Strong Rate of Tamed EulerMaruyama Approximation for Stochastic Differential Equations with H¨older Continuous Diffusion Coefficient

We study the strong rate of convergence of the tamed EulerMaruyama approximation for onedimensional stochastic differential equations with superlinearly growing drift and H¨older continuous diffusion coef ficients.

Strong Rate of Tamed Euler-Maruyama Approximation for Stochastic Differential

Coefficient

Hanoi National Univesity of Education

Abstract

We study the strong rate of convergence of the tamed Euler-Maruyama approximation for one-dimensional stochastic differential equations with superlinearly growing drift and H¨older continuous diffusion coef-ficients

Let us consider a stochastic differential equation (SDE for short)

Xt = x0+

Z t 0

b(s, Xs)ds +

Z t 0

σ(s, Xs)dWs, x0 ∈ R, t ∈ [0, T ], (1.1)

where (Wt)0≤t≤T is a standard Brownian motion defined on a filtered proba-bility space (Ω,F, (Ft)t≥0, P)

It is well-known that when b and σ are Lipschitz continuous, the standard Euler-Maruyama approximation scheme for (Xt) has a strong rate of conver-gence of order 1/2 (see Kloeden, P and Platen, E (1995)) Recently, there

∗ ngolong@hnue.edu.vn

† trongsphn@gmail.com

have been extensive studies on the strong approximations of SDE (1.1) with non-Lipschitz coefficients The rates of Euler-Maruyama scheme for SDEs with H¨older continuous diffusion coefficients have been investigated by Yan, B.L ((2002)), Gy¨ongy, I and R´asonyi, M (2011), Ngo, H-L and Taguchi,

D (2013) (see also Alfonsi, A (2005); Berkaoui, B., Bossy, M and Diop, A (2008); Dereich, S., Neuenkirch, A and Szpruch, L (2012) for many strong approximation schemes proposed for Cox-Ingersoll-Ross type model) Hairer, M., Hutzenthaler, M and Jentzen, A (2012) have proposed an example of SDE with globally bounded and smooth coefficients that although the stan-dard Euler-Maruyama approximation converge to the exact solution of the SDE in both strong and weak senses but there is no positive polynomial rate

of convergence Moreover, Hutzenthaler, M., Jentzen, A and Kloeden, P E (2011) have showed that if b is superlinear growth then the absolute moments

of the standard Euler-Maruyama approximated solution may diverge to in-finity while the ones of the true solution is finite Therefore, the standard Euler-Maruyama scheme may fail to convergence in Lp sense There are two methods to overcome this difficulty The first method named implicit Euler-Maruyama scheme was proposed by Higham, D J., Mao, X and Stuart, A

M (2002), and Hu, Y (1996) A drawback of the implicit scheme is that at each simulation step, one needs to solve an algebraic equation which may not have an explicit solution Hutzenthaler, M., Jentzen, A and Kloeden, P.E (2011)) and Sabanis, S (2013) have recently introduced the second method named tamed Euler scheme It is an explicit scheme in which the drift co-efficient is modified so that it is bounded It has been shown that when the diffusion σ is Lipschitz continuous and the drift b is superlinear growth and one-sided Lipschitz, the tamed Euler scheme has a strong rate of order 1/2

In this article we study the strong convergence rate of the tamed Euler-Maruyama schemes applied to the SDE (1.1) where b is superlinear growth and σ is H¨older continuous This partly generalizes the results in Gy¨ongy,

I and R´asonyi, M (2011); Hutzenthaler, M., Jentzen, A and Kloeden, P.E (2011)) in the one-dimensional framework The main contributions of the current paper are:

• Establishing a new sufficient condition for the existence and uniqueness

of strong solution for one-dimensional SDEs with locally H¨older con-tinuous diffusion coefficient and superlinearly growing drift coefficient;

• Showing the strong convergence (in Lp sense) of the tamed

Euler-• Obtaining the order of these approximation errors.

