volume 6, issue 1, article 17, 2005.

*Received 15 December, 2004;*

*accepted 1 February, 2005.*

*Communicated by:**S.S. Dragomir*

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**Journal of** **Inequalities in** **Pure and** **Applied** **Mathematics**

**A STOCHASTIC GRONWALL INEQUALITY AND ITS**
**APPLICATIONS**

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

c

2000Victoria University ISSN (electronic): 1443-5756 242-04

**A Stochastic Gronwall**
**Inequality and Its Applications**

Kazuo Amano

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**J. Ineq. Pure and Appl. Math. 6(1) Art. 17, 2005**

**Abstract**

In this paper, we show a Gronwall type inequality for Itô integrals (Theorems1.1 and1.2) and give some applications. Our inequality gives a simple proof of the existence theorem for stochastic differential equation (Example2.1) and also, the error estimate of Euler-Maruyama scheme follows immediately from our result (Example2.2).

*2000 Mathematics Subject Classification:*26D10, 26D20, 60H05, 60H35
*Key words: Gronwall inequality, Itô integral*

**Contents**

1 A Stochastic Gronwall type inequality. . . 3 2 Applications. . . 7

References

**A Stochastic Gronwall**
**Inequality and Its Applications**

Kazuo Amano

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**1.** **A Stochastic Gronwall type inequality**

Letw(t),t ≥0be a standard Brownian motion on a probability space(Ω,F, P)
andF_{t},t ≥0be the natural filtration ofF. For a positive numberT,M_{w}^{2}[0, T]
denotes the set of all separable nonanticipative functionsf(t)with respect toF_{t}
defined on[0, T]satisfying

E Z T

0

f^{2}(t)dt

<∞.

* Theorem 1.1. Assume that* ξ(t)

*and*η(t)

*belong to*M

_{w}

^{2}[0, T]. If there exist

*functions*a(t)

*and*b(t)

*belonging to*M

_{w}

^{2}[0, T]

*such that*

(1.1) |ξ(t)| ≤

Z t

0

a(s)ds+ Z t

0

b(s)dw(s)

*and if there are nonnegative constants*α_{0}*,*α_{1}*,*β_{0}*and*β_{1}*such that*
(1.2) |a(t)| ≤α_{0}|η(t)|+α_{1}|ξ(t)|, |b(t)| ≤β_{0}|η(t)|+β_{1}|ξ(t)|

*for* 0≤t ≤T*, then we have*
(1.3) Eξ^{2}(t)≤4 α_{0}√

t+β_{0}2

exp

4t(α_{1}√

t+β_{1})^{2}Z t
0

Eη^{2}(s)ds
*for* 0≤t ≤T*.*

*Proof. Since*

E Z t

0

b(s)dw(s) 2

=E Z t

0

b^{2}(s)ds,

**A Stochastic Gronwall**
**Inequality and Its Applications**

Kazuo Amano

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**J. Ineq. Pure and Appl. Math. 6(1) Art. 17, 2005**

(1.1) implies, by Minkowsky and Schwarz inequalities,
Eξ^{2}(t)^{1}_{2}

≤

t Z t

0

Ea^{2}(s)ds
^{1}_{2}

+ Z t

0

Eb^{2}(s)ds
^{1}_{2}

. Direct computation gives, by (1.2),

t

Z t

0

Ea^{2}(s)ds
^{1}_{2}

≤√
2tα_{0}

Z t

0

Eη^{2}(s)ds
^{1}_{2}

+√
2tα_{1}

Z t

0

Eξ^{2}(s)ds
^{1}_{2}

, Z t

0

Eb^{2}(s)ds
^{1}_{2}

≤√
2β_{0}

Z t

0

Eη^{2}(s)ds
^{1}_{2}

+√
2β_{1}

Z t

0

Eξ^{2}(s)ds
^{1}_{2}

. Combining the above estimates, we obtain

(1.4) Eξ^{2}(t)≤4(α_{0}√

t+β_{0})^{2}
Z t

0

Eη^{2}(s)ds+ 4(α_{1}√

t+β_{1})^{2}
Z t

0

Eξ^{2}(s)ds
for 0≤t ≤T.

