Table Of ContentA DISCRETE STOCHASTIC GRONWALL LEMMA
RAPHAELKRUSEANDMICHAELSCHEUTZOW
Abstract. We derive a discrete version of the stochastic Gronwall Lemma
found in [Scheutzow, IDAQP, 2013]. The proof is based on a corresponding
6 deterministicversionofthediscreteGronwallLemmaandaninequalitybound-
1 ingthesupremumintermsoftheinfimumfortimediscretemartingales. Asan
0 applicationtheproofofanaprioriestimateforthebackwardEuler-Maruyama
2 methodisincluded.
n
a
J
7 1. Introduction
2
The Gronwall Lemma is an often used tool in classical analysis for deriving a
] prioriandstabilityestimatesofsolutionstodifferentialequations. Itisnamedafter
R
T. H. Gr¨onwall and originated in its differential form from his work [5]. Besides
P
the integralversionin [3] many more variations of the GronwallLemma have been
.
h
introduced with a wide area of applications, for example, in ordinary differential
t
a equations,partialdifferentialequations,integralequations,andstochasticanalysis.
m
Similarly, discrete versions of the Gronwall Lemma are often applied in order to
[
estimate the growth of solutions to time discrete difference equations, such as nu-
1 merical approximations of differential equations. For instance, we refer to [4] and
v the references therein. The purpose of this paper is the derivation of the following
3
time discrete version of the stochastic Gronwall Lemma from [8]:
0
5
7 Theorem 1. Let (Mn)n∈N0 be an (Fn)n∈N0-martingale satisfying M0 =0 on a fil-
0 tered probability space (Ω,F,(Fn)n∈N0,P). Let (Xn)n∈N0, (Fn)n∈N0, and (Gn)n∈N0
1. be sequences of nonnegative and adapted random variables with E[X0] < ∞ such
0 that
6
n−1
1 (1) X ≤F +M + G X , for all n∈N .
: n n n k k 0
v Xk=0
i
X Then, for any p∈(0,1) and µ,ν ∈[1,∞] with 1 + 1 =1 and pν <1, it holds true
µ ν
r that
a
1 1 n−1
(2) E sup Xp ≤ 1+ ν (1+G )p E sup F p
(cid:2)0≤k≤n k(cid:3) (cid:16) 1−νp(cid:17) (cid:13)(cid:13)kY=0 k (cid:13)(cid:13)Lµ(Ω)(cid:0) (cid:2)0≤k≤n k(cid:3)(cid:1)
(cid:13) (cid:13)
for alln∈N0. Inparticular, if(Gn)n∈N0 is adeterministic sequenceofnonnegative
real numbers, then for any p∈(0,1) it holds true that
n−1
(3) E sup Xp ≤ 1+ 1 (1+G )p E sup F p
(cid:2)0≤k≤n k(cid:3) (cid:16) 1−p(cid:17)(cid:16)kY=0 k (cid:17)(cid:0) (cid:2)0≤k≤n k(cid:3)(cid:1)
for all n∈N .
0
2010 Mathematics Subject Classification. Primary: 60G46Secondary: 26D1560G4265C30.
Key words and phrases. discrete stochastic Gronwall Lemma, martingale inequality, a priori
estimate,backwardEuler-Maruyamamethod.
1
2 R.KRUSEANDM.SCHEUTZOW
The main novelty of Theorem 1 and its continuous time counter-part in [8] is
the presence of a martingale term on the right hand side of Equation (1). In this
situationdeterministicversionsoftheGronwallLemmausuallyrequiretofirsttake
expectation in Equation (1) in order to discard the centered martingale from the
inequality. However, this line of arguments then often results in weaker estimates
in the sense that taking the supremum with respect to k would occur outside the
expectation on the left hand side of Equations (2) and (3).
We emphasize that the estimates in Equations (2) and (3) are uniform with
respect to the martingale (Mn)n∈N0. The price we have to pay for this uniformity
is the restriction of the parameter p to the interval (0,1). As already indicated
in Remark 3 in [8] the martingale inequality in Lemma 3 cannot be extended to
p≥1. Instead one could try to apply, for instance, Burkholder-Davis-Gundy-type
inequalitiesresultingintheappearanceofthequadraticvariationofthemartingale
on the right hand side of the estimates.
