Journal of Inequalities in Pure and
Applied Mathematics
A STOCHASTIC GRONWALL INEQUALITY AND ITS
APPLICATIONS
volume 6, issue 1, article 17,
2005.
KAZUO AMANO
Department of Mathematics
Faculty of Engineering
Gunma University
Kiryu, Tenjin 1-5-1
376-8515 Japan
EMail: kamano@math.sci.gunma-u.ac.jp
Received 15 December, 2004;
accepted 1 February, 2005.
Communicated by: S.S. Dragomir
Abstract
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2000
Victoria University
ISSN (electronic): 1443-5756
242-04
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Abstract
In this paper, we show a Gronwall type inequality for Itô integrals (Theorems 1.1
and 1.2) and give some applications. Our inequality gives a simple proof of the
existence theorem for stochastic differential equation (Example 2.1) and also,
the error estimate of Euler-Maruyama scheme follows immediately from our
result (Example 2.2).
A Stochastic Gronwall
Inequality and Its Applications
2000 Mathematics Subject Classification: 26D10, 26D20, 60H05, 60H35
Key words: Gronwall inequality, Itô integral
Kazuo Amano
Contents
Title Page
1
A Stochastic Gronwall type inequality . . . . . . . . . . . . . . . . . . .
2
Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References
3
7
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J. Ineq. Pure and Appl. Math. 6(1) Art. 17, 2005
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1.
A Stochastic Gronwall type inequality
Let w(t), t ≥ 0 be a standard Brownian motion on a probability space (Ω, F, P )
and Ft , t ≥ 0 be the natural filtration of F. For a positive number T , Mw2 [0, T ]
denotes the set of all separable nonanticipative functions f (t) with respect to Ft
defined on [0, T ] satisfying
Z T
2
E
f (t) dt < ∞.
0
Theorem 1.1. Assume that ξ(t) and η(t) belong to Mw2 [0, T ]. If there exist
functions a(t) and b(t) belonging to Mw2 [0, T ] such that
Z t
Z t
(1.1)
|ξ(t)| ≤ a(s) ds +
b(s) dw(s)
0
0
and if there are nonnegative constants α0 , α1 , β0 and β1 such that
(1.2)
|a(t)| ≤ α0 |η(t)| + α1 |ξ(t)|,
|b(t)| ≤ β0 |η(t)| + β1 |ξ(t)|
for 0 ≤ t ≤ T , then we have
(1.3)
√
Eξ 2 (t) ≤ 4 α0 t + β0
2
Z t
√
2
exp 4t (α1 t + β1 )
Eη 2 (s) ds
A Stochastic Gronwall
Inequality and Its Applications
Kazuo Amano
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for 0 ≤ t ≤ T .
Page 3 of 11
Proof. Since
Z
E
0
t
2
Z t
b(s) dw(s) = E
b2 (s) ds,
0
J. Ineq. Pure and Appl. Math. 6(1) Art. 17, 2005
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(1.1) implies, by Minkowsky and Schwarz inequalities,
1
Eξ (t) 2 ≤
2
Z t
12 Z t
12
2
2
t
Ea (s) ds
+
Eb (s) ds .
0
0
Direct computation gives, by (1.2),
Z t
21
Z t
12
Z t
12
√
√
t
Ea2 (s) ds
≤ 2tα0
Eη 2 (s) ds
+ 2tα1
Eξ 2 (s) ds ,
0
Z
0
0
21
Z t
12
Z t
12
t
√
√
2
2
2
Eb (s) ds
≤ 2β0
Eη (s) ds
+ 2β1
Eξ (s) ds .
0
0
A Stochastic Gronwall
Inequality and Its Applications
Kazuo Amano
0
Combining the above estimates, we obtain
Z t
Z t
√
√
2
2
2
2
(1.4) Eξ (t) ≤ 4(α0 t + β0 )
Eη (s) ds + 4(α1 t + β1 )
Eξ 2 (s) ds
0
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0
for 0 ≤ t ≤ T .
Let us fix a nonnegative number t0 ≤ T arbitrarily. Then, for any δ > 0, the
last inequality (1.4) shows
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√
d
log 4(α0 t0 + β0 )2
dt
Z
0
t0
√
Eη 2 (s) ds + 4(α1 t0 + β1 )2
Z
t
Eξ 2 (s) ds + δ
0
√
≤ 4(α1 t0 + β1 )2
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almost everywhere in [0, t0 ]. Integrating this estimate from 0 to t0 with respect
to t, we get
!