Let us fix a nonnegative numbert_{0} ≤T arbitrarily. Then, for anyδ > 0, the
last inequality (1.4) shows

d dtlog

4(α0

√t0+β0)^{2}
Z t0

0

Eη^{2}(s)ds+ 4(α1

√t0+β1)^{2}
Z t

0

Eξ^{2}(s)ds+δ

≤4(α_{1}√

t_{0}+β_{1})^{2}

**A Stochastic Gronwall**
**Inequality and Its Applications**

Kazuo Amano

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almost everywhere in [0, t_{0}]. Integrating this estimate from0tot_{0} with respect
tot, we get

log 4(α_{0}√

t_{0}+β_{0})^{2}Rt0

0 Eη^{2}(s)ds+ 4(α_{1}√

t_{0}+β_{1})^{2}Rt0

0 Eξ^{2}(s)ds+δ
4(α0

√t0+β0)^{2}Rt0

0 Eη^{2}(s)ds+δ

!

≤4t_{0}(α_{1}√

t_{0}+β_{1})^{2}.
Therefore, by (1.4), we have

Eξ^{2}(t_{0})≤exp 4t_{0}(α_{1}√

t_{0} +β_{1})^{2}

4(α_{0}√

t_{0}+β_{0})^{2}
Z t0

0

Eη^{2}(s)ds+δ

. Now, lettingδ→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 anal- ysis of the numerical solutions of stochastic differential equations.

* Theorem 1.2. Assume that*ξ(t)

*and*η(t)

*belong to*M

_{w}

^{2}[0, T]

*and*ξ(t)

*is a step*

*function such that*

(1.5) ξ(t) = ξ(t_{n}) *when* t_{n}≤t < t_{n+1}

*for* n = 0,1,2, . . . , N − 1, where N *is a positive integer and* {t_{n}}^{N}_{n=0} *is a*
*partition of the interval* [0, T] *satisfying* 0 = t_{0} < t_{1} < t_{2} < · · · < t_{N}−1 <

t_{N} =T*. If there exist functions*a(t)*and*b(t)*belonging to*M_{w}^{2}[0, T]*such that*

(1.6) |ξ(t_{n})| ≤

Z tn

0

a(s)ds+ Z tn

0

b(s)dw(s)

**A Stochastic Gronwall**
**Inequality and Its Applications**

Kazuo Amano

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*is valid for each*n = 0,1,2, . . . , N *and if there are nonnegative constants*α_{0}*,*
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

Eξ^{2}(t_{n})≤4(α_{0}√

t_{n}+β_{0})^{2}
Z tn

0

Eη^{2}(s)ds+ 4(α_{1}√

t_{n}+β_{1})^{2}
Z tn

0

Eξ^{2}(s)ds
forn= 0,1,2, . . . , N; this implies, by (1.5),

Eξ^{2}(t)≤4(α_{0}√

t+β_{0})^{2}
Z t

0

Eη^{2}(s)ds+ 4(α_{1}√

t+β_{1})^{2}
Z t

0

Eξ^{2}(s)ds
for0≤t ≤T.

The remaining part of the proof is exactly same as that of Theorem1.1.

**A Stochastic Gronwall**
**Inequality and Its Applications**

Kazuo Amano

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**2.** **Applications**

Throughout this section, we assume that ξ(t) ∈ M_{w}^{2}[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) = ξ_{0}, where a(t, x) and b(t, x) are real-
valued 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|).

HereKandLare 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 {ξ_{n}(t)} defined by
ξ0(t) = ξ0 and

ξ_{n+1}(t) = ξ_{0}+
Z t

0

a(s, ξ_{n}(s))ds+
Z t

0

b(s, ξ_{n}(s))dw(s)
forn= 0,1,2, . . .. Then, we easily have

ξ_{n+1}(t)−ξ_{n}(t) =
Z t

0

a(s, ξ_{n}(s))−a(s, ξn−1(s))
ds
+

Z t

0

b(s, ξ_{n}(s))−b(s, ξ_{n−1}(s))
dw(s)

**A Stochastic Gronwall**
**Inequality and Its Applications**

Kazuo Amano

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and the Lipschitz continuity ofa(t, x)andb(t, x)implies
a(s, ξ_{n}(s))−a(s, ξ_{n−1}(s))

≤L|ξ_{n}(s)−ξ_{n−1}(s)|,
b(s, ξ_{n}(s))−b(s, ξn−1(s))

≤L|ξ_{n}(s)−ξn−1(s)|.