The rest of the paper is organized as follows The next section introduces some notations and assumptions All main results are presented in Section

3 while their proofs are given in Section 4

For integers n ≥ 1, we define ηn(s) : [0; T ] → [0; T ] by ηn(t) = kT

n := t(n)k

if t ∈ hkTn ;(k+1)Tn  The tamed Euler-Maruyama approximation of equation (1.1) is defined as follows

Xt(n)= x0+

Z t 0

bn(s, Xη(n)

n (s))ds +

Z t 0

σ(s, Xη(n)

n (s))dWs, t ∈ [0, T ] (2.1) with bn(t, x) = 1+nb(t,x)−λ |b(t,x)| for some λ ∈ 0;1

2 Note that if bn is replaced by

b in (2.1) then X(n) is called the standard Euler-Maruyama approximation

We will make the following assumptions:

A1 There exists a positive constant L such that

xb(t, x) ∨ |σ(t, x)|2 ≤ L(1 + |x|2) for any x ∈ R and t ∈ [0; T ]

A2 b is one-sided Lipschitz: there exists a positive constant L such that

(x − y)(b(t, x) − b(t, y)) ≤ L|x − y|2 for any x, y ∈ R and t ∈ [0; T ]

A3 There exist positive constants L and l such that

|b(t, x) − b(t, y)| ≤ L 1 + |x|l+ |y|l
|x − y|, and

|b(t, x)| ≤ L(1 + |x|l+1) for any x, y ∈ R and t ∈ [0; T ]

A4 σ is (α + 12)-H¨older continuous: there exist positive constants L and

α ∈ (0,12] such that

|σ(t, x) − σ(t, y)| ≤ L|x − y|1/2+α

for any x, y ∈ R and t ∈ [0; T ]

A5 b is locally Lipschitz and locally bounded: for any R > 0, there exists

a positive constant LR> 0 such that

|b(t, x) − b(t, y)| ≤ LR|x − y|

and |b(t, x)| ≤ LR for all |x| ∨ |y| ≤ R and t ∈ [0, T ]

A6 σ is locally (α + 12)-H¨older continuous: for any R > 0, there exist positive constants LR and α ∈ (0,12] such that

|σ(t, x) − σ(t, y)| ≤ LR|x − y|1/2+α

for all |x| ∨ |y| ≤ R and t ∈ [0, T ]

It is clear that the Assumptions A3 and A4 imply the Assumptions A5 and A6, respectively

The existence and uniqueness of solution for SDEs with H¨older continuous diffusion coefficient and bounded measurable drift has been established in Veretennikov, A.Yu (1981) (see also Gy¨ongy, I and Krylov, N.V (1996); Ya-mada, T and Watanabe, S (1971)) In the following, we show the existence and uniqueness of solution to equation (1.1) under Assumptions A1, A5 and A6

Theorem 3.1 (i) Suppose that A1, A5, A6 hold, and equation (1.1) has a solution (Xt)t∈[0,T ] then it is the unique solution

(ii) Suppose that A1, A5, A6 holds, and there exist positive constants C and

l such that

sup

t∈[0,T ]

|b(t, x)| ≤ C 1 + |x|l
,

Example 3.1 For clarity of exposition we consider the following SDE

Xt = x0+

Z t 0

(aXs− bXs3)ds +

Z t 0

σ|Xs− K|1/2+α, (3.1)

where b is a non-negative constant, α ∈ [0,12] and x0, a, σ, K ∈ R It is straightforward to verify that coefficients of this SDE satisfy Assumptions A1–A4 Therefore equation (3.1) has a unique strong solution If b > 0, it follows from Theorem 2.1 in Hutzenthaler, M., Jentzen, A and Kloeden, P

E (2011) that the standard Euler-Maruyama approximated solution of (3.1) does not have a finite moment of any order p ≥ 1 while Xt’s have finite mo-ments of all order (Lemma 4.1) It means that the standard Euler-Maruyama approximation for equation (3.1) does not converge in strong sense

In the following we always assume that the equation (1.1) has a unique strong solution Moreover, we choose λ ∈ [α,12] for the tamed Euler-Maruyama approximation (2.1)

Theorem 3.2 Suppose that A1–A4 hold, then there is a constant C > 0 independent of n such that

sup

τ ∈ TE|Xτ − Xτ(n)| ≤

(C

n α if α ∈ 0,12 ,

C log n if α = 0, where T is the set of all stopping times τ ≤ T

Corollary 3.3 Suppose that A1–A4 hold, then there is a constant C > 0 independent of n such that

E

 sup

0≤t≤T

|Xt− Xt(n)|



≤

( C

n 2α2 if α ∈ 0,12 ,

C

√ log n if α = 0

The following estimates for the moments of strong approximation errors play an important role in designing a Multilevel Monte Carlo scheme to esti-mate E[F (X)] for some function F defined on C[0, T ] (see Giles, M (2008)) Theorem 3.4 Suppose that A1-A4 hold, then for all p > 0, there is a constant Cp > 0 depend on p and independent of n such that