In addition, it is worth to take note of the following subtle difference between
Theorem 1 and its continuous time counter-part in [8]: On the right hand side of
Equations(2)and(3)wehavethep-thpoweroftheexpectationofsup F . In
0≤k≤n k
[8,Theorem4]theorderofthep-thpowerandtheexpectationisreversedresulting
inasharperestimate. Thereasonforthisdifferenceliesinthemartingaleinequality
inLemma 3whichfordiscrete time martingalesonly holdstrue inthe weakerform
used in this paper. Compare further with [8, Remark 3].
TheproofofTheorem1ismostlybasedontwoingredients: Thefirstisadiscrete
version of the classical Gronwall Lemma which is found in Lemma 2 below. The
second ingredient is an inequality stated in Lemma 3 that relates the Lp-norm,
p∈(0,1), of the supremum of a time discrete martingale to its infimum. Lemma 3
therefore is the discrete time counter-part of [8, Proposition 1]. A further version
of the latter with optimal constant is also found in [2]. For all details of the proof
we refer to Section 2.
As already mentioned, discrete versions of the Gronwall Lemma are often used
in order to derive a priori estimates for numerical approximations of differential
equations. TothisendwedemonstrateinSection3howTheorem1canbeappliedin
ordertoestimatetheLp-norm,p∈(0,2),ofthebackwardEuler-Maruyamamethod
for stochastic differential equations under rather mild conditions on the coefficient
functions, namely continuity and a global coercivity condition (see Equation (11)
below).
Notation: Throughoutthis paper we use the conventionthat sums overempty
index sets are equal to zero and products over empty index sets are equal to one.
Further, we let N := {1,2,...} denote the set of all positive integers and define
N :=N∪{0}. As usual, we write a∧b:= min(a,b) and a∨b:= max(a,b) for all
0
a,b∈R. Finally, for an arbitrary sequence (Fn)n∈N0 of random variables we set
F∗ := sup F .
n k
0≤k≤n
2. Proof of the discrete stochastic Gronwall Lemma
In this section we present a proof of Theorem 1. As already indicated in the
introduction, we first state a corresponding deterministic version of the discrete
GronwallLemma. For completeness we include a proofbased on a presentationby
A DISCRETE STOCHASTIC GRONWALL LEMMA 3
JohnM. Holte1. Then, we derive a discrete time versionof a martingaleinequality
from [8] that gives a bound for the supremum of the martingale in terms of its
infimum.
Lemma 2. Consider real-valued sequences (fn)n∈N0, (gn)n∈N0, and (yn)n∈N0. As-
sume that (gn)n∈N0 is nonnegative. If we have
n−1
(4) y ≤f + g y , for all n∈N ,
n n k k 0
X
k=0
then it also holds true that
n−1 n−1
(5) y ≤f + f g (1+g ), for all n∈N .
n n k k j 0
X Y
k=0 j=k+1
Proof. Obviously, the assertion is true for n = 0. Now let n > 0 and assume that
(5) is satisfied for all 0 ≤ k < n. Then, by inserting (5) into (4) for all k < n we
obtain
n−1
y ≤f + g y
n n k k
X
k=0
n−1 k−1 k−1
≤f + g f + f g (1+g )
n k k i i j
X (cid:16) X Y (cid:17)
k=0 i=0 j=i+1
n−1 n−1k−1 k−1
=f + g f + f g g (1+g )
n k k i i k j
X XX Y
k=0 k=0i=0 j=i+1
n−1 n−1 n−1 k−1
=f + g f + f g g (1+g )
n k k i i k j
X X X Y
k=0 i=0 k=i+1 j=i+1
n−1 n−1 i−1
=f + f g 1+ g (1+g ) .
n k k i j
X (cid:16) X Y (cid:17)
k=0 i=k+1 j=k+1
Thus, it suffices to show that
n−1 i−1 n−1
(6) 1+ g (1+g )= (1+g ), for all 0≤k ≤n−1.
i j j
X Y Y
i=k j=k j=k
But this follows from a telescopic sum argument as follows:
n−1 n−1 i−1 i
1− (1+g )= (1+g )− (1+g )
j j j
Y X(cid:16)Y Y (cid:17)
j=k i=k j=k j=k
n−1 i−1
= 1−(1+g ) (1+g )
i j
Xi=k(cid:16)(cid:0) (cid:1)jY=k (cid:17)
n−1 i−1
=− g (1+g ).
i j
X Y
i=k j=k
Rearranging the terms yields (6) and completes the proof. (cid:3)
Next, we introduce the discrete time counter-part of Proposition 1 in [8].
