/i
CZ
0
2hf .Xn
uCpn
u/f .Xm
uCpm
u/; Xn.u/ Xm.u/idu
CZt
0ˇˇˇˇˇˇg.Xn
uCpn
u/g.Xm
uCpm
u/ˇˇˇˇˇˇ2du:
Bereitgestellt von | Technische Universität Berlin
Angemeldet
Heruntergeladen am | 27.11.17 15:58
272 M.-K. von Renesse and M. Scheutzow
In order to use condition (M), note that by construction for s > 0 and sCu0
Xm
s.u/ DXn
s.u/ D'.s Cu/:
Hence, together with (3.3) and (3.4), the sum of the du-integrals on the r.h.s. can
be estimated from above by
Z
02hf .Xn
uCpn
u/f .Xm
uCpm
u/; pm
u.0/ pn
u.0/i
CKRXn
uCpn
u.Xm
uCpm
u/2du
Z
04C1.R/jpn
u.0/jCjpm
u.0/j
C4KRpn
u2Cpm
u2C2KRXn
uXm
u2du
Z
04C1.R/ C4KRe.R/pn
uCpm
u
C2KRsup
v2Œ0;u jXn.v/ Xm.v/j2du
provided m
R^n
R^rRDW . Lemma 5.4 applies to Z.s/ WD jXn.s ^/
Xm.s ^/j2with
M.s/ WD 2Zs^
0hXn.u/ Xm.u/; g.Xn
uCpn
u/g.Xm
uCpm
u/dW.u/i;
H.s/ DZs^
04C1.R/ C4K.R/e.R/pn
uCpm
udu
and TDrR.
Once we have shown that some moment of H.T / WD sup0sTH.s/ con-
verges to 0 as n; m ! 1, Lemma 5.4 implies that for all ">0,
lim
m;n!1
P®sup
s2Œ0;m
R^n
R^rRjXm.s/ Xn.s/j "¯D0: (3.5)
Since H.T / is bounded uniformly in !; n; m, it suffices to show that H.T /
converges to zero in probability as m; n ! 1 which can be verified as follows:
pn
s.u/ D´0; u r; u Csbnsc
n
RsCu
bnsc=n f .Xn
t/dtRsCu
bnsc=n g.Xn
t/dW.t/; u Csbnsc
n; u 0
implies
kpn
sk sup
bnsc=ntsˇˇˇZt
bnsc=n
f .Xn
u/duˇˇˇCsup
bnsc=ntsˇˇˇZt
bnsc=n
g.Xn
u/dW.u/ˇˇˇ;
Bereitgestellt von | Technische Universität Berlin
Angemeldet
Heruntergeladen am | 27.11.17 15:58
Well-posedness of stochastic functional differential equations 273
and hence – since fand gare bounded on CR–
E1¹n
Rsºkpn
sk ! 0as n! 1;uniformly in Œ0; rR:
Therefore, EH.T / converges to 0 and (3.5) follows.
By definition of Xmthis also yields
lim
m;n!1
P®sup
s2Œ0;m
R^n
R^rRkXm
sXn
sk "¯D0: (3.6)
Since f; g are uniformly continuous on the compact set CR,
lim
m;n!1
P®sup
s2Œ0;m
R^n
R^rR®jf .Xm
s/f .Xn
s/j_ˇˇˇˇˇˇg.Xm
s/g.Xn
s/ˇˇˇˇˇˇ¯"¯D0:
(3.7)
To further improve this statement, we apply Lemma 5.5 to
Xn.s ^m
R^n
R^rR/Xm.s ^m
R^n
R^rR/
D
s^m
R^n
R^rR
Z0
.F nFm/.u/ dZ.u/;
where for simplicity we write
Z.u/ D.u; W.u// 2RmC1; F n.u/ Df .Xn
u/; g.Xn
u/:
Together with (3.7) this allows to conclude that for all ">0
lim
m;n!1
P®Xm./Xn./1=4IŒ0;m
R^n
R^rR"¯D0: (3.8)
Let us select a subsequence, which will again be denoted by Xnsuch that
P®Xk.:/ Xl.:/1=4IŒ0;k
R^l
R^rR2.l^k/¯2.l^k/;(3.9)
and define
RDlim inf
n!1 n
R:
Due to (3.9), there is an .Ft/-adapted process Xdefined in Œ0; R/\Œ0; rRto
which Xnconverges P-almost surely locally in C1=4.Œ0; R/\Œ0; rRIRd/. From
(3.2), (3.5) and (3.7) and the continuity of fand gwe infer that Xmust be a
solution to equation (2.1) on Œ0; R/\Œ0; rR/.