E



sup

0≤t≤T

Xt− Xt(n)

p

≤







C p

n p/2 if α = 1

2,

C p

n α if α ∈ 0;1

2

and p ≥ 2,

C p

log n if α = 0 and p ≥ 2

We note here that that the strong rates of the tamed Euler-Maruyama approximation obtained in Theorems 3.2, 3.4 and Corollary 3.3 are the same as the ones of the standard Euler-Maruyama approximation applied

to SDEs with H¨older continuous diffusion coefficient and linear growth drift (see Gy¨ongy, I and R´asonyi, M (2011))

Finally we show the convergence of tamed Euler-Maruyama scheme under the local Lipschitz condition on b and local H¨older continuous condition on σ

Theorem 3.5 Suppose that A1, A5 and A6 hold Then for any p > 0,

lim

n→∞E

h sup

0≤t≤T

Xt− Xt(n)

pi

= 0

In the following we will first prove Theorems 3.2–3.5 The proof of Theorem 3.1 will be given at Sections 4.6 and 4.7

Throughout this section C denotes some positive constants which may depend on L, l, T, α and x0 but independent of n and t When C depend on

p or R we denote C by Cp or CR, respectively

Denote Ut(n) = Xt(n)− Xη(n)

n (t) and Yt(n) = Xt− Xt(n) We need the following bounds on moments of X and X(n)which are direct consequences of Lemmas 3.1–3.3 in Sabanis, S (2013)

Lemma 4.1 Suppose that A1 holds, then for any p > 0, there exists a constant Cp > 0 such that

E

 sup

0≤t≤T

|Xt|p



∨ sup

n≥1E

 sup

0≤t≤T

|Xt(n)|p



< Cp,

and

sup

0≤t≤TE

h

|Ut(n)|pi ≤ Cp

np/2 (4.1)

Lemma 4.2 Suppose that A1 and A3 hold, then for any p > 0, there exist

a constant Cp > 0 such that

Z T 0

E

h

b s, Xs(n)
− bs, Xη(n)

n (s)



pi

ds ≤ Cp

np/2 Proof It is enough to prove the lemma for p > 1 By A3 we have

b s, Xs(n)
− bs, Xη(n)

n (s)



p

≤ Cp



1 +

Xη(n)

n (s)

lp

+Us(n)

lp

Us(n)

p

According to Lemma 4.1,

E



Xη(n)

n (s)

lp

Us(n)

p

≤ 1

np/2E



Xη(n)

n (s)

2lp + np/2E

h

Us(n)

2pi

≤ Cp

np/2 The lemma is proved completely

We borrow the following result form Gy¨ongy, I and R´asonyi, M (2011) Lemma 4.3 Let (Xt)t≥0 be a nonnegative stochastic process and set Vt = sups≤tXs Assume that for some p > 0, q ≥ 1, ρ ∈ [1, q] and constants K and ∆ ≥ 0, it holds

E[Vtp] ≤ KE

Z t 0

Vsds

p

+ KE

Z t 0

Xsρ

p/q

+ ∆ < ∞ for any t ≥ 0 Then for each T ≥ 0, the following statements hold

(i) If ρ = q then there is a constant CT such that E[VTp] ≤ CT∆

(ii) If p ≥ q or both ρ < q and p > q + 1 − ρ hold, then the exist constants

C1 and C2 depending on K, T, ρ, q and p such that E[VTp] ≤ C1∆ +

C2R0T E[Xs]ds

We will repeatedly use the approximation technique of Yamada and Watan-abe (see Yamada, T and WatanWatan-abe, S (1971); Gy¨ongy, I and R´asonyi,

M (2011)) For each δ > 1 and ε > 0 there exists a continuous function

ψδε : R → R+ with suppψδε ⊂ [ε/δ; ε] such that Rε

ε/δψδε(z)dz = 1 and

0 ≤ ψδε(z) ≤ z log δ2 for z > 0 Define

φδε(x) :=

Z |x|

0

Z y 0

ψδε(z)dzdy, x ∈ R

It is easy to verify that φδ has the following useful properties: for any x ∈ R

(i) φ0δε(x) = |x|xφ0δε(|x|),

(ii) 0 ≤ |φ0δε(x)| ≤ 1,

(iii) |x| ≤ ε + φδε(x),

(iv) φ

0

δε (|x|)

|x| ≤ δ

ε, (v) φ00δε(|x|) = ψδε(|x|) ≤ |x| log δ2 1[ε

δ ;ε](|x|)