1http://homepages.gac.edu/~holte/publications/gronwallTALK.pdf
4 R.KRUSEANDM.SCHEUTZOW
Lemma 3. Let (Mn)n∈N0 be an (Fn)n∈N0-martingale with M0 = 0. Then, for
every p∈(0,1) and every n∈N we have
0
(7) E[( sup M )p]≤ 1 E[− inf M ] p
k k
0≤k≤n 1−p(cid:0) 0≤k≤n (cid:1)
or, equivalently,
(8) E[(sup M )p]≤ 1 E[− inf M ] p.
k k
k∈N0 1−p(cid:0) k∈N0 (cid:1)
Proof. The equivalence of (7) and (8) follows at once from the monotone con-
vergence theorem and from stopping the martingale at n, respectively. Hence, it
suffices to prove (7).
Since M =0 we get that
0
0=E[M ]=E[(M ∨0)]−E[(−M )∨0], for all n∈N ,
n n n 0
and, consequently,
E[M ∨0]=E[(−M )∨0]≤E sup (−M ) =E − inf M
n n k k
(cid:2)0≤k≤n (cid:3) (cid:2) 0≤k≤n (cid:3)
for all n∈N . Next, for n∈N we define a mapping ϕ : [0,∞)→[0,1] by
0 0 n
ϕ (x)=P sup M ≥x , for all x≥0.
n k
(cid:0)0≤k≤n (cid:1)
Now, fix x>0 and n∈N arbitrarily and define the stopping times
0
τ :=inf m∈N : M ≥x}∧n.
n 0 m
(cid:8)
We set
M :=M
k k∧τn
f
and note that (Mk)k∈N0 is again an (Fk)k∈N0-martingale with M0 = M0 = 0. In
addition, we have E[M ∨0]=E[(−M )∨0]≤E[−inf M ] for all k ∈N
f k k 0≤ℓ≤(n∧k)fℓ 0
and the same n ∈ N as above. Furthermore, note that {sup M ≥ x} =
f0 f 0≤k≤n k
{sup M ≥x}={M ≥x} and, therefore,
0≤k≤n k n
f f E[M ∨0]≥xϕ (x).
n n
Altogether, this implies f
1 1
ϕ (x)≤ E[M ∨0]≤ E − inf M .
n n ℓ
x x (cid:2) 0≤ℓ≤n (cid:3)
f
Finally, we obtain for every p∈(0,1)
∞
E ( sup Mk)p =Z P sup Mk ≥x1p dx
(cid:2) 0≤k≤n (cid:3) 0 (cid:0)0≤k≤n (cid:1)
∞
≤Z0 (cid:0)x−p1E(cid:2)−0≤inℓ≤fnMℓ(cid:3)(cid:1)∧1dx
= E − inf M p+ p E − inf M p
ℓ ℓ
(cid:0) (cid:2) 0≤ℓ≤n (cid:3)(cid:1) 1−p(cid:0) (cid:2) 0≤ℓ≤n (cid:3)(cid:1)
= 1 E − inf M p,
ℓ
1−p(cid:0) (cid:2) 0≤ℓ≤n (cid:3)(cid:1)
which is the assertion. (cid:3)
A DISCRETE STOCHASTIC GRONWALL LEMMA 5
Remark 4. The constant 1 in Lemma 3 is (most likely) not sharp but at most
1−p
off from the optimal constant C by a factor of 4. This can be seen as follows:
p π
Let W: [0,∞)×Ω → R be a standard Wiener process and define the stopping
time τ = inf{s ≥ 0 : W(s) = −1}. Set M(t) := W(t∧τ ) and h = 2−k for
−1 −1 k
k ∈ N . Then, for every k ∈ N we obtain a discrete time martingale by setting
0 c
Mk :=M(nh ). From the continuity of the trajectories of the Wiener process and
n k
the monotone convergence theorem it follows that
c
∞
lim E supMk p =E supM(t) p = P supM(t)≥xp1 dx
k→∞ (cid:2)(cid:0)ℓ∈N0 ℓ(cid:1) (cid:3) (cid:2)(cid:0)t≥0 (cid:1) (cid:3) Z0 (cid:0)t≥0 (cid:1)
= ∞ 1+x1/p −1dx=cπp = πp limcE − inf Mk p .