We remark that R> 0 almost surely, which can be seen as follows. For any
">0, using (3.9) we choose n0such that the set
AD°!ˇˇˇsup
kn0Xn0./Xk./1=4IŒ0;k
R^n0
R^rR<R
2±
Bereitgestellt von | Technische Universität Berlin
Angemeldet
Heruntergeladen am | 27.11.17 15:58
274 M.-K. von Renesse and M. Scheutzow
satisfies P.A/ 1". From Xn0
s./D'..s C/^0/ 2ˆfor s2Œ0; 1
n0, using
Lemma 5.5 for the SDE (3.2) solved by Xn0, it follows that
n0
R=2 WD inf °t0ˇˇˇkXn0./'.0/k1=4IŒ0;t R
2±^rR
is strictly positive. By construction of Ait holds on Athat n
R^rRn0
R=2 ^rR
for all nn0, hence in particular R> 0.
Next, we show that almost surely one of the two following events occurs:
¹RrRºor ®R< rR¯\°sup
t<RkX./'.0/k1=4IŒ0;t 3R
4±:(3.10)
In case ¹RrRº, using (2.1) for X./on Œ0; rR/and the uniform boundedness
of the coefficients on CRwe may extend X./on the closed interval Œ0; rRby
setting
X.rR/WD X.0/ CZrR
0
f .Xs/dsCZrR
0
g.Xs/dW.s/:
Together with (3.10) for
RWD inf °t2Œ0; R/\Œ0; rRjkX./'.0/k1=4IŒ0;t R
2±^rR
this gives a well-defined process t7! X.t/ for t2Œ0; Rwhich solves (2.1) up
to time Rin the sense of Definition 2.1. Moreover, (3.1) holds by construction.
To prove (3.10) we show that the set
BWD ¹R< rRº\°sup
t<RkX./'.0/k1=4IŒ0;t <3R
4±:
has vanishing P-measure. Assume the contrary, i.e. P.B/ Dp > 0. Then by
(3.9) and the definition of Rwe find some n02Nsuch that P.A/ > p
2, where
AWD 8
<
:
supkn0Xn0./Xk./1=4IŒ0;k
R^n0
R^rR<R
16 I
infnn0n
R< rRIsupt<RkX./'.0/k1=4IŒ0;t <3R
49
=
;:
We show that in fact P.A/ D0. To this aim note that w.l.o.g. we may assume that
Xnconverges to Xlocally in C1=4.Œ0; R// and
Œ0; rR^m
R3t7! kXm./k1=4IŒ0;t
is continuous for all m2N, for all !2A, where the latter is again a consequence
of Lemma 5.5. Now for !2Achoose mDm.!/ n0such that m
R< rR. Let
m
RWD inf ®t0jkXm./'.0/k1=4IŒ0;t R¯m
R;
Bereitgestellt von | Technische Universität Berlin
Angemeldet
Heruntergeladen am | 27.11.17 15:58
Well-posedness of stochastic functional differential equations 275
then by continuity m
7R=8 < m
15R=16 n
Rfor all nn0, hence m
7R=8 < R.
Again by continuity,
sup
t<m
7R=8 kXn./'.0/k1=4IŒ0;t 3R
4
for all nn0satisfying n
R> m
7R=8. In view of the convergence of Xnto Xin
C1=4Œ0; m
7R=8for n! 1 this yields a contradiction to
sup
t<RkX./'.0/k1=4IŒ0;t <3R
4:
Hence AD ;almost surely which proves (3.10).