Hence, Theorem1.1withα_{0} =β_{0} =Landα_{1} =β_{1} = 0shows
E|ξ_{n+1}(t)−ξ_{n}(t)|^{2} ≤4L^{2}(√

t+ 1)^{2}
Z t

0

E|ξ_{n}(s)−ξn−1(s)|^{2}ds
forn= 1,2,3, . . .; the recursive use of this estimate gives

E|ξ_{n+1}(t)−ξ_{n}(t)|^{2} ≤ 4L^{2}(√

t+ 1)^{2}tn

n! sup

0≤s≤t

E|ξ_{1}(s)−ξ_{0}(s)|^{2}.
Consequently, as is well-known, the convergence of{ξ_{n}(t)}follows.

By virtue of Theorem1.1withα_{0} =β_{0} = 0andα_{1} =β_{1} =L, the unique-
ness of the solution is clear.

**Example 2.2. The error estimate of the Euler-Maruyama scheme**

ξ_{n+1} =ξ_{n}+a(t_{n}, ξ_{n})∆t+b(t_{n}, ξ_{n})∆w_{n}, n= 0,1,2, . . . , N −1
*follows immediately from Theorem* *1.2, where* N *is a sufficiently large pos-*
*itive integer,* ∆t = T /N*,* tn = n∆t *and* ∆wn = w(tn+1) − w(tn) *for*
n = 0,1,2, . . . , N −1.

**A Stochastic Gronwall**
**Inequality and Its Applications**

Kazuo Amano

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Since

ξn+1 =ξn+ Z tn+1

tn

a(tn, ξn)ds+ Z tn+1

tn

b(tn, ξn)dw(s),
ξ(t_{n+1}) = ξ(t_{n}) +

Z tn+1

tn

a(s, ξ(s))ds+ Z tn+1

tn

b(s, ξ(s))dw(s), we have

ξn+1−ξ(tn+1) =ξn−ξ(tn) + Z tn+1

tn

a(tn, ξn)−a(s, ξ(s)) ds +

Z tn+1

tn

b(t_{n}, ξ_{n})−b(s, ξ(s))

dw(s).

Now, forn= 0,1,2, . . . , N −1, if we put
ε(s) =ξ_{n}−ξ(t_{n}),

f(s) =a(t_{n}, ξ_{n})−a(s, ξ(s)),
g(s) =b(tn, ξn)−b(s, ξ(s))
when t_{n}≤s < t_{n+1} and

ε(t_{N}) =ξ_{N} −ξ(t_{N}),

f(t_{N}) =a(t_{N}, ξ_{N})−a(t_{N}, ξ(t_{N})),
g(t_{N}) =b(t_{N}, ξ_{N})−b(t_{N}, ξ(t_{N})),

**A Stochastic Gronwall**
**Inequality and Its Applications**

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then we obtain

ε(tn) = Z tn

0

f(s)ds+ Z tn

0

g(s)dw(s)

forn= 0,1,2, . . . , N. The Lipschitz continuity ofa(t, x)andb(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(t_{n}) when t_{n} ≤ s < t_{n+1} for n = 0,1,2, . . . , N − 1 and
ξ(t˜ _{N}) = ξ(t_{N}). Hence, Theorem1.2withα_{0} =α_{1} =β_{0} =β_{1} =Lshows

Eε^{2}(t)≤4L^{2}(√

t+ 1)^{2} exp 4L^{2}(√

t+ 1)^{2}t
Z t

0

E ∆t+|ξ(s)−ξ(s)|˜ 2

ds.

It follows from the fundamental property of Itô integrals that
E|ξ(s)−ξ(s)|˜ ^{2} ≤C∆t,

whereCis a nonnegative constant depending only onT,K andL. Combining the above estimates, we obtain

Eε^{2}(t) =O(∆t)

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 stochas-

**A Stochastic Gronwall**
**Inequality and Its Applications**

Kazuo Amano

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**References**

*[1] A. FRIEDMAN, Stochastic Differential Equations and Applications, Vol-*
*ume I, Academic Press, 1975.*

[2] K. ITÔAND*H. P. MCKEAN, Diffusion Processes and Their Sample Paths,*
Springer-Verlag, Berlin, 1965.

[3] P.E. KLOEDEN AND *E. PLATEN, Numerical Solution of Stochastic Dif-*
*ferential Equations, Springer-Verlag, 1992.*

[4] D.W. STROOCK AND *S.R.S. VARADHAN, Multidimensional Diffusion*
*Processes, Springer, 1982.*