Strong approximation for non Lipschitz stochastic functional differential equations with distributed delays

We consider a class of stochastic functional differential equations with distributed delays whose coefficients are superlinear growth and H¨older continuous with respect to the delay components. We introduce an EulerMaruyama approximation scheme for these equations and study their strong rate of convergence

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Strong approximation for non-Lipschitz stochastic functional

differential equations with distributed delays

Ngo Hoang Long∗ Pham Thi Tuyen

Hanoi National University of Education

Abstract

We consider a class of stochastic functional differential equations with distributed delays whose coefficients are super-linear growth and H¨ older continuous with respect to the delay com-ponents We introduce an Euler-Maruyama approximation scheme for these equations and study their strong rate of convergence.

1 Introduction

We are concerned with strong approximation for a class of stochastic functional differential equations (SFDEs) with distributed delays and non-Lipschitz coefficients More precisely, we consider the following multidimensional equation on a filtered probability space (Ω, F , (Ft)t≥0, P),

dX(t) =b

X(t),

Z t 0

H(t, s, X(s − τ ))ds

dt + σ

X(t),

Z t 0

G(t, s, X(s − τ ))ds

dW (t), t ∈ [0, T ], (1)

with initial data X(s) = ξ(s), s ∈ [−τ, 0], where τ is a fixed positive constant, W = (Wt, Ft)t≥0 is

a standard m-dimensional Brownian motion

SFDEs with distributed delays have been extensively studied since they can be used to model the dynamical behavior of processes having long period memory, which appear in many application domains, such as population dynamics, economy and engineering (see [4] [11], [14], [9], [12] and the references therein) Since these equations are not usually analytically solvable, we need to approximate their solutions by some numerical schemes One of the most popular approximation scheme is the Euler-Maruyama (EM) scheme, which is stated as follows: For each integers n ≥ 1, the continuous EM approximation scheme with step size h = hn = T /n = τ /l associated with (1)

is defined by

dYn(t) = b

Yn(κn(t)),

Z t 0

H(t, s, Yn(κn(s) − τ ))ds

dt + σ

Yn(κn(t)),

Z t 0

G(t, s, Yn(κn(s) − τ ))ds

dW (t), t ∈ [0, T ], (2)

∗

Correspoding author Email: ngolong@hnue.edu.vn

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where Yn(s) = ξ(s) for s ∈ [−τ, 0], and κn(t) = ih for t ∈ [ih, (i + 1)h) The strong error of this approximation is

E[ sup

0≤t≤T

|X(t) − Yn(t)|p]

for some p > 0

Recently, there have been many studies on the rates of convergence of strong error of EM type approximation schemes for stochastic functional differential equations, e.g in [10], [13], [12], [1], [7], [8] for stochastic differential delay equations, and in [2], [3] for general SFDEs We note that all these papers mentioned above as well as most of other papers in literature consider approximation for SFDEs with either local Lipschitz continuous or one-sided Lipschitz continuous coefficients

In this paper, we study the strong rates of convergence of Yn under the assumption that the coefficients b and σ are H¨older continuous and super-linear growth with respect to the second com-ponent This work extend the result of recent papers [6] and [1], which consider the strong rate

of approximation for ordinary stochastic differential equations with H¨older continuous diffusion coefficients and for stochastic delay differential equation with super-linear growth coefficients, re-spectively It is worth mentioning that the strong rates is a key factor to establish a Multi-level Monte Carlo scheme which is very effective method to approximate expectations of functional of X (see [5])

The paper is organized as follows In Section 2 we introduce some conditions on the coefficients

of equation (1) and prove the existence and uniqueness of its solution under these conditions In Section 3 we investigate the strong rate of convergence of the EM approximation scheme for equation (1)

2 Framework

For a positive integer d, let (Rd, h·, ·i, |.|) be the d-dimensional Euclidean space and ||A|| :=√traceA∗A the Hilbert-Schmidt norm for a matrix A, where A∗ is its transpose

We introduce the following assumptions

(A1) b : Rd× Rd 1 → Rd and σ : Rd× Rd 2 → Rd×m are measurable function and there exist L > 0 and α, β ∈ [12, 1] such that

|b(x1, y1) − b(x2, y2)| ≤ L|x1− x2| + V (|y1|, |y2|)|y1− y2|α, and

||σ(x1, z1) − σ(x2, z2)|| ≤ L|x1− x2| + V (|z1|, |z2|)|z1− z2|β, for all xi∈ Rd, yi ∈ Rd 1, zi ∈ Rd 2, i = 1, 2, where V : R+× R+→ R+ is polynomial bounded, i.e., there exist K > 0 and q ≥ 1 such that