To show uniqueness of a local solution, assume Xand Q
Xare two solutions
defined up to a stopping time QR. Applying Itô’s formula to the square of the
norm of the difference of the solutions and using condition (M), Lemma 5.2 (with
CD0) shows that the solutions agree on Œ0; Q almost surely.
Proof of Theorem 2.2.First we remark that it is sufficient to prove both the exis-
tence and uniqueness assertion of the theorem under the stronger assumption that
P.' 2ˆ/ D1for any fixed compact subset ˆC. In fact, since any probability
measure on the Polish space Cis tight, in both cases the general statement follows
by approximation in P-measure by initial conditions 'nD1ˆn.'/ ', where e.g.
the compact subsets ˆnCare chosen such that P.' 62 ˆn/1
n.
The proof of the existence statement is based on iterative use of Lemma 3.1.
Recall for R > 0,rRdenotes the constant rCin condition (M) when CDCˆ;R.
We may assume w.l.o.g. that the function R7! rRis non-increasing and we may
select a sequence R.k/ % 1,k2N, such that PkrR.k/ D 1.
Lemma 3.1 with ˆDW ˆ.1/ and RWD R.1/ for initial condition 'DW '.1/ 2
ˆ.1/ guarantees the existence of a process t7! X.t/ DW X.1/.t/,t2Œ0; .1/,
with an F-stopping time .1/ WD R.1/ rR.1/ which is a local solution to (2.1)
on Œ0; .1//.
Next we may apply Lemma 3.1 to the same equation (2.1), now in the situation
when Rand Ware chosen to be R.2/ and W.2/
tDW..1/ Ct/ W..1//on
.; F;P/respectively, with F.2/
tDFW.2/
t_NF,.t 0/, and F.2/
-
independent initial condition '.2/ WD X.1/
.1/ 2Cˆ;R1DW ˆ.2/. This yields an
F.2/
-stopping time Q.2/ rR.2/ and a process t7! Q
X.2/,Œ0; Q.2/, solving (2.1)
on t2Œ0; Q.2//. (Note that here we have used the simple fact that CCˆ;R1;R2D
Bereitgestellt von | Technische Universität Berlin
Angemeldet
Heruntergeladen am | 27.11.17 15:58
276 M.-K. von Renesse and M. Scheutzow
Cˆ;R2for R2R1.) Hence, by continuation
t7! X.2/.t/ D´X.1/.t/ if t2Œr; .1/
Q
X.2/.t .1//if t2..1/; .1/ C Q.2/
we obtain an F-adapted process which is a local solution to equation (2.1) up to
the F-stopping time .2/ D.1/ C Q.2/ in the sense of Definition 2.1.
For general nthis construction is repeated inductively, furnishing a local solu-
tion .X; / to equation (2.1) in the sense of Definition 2.1 where
Dlim
n!1 .n/:
To prove that .X; / is maximal using the continuity of fand git suffices to
prove that the set
†D®sup
t2Œ0;Œ
.jf .Xt/j_jjjg.Xt/jjj/ < 1¯\® < 1¯(3.11)
has zero P-measure. Now from the second statement in Lemma 3.1, from the
construction of Xand from the property PkrR.k/ D 1 it follows that
sup
s2Œ0;Œ kX./X..k1//k1=4IŒ0;s R.k/
2P-a.s.
for infinitely many k2Non ¹ < 1º, i.e.
P.†/ DP < 1I sup
s2Œ0;/
.jf .Xs/j_jjjg.Xs/jjj/ < 1I
sup
s2Œ0;/ kX./k1=4IŒ0;s D 1:
Since Xsolves (2.1), due to e.g. Lemma 5.5, the r.h.s. is zero.