Z0 (cid:0) (cid:1) sin(πp) sin(πp)k→∞ (cid:2)(cid:0) ℓ∈N0 ℓ(cid:1) (cid:3)
Hence,
πp 1
≤C ≤ .
p
sin(πp) 1−p
TheratioR := 1 sin(πp) iseasilyseentoobtainitsmaximumvalue 4 atp=1/2.
p 1−p πp π
Note that
limR =limR =1,
p p
p↓0 p↑1
so the constant 1 in Lemma 3 becomes optimal in the limits p→0 and p→1.
1−p
Now we are well-prepared for the proof of Theorem 1:
Proof of Theorem 1. We first apply Lemma 2 ω-wise and obtain
n−1 n−1
X ≤F +M + F +M G (1+G )
n n n k k k j
Xk=0(cid:0) (cid:1) j=Yk+1
n−1 n−1 n−1 n−1
=F + F G (1+G )+M + M G (1+G ).
n k k j n k k j
X Y X Y
k=0 j=k+1 k=0 j=k+1
Now, since F ≤F∗ for all k≤n we have
k n
n−1 n−1 n−1 n−1
F + F G (1+G )≤F∗ 1+ G (1+G )
n k k j n k j
X Y (cid:16) X Y (cid:17)
k=0 j=k+1 k=0 j=k+1
n−1
=F∗ (1+G ),
n j
Y
j=0
where we applied (6). Moreover,it holds true that
n−1 n−1 n−1 n−1
M G (1+G )= M (1+G −1) (1+G )
k k j k k j
X Y X Y
k=0 j=k+1 k=0 j=k+1
n−1 n−1 n−1
= M (1+G )− (1+G ) .
k j j
X (cid:16)Y Y (cid:17)
k=0 j=k j=k+1
Hence, since M =0 we get by summation by parts
0
n−1 n−1 n−1 n−1
M + M G (1+G )= M −M (1+G )
n k k j k+1 k j
kX=0 j=Yk+1 kX=0(cid:0) (cid:1)j=Yk+1
n−1
=L (1+G ),
n i
Y
i=0
6 R.KRUSEANDM.SCHEUTZOW
where
n−1 k
L := (M −M ) (1+G )−1
n k+1 k j
X Y
k=0 j=0
is a further (Fn)n∈N0-martingale. Altogether, we have shown that
n−1
(9) X ≤ F∗+L (1+G ).
n n n i
(cid:0) (cid:1)iY=0
Hence, Ho¨lder’s inequality with 1= 1 + 1 yields
µ ν
n−1
E sup Xp ≤ (1+G )p (F∗+L∗)p
(cid:2)0≤k≤n k(cid:3) (cid:13)(cid:13)iY=0 i (cid:13)(cid:13)Lµ(Ω)(cid:13)(cid:13) n n (cid:13)(cid:13)Lν(Ω)
(cid:13) (cid:13)
n−1 1
≤ (1+G )p E (F∗)νp +E (L∗)νp ν.
i n n
(cid:13)(cid:13)iY=0 (cid:13)(cid:13)Lµ(Ω)(cid:16) (cid:2) (cid:3) (cid:2) (cid:3)(cid:17)
(cid:13) (cid:13)
Moreover,sinceX ≥0itfollowsfrom(9)that−L ≤F∗foralln∈N . Therefore,
n n n 0
we have −inf L ≤F∗. Thus, after applying the martingale inequality from
0≤k≤n k n
Lemma 3 to E (L∗)νp we conclude
n
(cid:2) (cid:3)
E[ sup Xp]≤ n−1(1+G )p E (F∗)νp + 1 E F∗ νp ν1.