Rt
Rt
√
√
4(α0 t0 + β0 )2 0 0 Eη 2 (s) ds + 4(α1 t0 + β1 )2 0 0 Eξ 2 (s) ds + δ
log
Rt
√
4(α0 t0 + β0 )2 0 0 Eη 2 (s) ds + δ
√
≤ 4t0 (α1 t0 + β1 )2 .
Therefore, by (1.4), we have
2
√
2
Eξ (t0 ) ≤ exp 4t0 (α1 t0 + β1 )
√
2
Z
t0
4(α0 t0 + β0 )
2
Eη (s) ds + δ .
A Stochastic Gronwall
Inequality and Its Applications
Kazuo Amano
0
Now, letting δ → 0, we obtain (1.3).
In case ξ(t) is a step function, a weak assumption (1.6) will be enough to
show the inequality (1.3), which would play an important role in the error analysis of the numerical solutions of stochastic differential equations.
Theorem 1.2. Assume that ξ(t) and η(t) belong to
function such that
Mw2 [0, T ]
and ξ(t) is a step
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(1.5)
ξ(t) = ξ(tn )
when tn ≤ t < tn+1
Close
{tn }N
n=0
for n = 0, 1, 2, . . . , N − 1, where N is a positive integer and
is a
partition of the interval [0, T ] satisfying 0 = t0 < t1 < t2 < · · · < tN −1 <
tN = T . If there exist functions a(t) and b(t) belonging to Mw2 [0, T ] such that
Z tn
Z tn
(1.6)
|ξ(tn )| ≤ a(s) ds +
b(s) dw(s)
0
0
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J. Ineq. Pure and Appl. Math. 6(1) Art. 17, 2005
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is valid for each n = 0, 1, 2, . . . , N and if there are nonnegative constants α0 ,
α1 , β0 and β1 satisfying (1.2) for 0 ≤ t ≤ T , then we have (1.3) for 0 ≤ t ≤ T .
Proof. As in the proof of Theorem 1.1, we have
Z tn
Z
√
√
2
2
2
2
Eξ (tn ) ≤ 4(α0 tn + β0 )
Eη (s) ds + 4(α1 tn + β1 )
0
tn
Eξ 2 (s) ds
0
for n = 0, 1, 2, . . . , N ; this implies, by (1.5),
Z t
Z t
√
√
2
2
2
2
Eξ (t) ≤ 4(α0 t + β0 )
Eη (s) ds + 4(α1 t + β1 )
Eξ 2 (s) ds
0
0
for 0 ≤ t ≤ T .
The remaining part of the proof is exactly same as that of Theorem 1.1.
A Stochastic Gronwall
Inequality and Its Applications
Kazuo Amano
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J. Ineq. Pure and Appl. Math. 6(1) Art. 17, 2005
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2.
Applications
Throughout this section, we assume that ξ(t) ∈ Mw2 [0, T ] is a solution of the
stochastic differential equation
dξ(t) = a(t, ξ(t)) dt + b(t, ξ(t)) dw(t),
0≤t≤T
satisfying the initial condition ξ(0) = ξ0 , where a(t, x) and b(t, x) are realvalued functions defined in [0, T ] such that
|a(t, x)|, |b(t, x)| ≤ K(1 + |x|),
|a(t, x) − a(s, y)|, |b(t, x) − b(s, y)| ≤ L(|t − s| + |x − y|).
A Stochastic Gronwall
Inequality and Its Applications
Kazuo Amano
Here K and L are nonnegative constants.
Example 2.1. Theorem 1.1 gives a simple proof of the existence theorem for
stochastic differential equations.
We use Picard’s method. Let us consider a sequence {ξn (t)} defined by
ξ0 (t) = ξ0 and
Z t
Z t
ξn+1 (t) = ξ0 +
a(s, ξn (s)) ds +
b(s, ξn (s)) dw(s)
0
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for n = 0, 1, 2, . . . . Then, we easily have
Z t
ξn+1 (t) − ξn (t) =
a(s, ξn (s)) − a(s, ξn−1 (s)) ds
0
Z t
+
b(s, ξn (s)) − b(s, ξn−1 (s)) dw(s)
0
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and the Lipschitz continuity of a(t, x) and b(t, x) implies
a(s, ξn (s)) − a(s, ξn−1 (s)) ≤ L|ξn (s) − ξn−1 (s)|,
b(s, ξn (s)) − b(s, ξn−1 (s)) ≤ L|ξn (s) − ξn−1 (s)|.