As for the uniqueness statement let .Y; / be another maximal solution with
an associated sequence of announcing stopping times .n/. The construction of X
above yields a sequence of announcing stopping times .n/ for and compact sets
CnCsuch that Xt^.n/ 2Cn. Hence, by the same argument as in the proof
of Lemma 3.1 one obtains that X.n/^.n/^ and Y.n/^.n/^ are indistinguishable.
Moreover, the maximality of the pair .Y; / implies that .n/ < for all n2N,
i.e. almost surely. Conversely, the maximality of implies > n, i.e.
, which completes the proof.
Bereitgestellt von | Technische Universität Berlin
Angemeldet
Heruntergeladen am | 27.11.17 15:58
Well-posedness of stochastic functional differential equations 277
4 Proof of Theorem 2.3
Proof of Theorem 2.3.Let .X; / be the maximal strong solution of equation (2.1).
We want to show that D 1 almost surely. Since fand gare bounded on
bounded subsets of C, it follows from (3.11) that lim supt%jX.t/jD1almost
surely on the set ¹ < 1º. For a stopping time 0 < , Itô’s formula implies
that
X2./ X2.0/ D
Z0
2hf .Xu/; X.u/iCjjjg.Xu/jjj2du
C2
Z0
hX.u/; g.Xu/dW.u/i
Z0
.kXuk2/duCM./;
where Mis a continuous local martingale. Applying Lemma 5.1 to Z.t/ WD X2.t/
finishes the proof.
5 Appendix
We start by proving three lemmas which could be called stochastic Gronwall lem-
mas. We use them in the proof of Theorems 2.2 and 2.3. Then we prove a result
about the tails of Hölder norms of stochastic integrals which we owe to Steffen
Dereich (TU Berlin). We believe that all these results are of independent inter-
est. In all lemmas, we assume that a filtered probability space .; F; .Ft/t0;P/
is given and that it satisfies the usual conditions. Throughout, we will use the
notation Z.T / Dsup0tTZ.t/ for a real-valued process Z.
Lemma 5.1. Let > 0 be a stopping time and let Zbe an adapted non-negative
stochastic process with continuous paths defined on Œ0; / which satisfies the in-
equality
Z.t/ Zt
0
.Z.u// duCM.t/ CC;
and limt"Z.t/ D 1 on ¹ < 1º almost surely. Here, C0and Mis a
continuous local martingale defined on Œ0; /,M.0/ D0and WŒ0; 1/!.0; 1/
is non-decreasing, and R1
01=.u/ duD 1. Then D 1 almost surely.
Bereitgestellt von | Technische Universität Berlin
Angemeldet
Heruntergeladen am | 27.11.17 15:58
278 M.-K. von Renesse and M. Scheutzow
Proof. Let Ybe the unique (maximal) solution of the equation
Y.t/ DZt
0
.Y .u// duCM.t/ CC:
Clearly, Y.t/ Z.t/ for all tfor which Yis defined and therefore it suffices to
prove the claim for Yinstead of Z. For a > C , define aWD inf¹t0jY.t/ aº.
For C < a < b and ı > 0 we get
P¹baıjFaº P®baı.b/ Csup
t2Œa;b^.aCı/
M.t/ M.a/ˇˇFa¯
on the set ¹a<1º. Note that on ¹a<1º we have
M.t/ M.a/Y.t/ Y.a/.t a/.b/ aı.b/ (5.1)
for atb^.aCı/ since Yis non-negative. For
WD inf¹tajM.t/ M.a/baı.b/º^b^.aCı/
we therefore get
0DE.M./ M.a/jFa/.b aı.b//p .a Cı.b//.1 p/;
where pWD P¹M./ M.a/baı.b/ jFaº. Hence
P°baıˇˇˇFa±paCı.b/
bon ¹a<1º:(5.2)
Fix a > C . Then
DaC
1
X
kD12ka2k1a:
We show that the sum diverges almost surely. To ease notation, we write kinstead
of 2ka. For ık> 0,k2N, (5.2) implies that
P¹kk1ıkjFk1º 1
2ık
.2ka/
2ka
on the set ¹k1<1º. Now
1
X
kD1
.kk1/
1
X
kD1
ık1¹kk1ıkº:(5.3)
Bereitgestellt von | Technische Universität Berlin
Angemeldet
Heruntergeladen am | 27.11.17 15:58
Well-posedness of stochastic functional differential equations 279
We choose
ıkWD 1
4
2ka
.2ka/; k 2N:
Since is non-decreasing we have
1
X
kD1
ık1
4Z1
2a
1
.u/ duD 1
and
P¹kk1ıkjFk1º 1
4on ¹k1<1º:
It follows (e.g. from Kolmogorov’s three series theorem) that the right hand side
of (5.3) diverges on the set ¹k<1for all k2Nº. On the complement of this
set, is also infinite, i.e. the proof of the lemma is complete.