0≤k≤n k (cid:13)(cid:13)iY=0 i (cid:13)(cid:13)Lµ(Ω)(cid:16) (cid:2) n (cid:3) 1−νp(cid:0) (cid:2) n(cid:3)(cid:1) (cid:17)
(cid:13) (cid:13)
An application of Jensen’s inequality completes the proof of Equation (2). The
proof of Equation (3) follows from the same steps but with µ=∞. (cid:3)
3. Application to numerical schemes
In this section we prove an a priori estimate for the backward Euler-Maruyama
approximation of solutions to stochastic differential equations, whose coefficient
functions satisfy a coercivity condition.
To be more precise let T > 0 and d,m ∈ N. Consider the stochastic ordinary
differential equation
dX(t)=f(X(t))dt+g(X(t))dW(t), t∈[0,T],
(10)
X(0)=X ,
0
where f: Rd → Rd and g: Rd → Rd×m denote the drift and diffusion coefficient
functions, respectively. Further, W: [0,T]×Ω→Rm is a standardWiener process
on a filtered probability space (Ω,F,(F ) ,P). For simplicity, let the initial
t t∈[0,T]
condition X ∈Rd be deterministic.
0
We assume that f and g are continuous and satisfy the following coercivity
condition: There exists L≥0 such that
1
(11) hf(x),xi+ |g(x)|2 ≤L 1+|x|2
2
(cid:0) (cid:1)
for all x∈Rd, where we let |·| denote the Euclidean norms on Rd and Rm as well
as the Frobenius norm if applied to matrices from Rd×m.
An often considered numerical method for the approximation of the solution X
to (10) is the backward Euler-Maruyama method, see for instance [6, 7], given by
Yj+1 =Yj +hf(Yj+1)+g(Yj)∆ Wj+1, j =1,...,N ,
h h
(12)
Y0 =X ,
0
A DISCRETE STOCHASTIC GRONWALL LEMMA 7
where h ∈ (0,1) denotes the equidistant step size and N ∈ N is determined
h
by N h ≤ T < (N + 1)h. The stochastic increment is given by ∆ Wj+1 =
h h h
W(t )−W(t ), where t =jh.
j+1 j j
Ouraimistoprovethefollowingaprioriestimate on(Yj)Nh ,whichisasharper
j=0
version of Theorem 4.2 in [1] in the sense that taking the supremum now occurs
inside the expectation but only with respect to the L2p-norm for p∈(0,1).
Proposition 5. Let h ∈ (0,(2L)−1) denote an upper step size bound. For every
0
p ∈ (0,1) and for every (Fnh)n∈N0-adapted process (Yn)n∈N0 satisfying (12) with
h∈(0,h ) we have
0
E sup |Yj|2+h|g(Yj)|2 p ≤ 1+ 1 exp p(1−2h L)−12LT
0
(cid:2)0≤j≤Nh(cid:0) (cid:1) (cid:3) (cid:16) 1−p(cid:17) (cid:0) (cid:1)
p
× |X |2+(1−2h L)−1 h |g(X )|2+2LT .
0 0 0 0
(cid:16) (cid:0) (cid:1)(cid:17)
In particular, this bound is independent of the step size h.
Proof. Let(Yn)n∈N0 beanadaptedprocesssatisfying(12)withstepsizeh∈(0,h0).
For every j ∈ {0,...,N −1} we get from the polarization identity ha−b,ai =
h
1(|a|2−|b|2+|a−b|2), which is valid for all a,b∈Rd, that
2
|Yj+1|2−|Yj|2+|Yj+1−Yj|2 =2hYj+1−Yj,Yj+1i
=2hhf(Yj+1),Yj+1i+2hg(Yj)∆ Wj+1,Yj+1i,
h
sinceYj+1 satisfies(12). Now,anapplicationofthecoercivitycondition(11)yields
|Yj+1|2−|Yj|2+|Yj+1−Yj|2
(13) ≤2hL 1+|Yj+1|2 −h|g(Yj+1)|2+2hg(Yj)∆ Wj+1,Yj+1−Yji
h
(cid:0) (cid:1)
+2hg(Yj)∆ Wj+1,Yji.
h
From the Cauchy-Schwarzand Young inequalities we deduce
2hg(Yj)∆ Wj+1,Yj+1−Yji≤|g(Yj)∆ Wj+1|2+|Yj+1−Yj|2.
h h
Notethatthe secondtermalsoappearsonthe lefthandside ofthe inequality (13).