V (u1, u2) ≤ K(1 + uq1+ uq2), for all u1, u2 ∈ R+ (3) (A2) H : [0, T ] × [0, T ] × Rd−→ Rd 1 and G : [0, T ] × [0, T ] × Rd−→ Rd 2 are measurable function and there exist L > 0 and γ, µ ∈ [12, 1] such that

|H(t, s, x) − H(t, s, y)| ≤ L|x − y|γ,

|G(t, s, x) − G(t, s, y)| ≤ L|x − y|µ and

sup

0≤s≤t≤T

|H(t, s, 0)| ∨ sup

0≤s≤t≤T

|G(t, s, 0)| ≤ L, for all x, y ∈ Rd

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We denote Cb

F 0([−τ, 0]; Rd) the class of all continuous process ξ defined on [−τ, 0] such that supt∈[−τ,0]|ξ(t)| < C < ∞ and ξ(t) are F0-measurable for all t ∈ [−τ, 0] The following theorem establishes the existence of a unique strong solution to equation (1)

Theorem 2.1 Assume that (A1) holds Then, for any initial data ξ ∈ CFb

0([−τ, 0]; Rd) there exists

a unique strong solution X(t) to equation (1)

This theorem can be proven by following the argument of the proof of Theorem 3.1 in [12] and will be skipped

Throughout the paper, C > 0 denotes a generic constant whose values may change from line to line

3 Strong rates of Euler-Maruyama approximation

Let Yn be defined by (2) We first prepare a lemma

Lemma 3.1 Assume that (A1), (A2) holds and the initial data ξ ∈ CFb

0([−τ, 0]; Rn) Then, for any p ≥ 2 there exists a constant C > 0 independent of h such that

E[ sup

0≤t≤T

|X(t)|p] ∨ E[ sup

0≤t≤T

|Yn(t)|p] ≤ C (4)

and

E

h

|Yn(t) − Yn(κn(t))|p

i

Proof We first show that

E

h sup

0≤t≤T

|X(t)|pi< ∞ (6)

For each integer N > 1, we define the stopping time

τN = T ∧ inf {t ∈ [−τ, T ] : |X(t)| ≥ N }

Clearly, |X(s)| ≤ N for all s ∈ [−τ, τN] and τN ↑ T Set XN(t) = X(t ∧ τN) for t ∈ [0, T ] Then

XN(t) satisfies the equation

XN(t) = ξ(0) +

t

Z

0

b

XN(s),

Z s 0

H(s, r, XN(r − τ ))dr

I[0≤s≤τN]ds

+

t

Z

0

σ

XN(s),

Z s 0

G(s, r, XN(r − τ ))dr

I[0≤s≤τ N ]dW (s), t ∈ [0, T ]

Using the elementary inequality |a + b + c|p ≤ 3p−1(|a|p+ |b|p+ |c|p), we can show that for t ∈ [0, T ]

UtN,p:= E

sup

0≤s≤t

|XN(s)|p

≤ C + CE

sup

0≤s≤t

Z s 0

b

XN(s),

Z s 0

H(s, r, XN(r − τ ))du

I[0≤r≤τ N ]dr

p

+ CE

sup

0≤s≤t

Z s 0

σ

XN(s),

Z s 0

I[0≤r≤τ N ]dW (r)

p

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By the H¨older and Burkholder-Davis-Gundy inequalities that

UtN,p≤ C

1 + E

Z t 0

b

XN(s),

Z s 0

H(s, r, XN(r − τ ))dr

p

ds +

Z t 0

σ

XN(s),

Z s 0

p

ds

, for all t ∈ [0, T ] It follows from assumption (A1) that

UtN,p≤ C

E

Z t 0

|XN(s)|pds +

Z t 0

Z s 0

E

|XN(r − τ )|pαγ+ |XN(r − τ )|pβµdrds

+

Z t 0

Z s 0

E

|XN(r − τ )|p(q+α)γ+ |XN(r − τ )|p(q+β)µ

drds + 1

Using the elementary inequality ap+ p − 1 ≥ pa for a > 0 and p ≥ 1, we have

UtN,p≤ C

1 + E

Z t 0

|XN(s)|pds +

Z t 0

Z s 0

E

drds

This, together with the Gronwall inequality, yields that for p ≥ 2 and t ∈ [0, T ],