While the previous lemma was concerned with non-blow up of Z, the follow-
ing lemma shows that Zremains small in case the initial condition is small. In
principle we could formulate the following lemma also using a function as in the
previous one but we prefer not to in order to obtain a reasonably explicit formula
for moments of Z.T /.
Lemma 5.2. Let Zbe an adapted non-negative stochastic process with continuous
paths defined on Œ0; 1/which satisfies the inequality
Z.t/ KZt
0
Z.u/ duCM.t/ CC;
where C0,K > 0 and Mis a continuous local martingale with M.0/ D0.
Then for each 0 < p < 1, there exist universal finite constants c1.p/; c2.p/
(not depending on K; C; T and M) such that
E.Z.T //pCpc2.p/ exp¹c1.p/KT ºfor every T0:
Proof. Let Ybe the unique solution of the equation
Y.t/ DKZt
0
Y.u/ duCM.t/ CC:
Clearly, Y.t/ Z.t/ for all t0and therefore it suffices to prove the claim for
Yinstead of Z. Let aWD inf¹t0WY.t/ aº. Like in the proof of Lemma 5.1,
we obtain for ˇ2.0; 1/ and b > a C
P°baˇ
KˇˇˇFa±aCˇb
bon ¹a<1º:(5.4)
Bereitgestellt von | Technische Universität Berlin
Angemeldet
Heruntergeladen am | 27.11.17 15:58
280 M.-K. von Renesse and M. Scheutzow
For T > 0,m2N, > .1 ˇ/1we get
P¹Y.T / mCº D P¹mCTº D P°m
X
iD1
iCi1CT±:
By (5.4), the last sum is stochastically larger than ˇ=K times a binomial variable
Vwith parameters mand ˛WD 11
ˇ. Therefore, for >0and NWD dK T
ˇe
we get
P¹Y.T / mCº P¹VNº D P¹eV eN º:
Applying Markov’s inequality, representing Vas a sum of mindependent Ber-
noulli(˛) variables and optimizing over > 0 as usual, we obtain for m
dN
˛e DW m0
P¹Y.T / mCº exp °.m N / log m
mNC.m N / log.1 ˛/
CNlog ˛CNlog m
N±:
Assume that plog Clog.1 ˛/ < 0 (which requires p < 1 since 1˛D
1
Cˇ > 1
) and fix q > 0 such that plog Clog.1 ˛/ Cq1< 0. Then
EY.T /pDZ1
0
P¹Y.T / s1=pºds
m0pCpC
1
X
mDm0
Cppm. 1/ exp °.m N / log m
mN
C.m N / log.1 ˛/ CNlog ˛CNlog m
N±
m0pCpCCp. 1/ exp °Nlog ˛q
1˛±
1
X
mDm0
exp ®m.p log Clog.1 ˛/ Cq1/¯
DCpm0pC. 1/ exp °Nlog ˛q
1˛±
exp¹m0.p log Clog.1 ˛/ Cq1/º
1exp¹plog Clog.1 ˛/ Cq1º;
where we used the inequalities
log.1 Cx/ xfor xDN
mN
and
log xlog qCq1.x q/ for xDm
N
Bereitgestellt von | Technische Universität Berlin
Angemeldet
Heruntergeladen am | 27.11.17 15:58
Well-posedness of stochastic functional differential equations 281
in the last “”. Observing that
m0kT
ˇC11
˛C1and NKT
ˇC1;
the claim follows.