After cancelling and some rearranging we therefore get
(14) |Yj+1|2+h|g(Yj+1)|2 ≤|Yj|2+h|g(Yj)|2+2hL 1+|Yj+1|2 +Zj+1,
(cid:0) (cid:1)
where
(15) Zj+1 :=|g(Yj)∆ Wj+1|2−h|g(Yj)|2+2hg(Yj)∆ Wj+1,Yji.
h h
By iterating the inequality we arrive at
n−1 n−1
|Yn|2+h|g(Yn)|2 ≤|Y0|2+h|g(Y0)|2+2hL 1+|Yj+1|2 + Zj+1,
Xj=0(cid:0) (cid:1) Xj=0
or, equivalently,
(1−2hL)|Yn|2+h|g(Yn)|2
(16) n−1 n−1
≤(1−2hL)|Y0|2+h|g(Y0)|2+2Lt + Zj+1+2hL |Yj|2.
n
X X
j=0 j=0
8 R.KRUSEANDM.SCHEUTZOW
Next, note that 1 ≥ (1−2hL) ≥ (1−2h L) > 0. From this we finally obtain the
0
relationship
n−1
X ≤F +M + G X , for all n∈N ,
n n n j j 0
X
j=0
where
X :=|Yn|2+h|g(Yn)|2,
n
F :=|Y0|2+(1−2h L)−1 h |g(Y0)|2+2Lt ,
n 0 0 n
(cid:0) (cid:1)
n−1
M :=(1−2h L)−1 Zj+1,
n 0
X
j=0
G :=(1−2h L)−12hL,
n 0
foralln∈N0. Clearly,theprocesses(Xn)n∈N0,(Fn)n∈N0,and(Gn)n∈N0 satisfythe
assumptionsofTheorem1. Henceitremainstoshowthat(Mn)n∈N0 isamartingale
with respect to the filtration (Ftn)n∈N0.
Forthisfirstnotethat(Mn)n∈N0 isadaptedandsatisfiesM0 =0. Then,weshow
inductively that M = (1−2h L)−1 n−1Zj+1 as well as the random variables
n 0 j=0
|Yn|2, |g(Yn)|2 are integrable: For n =P0 this is evident. Assume now that |Yj|2,
|g(Yj)|2 areintegrableforall0≤j <n. Then,from(15),the Cauchy-Schwarzand
the Young inequality, we obtain the estimate
E |Zn| ≤2E |g(Yn−1)∆ Wn|2 +hE |g(Yn−1)|2 +E |Yn−1|2 .
h
(cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3)
The first term is bounded by the Ito¯ isometry by
2E |g(Yn−1)∆ Wn|2 =2hE |g(Yn−1)|2 ,
h
(cid:2) (cid:3) (cid:2) (cid:3)
since Yn−1 is independent of ∆ Wn. The latter two terms are bounded by the
h
inductionhypothesis. Altogether,thisshowsthatZn and,hence,M areintegrable
n
random variables. Further, we have
E Zn =0.
(cid:2) (cid:3)
Thus, taking expectation in (16) yields that |Yn|2, |g(Yn)|2 are also integrable.
Finally, as above we get
E Zn|F =0,
tn−1
(cid:2) (cid:3)
which proves the martingale property for (Mn)n∈N0. Theorem 1 is therefore appli-
cable and yields the assertion (together with the inequality 1+x≤ex). (cid:3)
Acknowledgement
Thefirstauthorgratefullyacknowledgesfinancialsupportbytheresearchcenter
Matheon.
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RaphaelKruse,Technische Universita¨t Berlin, Institut fu¨r Mathematik,Secr. MA
5-3,Straßedes 17.Juni136,DE-10623Berlin,Germany
E-mail address: [email protected]
MichaelScheutzow, Technische Universita¨t Berlin,Institut fu¨rMathematik,Secr.
MA7-5,Straßedes 17.Juni136,DE-10623Berlin,Germany
E-mail address: [email protected]