E

sup

0≤s≤t

|XN(s)|p

≤ C

1 +

Z t 0

Z s 0

E

drds

Set ρ := [(q + α)γ] ∨ [(q + β)µ] and pi:= ([T /τ ] + 2 − i)pρ[T /τ ]+1−i, for i = 1, 2, , [T /τ ] + 1, where [a] denotes the integer part of the real number a Thus, due to ρ ≥ 1 and p ≥ 2, it is easy to see that pi ≥ 2, pi+1ρ ≤ pi and p[T /τ ]+1 = p Since X(s) = ξ(s) for s ∈ [−τ, 0] and ξ ∈ CFb0([−τ, 0]; Rn),

we obtain that

E

sup

0≤s≤τ

|XN(s)|p1

Furthermore, it follows from (7), the fact p2ρ < p1 and the Liapunov inequality that

E

sup

0≤s≤2τ

|XN(s)|p2

≤ C

1 +

Z τ 0

Z s 0

[E|XN(r − τ )|p1]p2(q+α)γp1 drds +

Z τ 0

Z s 0

[E|XN(r − τ )|p1]p2(q+β)µp1 drds

Thanks to (8), we get E

sup

0≤s≤2τ

|XN(s)|p2

≤ C Repeating the previous procedures gives

E

"

sup

0≤s≤T

|XN(s)|p

#

= E

"

sup

0≤s≤T

|XN(s)|p[T /τ ]+1

#

≤ C

Since XN(t)−→ X(t), applying the Fatou Lemma we get (6) By a similar argument we also havea.s

E

"

sup

0≤s≤T

|Yn(s)|p

#

≤ C This concludes (4)

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Next we will show (5) Since

Yn(t) − Yn(κn(t)) =

Z t

κ n (t)

b

Yn(κn(s)),

Z s 0

H(s, r, Yn(κn(r) − τ ))dr

ds

+

Z t

κ n (t)

σ

Yn(κn(s)),

Z s 0

G(s, r, Yn(κn(r) − τ ))dr

dW (s), (9)

by the elementary inequality |a + b|p ≤ 2p−1(|a|p+ |b|p), the H¨older inequality and the Burkholder-Davis-Gundy inequality, we have

E

h

|Yn(t) − Yn(κn(t))|p

i

≤ 2p−1(t − κn(t))p−1

Z t

κ n (t)

E

b

Yn(κn(s)),

Z s 0

p ds

+ 2p−1C(t − κn(t))p2 −1Z t

κ n (t)

E

σ

Yn(κn(s)),

Z s 0

p ds

Thanks to conditions A1, A2, estimate (4) and the fact that t − κn(t) ≤ h, we obtain (5)

We are now in a position to state the main theorem of this paper

Theorem 3.2 Suppose (A1), (A2) hold and ξ ∈ Cb

F 0([−τ, 0]; Rn) For any p ≥ 2 there exists C > 0 independent of h and n such that

E

"

sup

0≤s≤T

|X(s) − Yn(s)|p

#

≤ Chp2 ν [T /τ ]+1

,

where ν = (αγ) ∧ (βµ)

Proof We use the method of Yamada and Watanabe to approximate the function φ(x) = |x| Let

δ > 1 and ∈ (0, 1) SinceR

/δ

1

xdx = lnδ, there is a continuous nonnegative function ψδ(x), x ≥ 0, which is zero outside [/δ, ] satisfiesR/δ ψδ(x) = 1 and ψδ(x) ≤ xlnδ2 Define

φδ(x) :=

Z x 0

Z y 0

ψδ(z)dzdy, x > 0

Then φδ ∈ C2(R+; R+) possesses the following properties:

x − ≤ φδ(x) ≤ x, x > 0

and

0 ≤ φ0δ(x) ≤ 1, φ00δ(x) ≤ 2

xlnδ1[/δ,](x), x > 0.