Hence, Theorem 1.1 with α0 = β0 = L and α1 = β1 = 0 shows
Z t
√
2
2
2
E|ξn (s) − ξn−1 (s)|2 ds
E|ξn+1 (t) − ξn (t)| ≤ 4L ( t + 1)
0
A Stochastic Gronwall
Inequality and Its Applications
for n = 1, 2, 3, . . . ; the recursive use of this estimate gives
√
2
2 n
4L
(
t
+
1)
t
E |ξn+1 (t) − ξn (t)|2 ≤
sup E|ξ1 (s) − ξ0 (s)|2 .
n!
0≤s≤t
Consequently, as is well-known, the convergence of {ξn (t)} follows.
By virtue of Theorem 1.1 with α0 = β0 = 0 and α1 = β1 = L, the uniqueness of the solution is clear.
Example 2.2. The error estimate of the Euler-Maruyama scheme
ξn+1 = ξn + a(tn , ξn )∆t + b(tn , ξn )∆wn ,
n = 0, 1, 2, . . . , N − 1
follows immediately from Theorem 1.2, where N is a sufficiently large positive integer, ∆t = T /N , tn = n∆t and ∆wn = w(tn+1 ) − w(tn ) for
n = 0, 1, 2, . . . , N − 1.
Kazuo Amano
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Since
Z
tn+1
ξn+1 = ξn +
Z
tn+1
a(tn , ξn )ds +
tn
b(tn , ξn )dw(s),
tn
Z
tn+1
ξ(tn+1 ) = ξ(tn ) +
Z
tn+1
a(s, ξ(s))ds +
tn
b(s, ξ(s))dw(s),
tn
we have
Z
tn+1
ξn+1 − ξ(tn+1 ) = ξn − ξ(tn ) +
tn
a(tn , ξn ) − a(s, ξ(s)) ds
Z tn+1
+
b(tn , ξn ) − b(s, ξ(s)) dw(s).
tn
Now, for n = 0, 1, 2, . . . , N − 1, if we put
ε(s) = ξn − ξ(tn ),
f (s) = a(tn , ξn ) − a(s, ξ(s)),
g(s) = b(tn , ξn ) − b(s, ξ(s))
when tn ≤ s < tn+1 and
ε(tN ) = ξN − ξ(tN ),
f (tN ) = a(tN , ξN ) − a(tN , ξ(tN )),
g(tN ) = b(tN , ξN ) − b(tN , ξ(tN )),
A Stochastic Gronwall
Inequality and Its Applications
Kazuo Amano
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then we obtain
Z
ε(tn ) =
tn
Z
tn
f (s) ds +
0
g(s) dw(s)
0
for n = 0, 1, 2, . . . , N . The Lipschitz continuity of a(t, x) and b(t, x) shows
˜
|f (s)| ≤ L ∆t + |ξ(s) − ξ(s)|
+ |ε(s)| ,
˜
|g(s)| ≤ L ∆t + |ξ(s) − ξ(s)|
+ |ε(s)| ,
˜
where ξ(s)
= ξ(tn ) when tn ≤ s < tn+1 for n = 0, 1, 2, . . . , N − 1 and
˜
ξ(tN ) = ξ(tN ) . Hence, Theorem 1.2 with α0 = α1 = β0 = β1 = L shows
Z
√
√
t
2
2
2
2
2
2
˜
Eε (t) ≤ 4L ( t + 1) exp 4L ( t + 1) t
E ∆t + |ξ(s) − ξ(s)|
ds.
0
It follows from the fundamental property of Itô integrals that
˜ 2 ≤ C∆t,
E |ξ(s) − ξ(s)|
where C is a nonnegative constant depending only on T , K and L. Combining
the above estimates, we obtain
Eε2 (t) = O(∆t)
A Stochastic Gronwall
Inequality and Its Applications
Kazuo Amano
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for any 0 ≤ t ≤ T when ∆t → 0; the error estimate of the Euler-Maruyama
scheme is proved.
Our Gronwall type inequality works for other numerical solutions of stochastic differential equations.
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References
[1] A. FRIEDMAN, Stochastic Differential Equations and Applications, Volume I, Academic Press, 1975.
[2] K. ITÔ AND H. P. MCKEAN, Diffusion Processes and Their Sample Paths,
Springer-Verlag, Berlin, 1965.
[3] P.E. KLOEDEN AND E. PLATEN, Numerical Solution of Stochastic Differential Equations, Springer-Verlag, 1992.
[4] D.W. STROOCK AND S.R.S. VARADHAN, Multidimensional Diffusion
Processes, Springer, 1982.
A Stochastic Gronwall
Inequality and Its Applications
Kazuo Amano
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