Remark 5.3. It is clear that the previous lemma does not hold for p > 1: just
consider a scalar geometric Brownian motion starting with C. Its pth moment for
p > 1 at time 1 (say) is unbounded with respect to the volatility . We don’t know
whether the lemma holds true for pD1but we conjecture that it doesn’t.
Lemma 5.4. Let Zbe an adapted non-negative stochastic process with continuous
paths defined on Œ0; 1/which satisfies the inequality
Z.t/ KZt
0
Z.u/ duCM.t/ CH.t/;
where K > 0,Mis a continuous local martingale with M.0/ D0, and His
an adapted process with continuous paths satisfying H.0/ D0. Then, for each
0<p<1and ˛ > 1Cp
1p, there exist constants c3; c4depending on p; ˛ only such
that
E.Z.T //pc3exp¹c4KT º.EH.T /˛/p=˛ for every T0:
Proof. Fix T > 0 and for i2Nlet Xibe the unique solution of
Xi.t/ DKZt
0
X
i.u/ duCM.t/ Ci:
Hence, ZXion Œ0; T iwhere
iWD ®!Wsup
0tT
H.t/ i¯:
Let s2.1
1p;˛
1Cp/and let r > 1 be defined by r1Cs1D1. Then pr < 1
and Lemma 5.2 and Hölder’s inequality imply
E.Z.T //p
1
X
iD1
E.X
i.T //p1ini1
1
X
iD1
.E.X
i.T //pr /1=r P¹ini1º1=s
Bereitgestellt von | Technische Universität Berlin
Angemeldet
Heruntergeladen am | 27.11.17 15:58
282 M.-K. von Renesse and M. Scheutzow
1
X
iD1
ipc2.pr/1=r exp¹KT c1.pr/=rºP¹H.T / i1º1=s
exp¹KT c1.pr/=rºc2.pr/1=r
.EH.T /˛/1=s
1
X
iD2
ip.i 1/˛=s C1;
where we used Markov’s inequality in the last step.
For each > 0, the inequality in the assumption of the lemma remains true if
H; M; and Zare multiplied by . Therefore, the inequality
E.Z.T //pexp¹KT c1.pr/=rºc2.pr/1=r
˛
sp.EH.T /˛/1=s
1
X
iD2
ip.i 1/˛=s Cp
follows. Optimizing the right hand side over > 0 yields the assertion of the
lemma.
Lemma 5.5 (S. Dereich). For m; d 2N,˛2.0; 1
2/and t0> 0 there exist some
universal strictly positive constants ciDci.d; m; ˛; t0/,iD1; 2; 3, such that for
Z.t/ D.t; W.t// 2RmC1with an Rm-valued Brownian motion W
PZ./
FdZ˛IŒ; uc1ec2u2=v2T
for u
v.T CT1˛/c3; T t0
for any pair of finite .Ft/-stopping times with Tand any .Ft/-
predictable RRdm-valued process .F.t// satisfying sups2Œ; jjjF.s/jjj v
P-almost surely.