Define Vδ(x) := φδ(|x|), x ∈ Rd, (Vδ)x(x) := (∂Vδ (x)

∂x 1 , ,∂Vδ (x)

∂x d ), and (Vδ)xx(x) := (∂2Vδ (x)

∂x i ∂x j )d×d, x ∈

Rd By a straightforward computation, we deduce that

0 ≤ |(Vδ)x(x)| ≤ 1, ||(Vδ)xx(x)|| ≤ 2d(1 + 1

ln σ) 1

|x|1[/δ,](|x|). (10)

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For any t ∈ [−τ, T ], set Z(t) := X(t) − Yn(t), Z(t) := Yn(t) − Yn(κn(t)) Applying the Itˆo formula, we get Vδ(Z(t)) = I1(t) + I2(t) + I3(t) where

I1(t) =

Z t 0

D (Vδ)x(Z(s)), b

X(s),

Z s 0

H(s, r, X(r − τ ))dr

− bYn(κn(s)),

Z s 0

H(s, r, Yn(κn(r) − τ ))drEds

I2(t) =1

2

Z t 0

trace

σX(s),

Z s 0

G(s, r, X(r − τ ))dr

−σYn(κn(s)),

Z s 0

∗

(Vδ)xx(Z(s))

σX(s),

Z s 0

−σYn(κn(s)),

Z s 0

ds

I3(t) =

Z t 0

D (Vδ)x(Z(s)),

σX(s),

Z s 0

−σYn(κn(s)),

Z s 0

E

dW (s)

E

sup

0≤s≤t

|I 1 (s)|p

≤ C

Z t

0

E

b

X(s),

Z s 0

H(s, r, X(r − τ ))dr

− b

Yn(κ n (s)),

Z s 0

H(s, r, Yn(κ n (r) − τ ))dr

p

ds

≤ C

Z t

0

E

|Z(s)| p + |Z(s)|p+ Vp

Z s 0

H(s, r, X(r − τ ))dr,

Z s 0

×

Z s

0

H(s, r, X(r − τ ))dr −

Z s 0

H(s, r, Yn(r − τ ))dr)

pα

+Vp

Z s

0

H(s, r, X(r − τ ))dr,

Z s 0

×

Z s

0

H(s, r, Yn(r − τ ))dr −

Z s 0

H(s, r, Yn(κn(r) − τ ))dr)

pα

ds.

By (A1), (A2) and (4), we have

E

V2p

Z s 0

H(s, r, X(r − τ ))dr,

Z s 0

≤ C

This fact together with assumption A2 and H¨older’s inequality implies that

E

h

sup

0≤s≤t

|I1(s)|p

i

≤ C

Z t 0

E|Z(s)|p+ E|Z(s)|p+

hZ s 0

E|Z(r − τ )|2pαγdr

i1

ds +

hZ s 0

i1 2

Similarly, thanks to (10), condition A1 and the H¨older inequality, we have

E

h

sup

0≤s≤t

|I2(s)|pi≤ CE

Z t 0

1

|Z(s)|p1[/δ,](|Z(s)|)h|Z(s)|2p+ |Z(s)|2p

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s 0

G(s, r, X(r − τ ))dr,

s 0

×

Z s 0

G(s, r, X(r − τ ))dr −

Z s 0

2pβi ds

This implies

E

h sup

0≤s≤t

|I2(s)|p

i

≤ C

Z t 0

E|Z(s)|p+ 1

pE|Z(s)|2p

+1

p

hZ s 0

E|Z(r − τ )|4pβµdri

1 2

+1

p

hZ s 0

E|Z(s − τ )|4pβµdri

1 2

ds (12)

By (10), the Burkholder-Davis-Gundy inequality, the inequality H¨older, we obtain that

E

h

sup

0≤s≤t

|I3(s)|pi

≤ C

"

Z t

0

E

σX(s),

Z s 0

G(s, r, X(r − τ ))dr−σYn(κn(s)),

Z s 0

2 # p/2

ds

≤ C

Z t

0

E

σX(s),

Z s 0

G(s, r, X(r − τ ))dr−σYn(κn(s),

Z s 0

p

ds.

