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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

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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

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_w²[0, T] denotes the set of all separable nonanticipative functionsf(t)with respect toF_t defined on[0, T]satisfying

E Z T

0

f²(t)dt

<∞.

Theorem 1.1. Assume that ξ(t) and η(t) belong to M_w²[0, T]. If there exist functionsa(t)andb(t)belonging toM_w²[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 constantsα₀,α₁,β₀andβ₁such 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) Eξ²(t)≤4 α₀√

t+β₀2

exp

4t(α₁√

t+β₁)²Z t 0

Eη²(s)ds for 0≤t ≤T.

Proof. Since

E Z t

0

b(s)dw(s) 2

=E Z t

0

b²(s)ds,

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(1.1) implies, by Minkowsky and Schwarz inequalities, Eξ²(t)¹₂

≤

t Z t

0

Ea²(s)ds ¹₂

+ Z t

0

Eb²(s)ds ¹₂

. Direct computation gives, by (1.2),

t

Z t

0

Ea²(s)ds ¹₂

≤√ 2tα₀

Z t

0

Eη²(s)ds ¹₂

+√ 2tα₁

Z t

0

Eξ²(s)ds ¹₂

, Z t

0

Eb²(s)ds ¹₂

≤√ 2β₀

Z t

0

Eη²(s)ds ¹₂

+√ 2β₁

Z t

0

Eξ²(s)ds ¹₂

. Combining the above estimates, we obtain

(1.4) Eξ²(t)≤4(α₀√

t+β₀)² Z t

0

Eη²(s)ds+ 4(α₁√

t+β₁)² Z t

0

Eξ²(s)ds for 0≤t ≤T.

Let us fix a nonnegative numbert₀ ≤T arbitrarily. Then, for anyδ > 0, the last inequality (1.4) shows

d dtlog

4(α0

√t0+β0)² Z t0

0

Eη²(s)ds+ 4(α1

√t0+β1)² Z t

0

Eξ²(s)ds+δ

≤4(α₁√

t₀+β₁)²

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almost everywhere in [0, t₀]. Integrating this estimate from0tot₀ with respect tot, we get

log 4(α₀√

t₀+β₀)²Rt0

0 Eη²(s)ds+ 4(α₁√

t₀+β₁)²Rt0

0 Eξ²(s)ds+δ 4(α0

√t0+β0)²Rt0

0 Eη²(s)ds+δ

!

≤4t₀(α₁√

t₀+β₁)². Therefore, by (1.4), we have

Eξ²(t₀)≤exp 4t₀(α₁√

t₀ +β₁)²

4(α₀√

t₀+β₀)² Z t0

0

Eη²(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 thatξ(t)andη(t)belong toM_w²[0, T]andξ(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 = t₀ < t₁ < t₂ < · · · < t_N−1 <

t_N =T. If there exist functionsa(t)andb(t)belonging toM_w²[0, T]such that

(1.6) |ξ(t_n)| ≤

Z tn

0

a(s)ds+ Z tn

0

b(s)dw(s)

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is valid for eachn = 0,1,2, . . . , N and if there are nonnegative constantsα₀, α1,β0andβ1satisfying (1.2) for 0≤t≤T , then we have (1.3) for 0≤t≤T . Proof. As in the proof of Theorem1.1, we have

Eξ²(t_n)≤4(α₀√

t_n+β₀)² Z tn

0

t_n+β₁)² Z tn

0

Eξ²(s)ds forn= 0,1,2, . . . , N; this implies, by (1.5),

Eξ²(t)≤4(α₀√

t+β₀)² Z t

0

t+β₁)² Z t

0

Eξ²(s)ds for0≤t ≤T.

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

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2. Applications

Throughout this section, we assume that ξ(t) ∈ M_w²[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 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) = ξ₀+ 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)

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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, Theorem1.1withα₀ =β₀ =Landα₁ =β₁ = 0shows E|ξ_n+1(t)−ξ_n(t)|² ≤4L²(√

t+ 1)² Z t

0

E|ξ_n(s)−ξn−1(s)|²ds forn= 1,2,3, . . .; the recursive use of this estimate gives

E|ξ_n+1(t)−ξ_n(t)|² ≤ 4L²(√

t+ 1)²tn

n! sup

0≤s≤t

E|ξ₁(s)−ξ₀(s)|². Consequently, as is well-known, the convergence of{ξ_n(t)}follows.

By virtue of Theorem1.1withα₀ =β₀ = 0andα₁ =β₁ =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.

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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)),

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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, Theorem1.2withα₀ =α₁ =β₀ =β₁ =Lshows

Eε²(t)≤4L²(√

t+ 1)² exp 4L²(√

t+ 1)²t 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

Eε²(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-

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References

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

[2] K. ITÔANDH. 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.