Proof. It suffices to treat the case when D0and mDdD1, where we have to
deal with real-valued semimartingales of the form
t7! Zt
0
F.s/ dsDW A.t/ or t7! Zt
0
F.s/ dW.s/ DW M.t/
with integrands satisfying sups2Œ0;T jF.s/j valmost surely. The first case is
easy: the map t7! A.t/ is Lipschitz with constant (at most) vand therefore
kA./k˛IŒ0; v .T CT1˛/almost surely, so the claim follows in this case. Let
Bereitgestellt von | Technische Universität Berlin
Angemeldet
Heruntergeladen am | 27.11.17 15:58
Well-posedness of stochastic functional differential equations 283
us consider M. The Gaussian isoperimetric inequality, cf. e.g. [2, Section 4.3],
implies the existence of some universal positive constants kiDki.˛/; i D1; 2
such that
PW./˛IŒ0;1 uk1ek2u2for u0:
We choose an independent Brownian motion W0and let
F0.s/ Dqv2F2.s/:
Then both processes
t7! B.j /.t/ DZt
0
F.s/ dW.s/ .1/jZt
0
F0.s/ dW0.s/; j D1; 2;
have the same distribution as t7! vW.t/. From
B.1/.t/ CB.2/.t/ D2Zt
0
F.s/ dW.s/
and the triangle inequality in C˛one gets
PZ./
0
F.s/ dW.s/˛IŒ0; u2PvW./˛IŒ0;T u
2PW./˛IŒ0;1 u
vpT .T ˛_1/
2k1exp °k2
u2
v2T .T ˛_1/2±;
which yields the claim of the lemma.
Remark. Alternatively, the previous lemma can be proved using the fact that each
continuous local martingale starting at 0 can be represented as a time-changed
Brownian motion.
Acknowledgments. This work was partially supported by the DFG-Forschergrup-
pe 718 “Analysis and Stochastics in Complex Physical Systems”.
Bibliography
[1] L. A. Alyushina, Euler polygonal lines for Itô equations with monotone coefficients,
Teor. Veroyatnost. i Primenen. 32 (1987), 367–373.
[2] V. I. Bogachev, Gaussian Measures, American Mathematical Society, Providence,
R.I., 1998.
Bereitgestellt von | Technische Universität Berlin
Angemeldet
Heruntergeladen am | 27.11.17 15:58
284 M.-K. von Renesse and M. Scheutzow
[3] E. Buckwar, R. Kuske, S.-E. Mohammed and T. Shardlow, Weak convergence of the
Euler scheme for stochastic differential delay equations, LMS J. Comput. Math. 11
(2008), 60–99.
[4] E. Clément, A. Kohatsu-Higa and D. Lamberton, A duality approach for the weak
approximation of stochastic differential equations, Ann. Appl. Probab. 16 (2006),
1124–1154.
[5] P. E. Kloeden and A. Neuenkirch, The pathwise convergence of approximation
schemes for stochastic differential equations, LMS J. Comput. Math. 10 (2007), 235–
253.
[6] N. V. Krylov, A simple proof of the existence of a solution to the Ito equation with
monotone coefficients, Teor. Veroyatnost. i Primenen. 35 (1990), 576–580.
[7] N. V. Krylov, On Kolmogorov’s equations for finite-dimensional diffusions, in:
Stochastic PDE’s and Kolmogorov Equations in Infinite Dimensions (Cetraro, 1998),
Lecture Notes in Mathematics 1715, pp. 1–63, Springer, Berlin, 1999.
[8] U. Küchler and E. Platen, Strong discrete time approximation of stochastic differen-
tial equations with time delay, Math. Comput. Simulation 54 (2000), 189–205.
[9] X. Mao Stochastic Differential Equations and Applications, Horwood, Chichester,
1997.
[10] S. E. A. Mohammed, Stochastic Functional Differential Equations, Pitman, Boston,
1984.
[11] C. Prévôt and M. Röckner, A Concise Course on Stochastic Partial Differential
Equations, Lecture Notes in Mathematics 1905, Springer, Berlin, 2007.
[12] D. Xu, Z. Yang and Y. Huang, Existence–uniqueness and continuation theorems
for stochastic functional differential equations, J. Differential Equations 245 (2008),
1681–1703.
Received June 1, 2009; accepted November 4, 2009.
Author information
Max-K. von Renesse, Institute of Mathematics, Technische Universität Berlin, Germany.
E-mail: [email protected]
Michael Scheutzow, Institute of Mathematics, Technische Universität Berlin, Germany.
E-mail: [email protected]
Bereitgestellt von | Technische Universität Berlin
Angemeldet
Heruntergeladen am | 27.11.17 15:58