Thanks to condition (A1),

E

h

sup

0≤s≤t

|I3(s)|p

i

≤ C

Z t

0

E

"

|Z(s)|p+ |Z(s)|p+ Vp

Z s 0

G(s, r, X(r − τ ))dr,

Z s 0

×

Z s

0

G(s, r, X(r − τ ))dr −

Z s 0

pβ#

ds

Therefore, we also have

E

h

sup

0≤s≤t

|I3(s)|pi≤ C

Z t 0

( E|Z(s)|p+ E|Z(s)|p+

Z s 0

E|Z(r − τ )|2pβµdr

1 2

+

Z s 0

E|Z(r − τ ))|2pβµdr

1 2

)

Since

E

h

sup

0≤s≤t

|Z(s)|pi≤h + sup

0≤s≤t

Vδ(Z(s))ip≤ 2p−1

p+ Ehsup

0≤s≤t

Vδp(Z(s))i

, combining (11), (12), (13), (5), we obtain that for any t ∈ [0, T ] and any p ≥ 2,

E

h

sup

0≤s≤t

|Z(s)|pi≤ C

p+ hp2 + hpαγ2 + 1

php+ 1

phpβµ+ hpβµ2

+

Z t 0

E|Z(s)|p+

hZ s 0

i1

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p

h s 0

i +

h s 0

i

ds This, together with the Gronwall inequality, further implies that

E

h

sup

0≤s≤t

|Z(s)|pi≤ C

(

p+ hp2 + hpαγ2 + 1

php+ 1

phpβµ+ hpβµ2

+

Z t 0

hZ s 0

i1

+1

p

hZ s 0

i1 2

+

hZ s 0

i1 2

ds

) (14)

We fix p ≥ 2, and denote pi:= ([T /τ ] + 2 − i)p4[T /τ ]+1−i, i = 1, 2, , [T /τ ] + 1 We have 4pi+1< pi, and p[T /τ ]+1 = p, i = 1, 2, , [T /τ ] For s ∈ [0, τ ], we have Z(s − τ ) = 0 and taking = hν2,

ν = (αγ) ∧ (βµ) It follows from (14) that

E

sup

0≤s≤τ

|Z(s)|p1

≤ Chp12 ν (15)

Next, taking = ∆ν22 , by using (14) and the Liapunov inequality, we obtain that

E

h

sup

0≤s≤2τ

|Z(s)|p2

i

≤ C

(

hp22 ν 2

+ hp22 + hp2αγ2 + hp2 βµ−p2ν22 + hp2βµ2

+

Z 2τ 0

hZ s 0

E|Z(r − τ )|2p2αγdri

1 2

+ 1

p 2

hZ s 0

E|Z(r − τ )|4p2βµdr

i1

+

hZ s 0

E|Z(r − τ )|2p2βµdr

i1

ds )

≤ C

(

hp22 ν 2

+

Z τ 0

hZ s 0

E|Z(r)|2p2αγdr

i1

+ 1

p 2

hZ s 0

E|Z(r)|4p2βµdr

i1

+

hZ s 0

E|Z(r)|2p2βµdr

i1

ds )

≤ C

(

hp22 ν 2

+

Z τ 0

hZ s 0

E|Z(r)|p1

2p2αγ p1 dr

i1 2

+ 1

p 2

hZ s 0

E|Z(r)|p1

4p2βµ p1 dr

i1 2

+

hZ s 0

E|Z(r)|p1

2p2βµ p1 dr

i1

2

ds )

≤ Cnhp22 ν 2

+ hp22 ναγ+ h2p22 νβµ−p2ν22 + hp22 νβµo

Therefore, it follows that

E

h sup

0≤s≤2τ

|Z(s)|p2

i

≤ Chp22 ν 2

, where ν = (αγ)∧(βµ) The desired assertion then follows by repeating the previous procedures

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This research was supported by the Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant Number 101.03-2014.14 This work was done during a stay of the first author at Vietnam Institute for Advance Study in Mathematics He wishes to express his gratitude to the institute for the support

References

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J Math Kyoto Univ 11 (1971) 155-167

drds

This, together with the Gronwall inequality, yields that for p ≥ and t ∈ [0, T ],

E

sup

0≤s≤t... class="page_container" data-page="6">

For any t ∈ [−τ, T ], set Z(t) := X(t) − Yn(t), Z(t) := Yn(t) − Yn(κn(t)) Applying the Itˆo formula, we get Vδ(Z(t))... XN(r − τ ))dr

p

ds

, for all t ∈ [0, T ] It follows from assumption (A1) that

UtN,p≤ C

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