Applying Itˆo’s formula for φδε(|Yt(n)) and using (iii), we get

|Yt(n)| ≤ ε +

Z t 0

φ0δε(Ys(n))hb(s, Xs) − bns, Xη(n)

n (s)

i ds

+ 1 2

Z t 0

φ00δε(Ys(n))

h σ(s, Xs) − σ



s, Xη(n)

n (s)

i2

ds

+

Z t 0

φ0δε(Ys(n))hσ(s, Xs) − σs, Xη(n)

n (s)

i

dWs (4.2)

We rewrite the first integral in (4.2) as

S1 =

Z t 0

φ0δε

Ys(n)



Ys(n)

Xs− X(n)

s
b(s, Xs) − b(s, Xs(n)) ds

+

Z t 0

φ0δε(Ys(n))hb s, Xs(n)
− bs, Xη(n)

n (s)

i ds

+

Z t 0

φ0δε(Ys(n))hbs, Xη(n)

n (s)



− bns, Xη(n)

n (s)

i ds

It follows from Conditions A2, A3, and the estimate (ii) that

S1 ≤ L

Z t

0

φ0δε

Ys(n)



Ys(n)

Ys(n)

2

ds +

Z t 0

b s, Xs(n)
− bs, Xη(n)

n (s)

 ds

+

Z t

0

n−λ

bs, Xη(n)

n (s)



2

1 + n−λ

b



s, Xη(n)

n (s)

 ds

≤ L

Z t

0

Ys(n)ds +

Z T 0

b s, Xs(n)
− bs, Xη(n)

n (s)



ds + 2L

2

nλ

Z T 0



1 +

Xη(n)

n (s)

2l+2 ds (4.3)

Next, thanks to condition A4 and the estimate (v), the second integral in

(4.3) is bounded by

Z t

0

22αL2

Ys(n)log δ

1h

ε ≤

Y s(n)

≤εi

h

Ys(n)

1+2α

+Us(n)

1+2αi ds

≤ 2

2αL2ε2αT log δ +

Z T 0

22αL2δ

ε log δ

Us(n)

1+2α

It follows from (4.2), (4.3) and (4.4) that

Yt(n)

≤ ε + L

Z t 0

Ys(n)ds +

Z T 0

b s, Xs(n)
− bs, Xη(n)

n (s)

 ds +2L

2

nλ

Z T 0



1 +

Xη(n)

n (s)

2l+2

ds + 22α+1L2 ε2αT

log δ +

Z T 0

δ

ε log δ

Us(n)

1+2α

ds



+

Z t

0

φ0δε Ys(n)
hσ(s, Xs) − σs, Xη(n)

n (s)

i

dWs (4.5)

Let Zt(n):=

Yt∧τ(n)

for any stopping time τ ≤ T It implies from (4.5) that

E[Zt(n)] ≤ ε + C

Z t 0

E[Zs(n)]ds +

Z T 0

E

h

b s, Xs(n)
− bs, Xη(n)

n (s)



i ds



+ Cn−λ

Z T 0



1 + E



Xη(n)

n (s)

2l+2

dsi+ Ch ε

2α

log δ +

Z T 0

δ

ε log δE



Us(n)

1+2α

ds



Thanks to Lemmas 4.1 and 4.2, we have

E

h

Zt(n)i ≤ ε + C

Z t 0

EZ(n)

s  ds + 1

nλ + ε

2α

log δ +

δ

ε log δ

1

n1/2+α



By Gronwall’s inequality

E

h

Zt(n)i ≤ C



ε + 1

nλ + ε

2α

log δ +

δ

ε log δ

1

n1/2+α



Case 1: α ∈ 0;12 By choosing ε = √1

n and δ = 2, we obtain EhZt(n)i≤ C

n α Let t ↑ T then

sup

τ ∈ TE



Xτ − X(n)

τ

 ≤ C

nα Case 2: α = 0 By choosing ε = log n1 and δ = n1/3, we obtain E

h

Zt(n)

i

≤ C log n Let t ↑ T then

sup

τ ∈ TE



Xτ − Xτ(n) ≤ C

log n, and thus we get the desired result

This proof is similar to the one of Corollary 2.3 in Gy¨ongy, I and R´asonyi,

M (2011) and will be omitted

4.4 Proof of Theorem 3.4

It is enough to prove the theorem for p ≥ 2 By (4.5), H¨older’s inequality,

we have

E

h

sup

0≤u≤t

Yu(n)

pi

≤ Cpεp+ Cp

 E

Z t 0

sup

0≤u≤s

Yu(n)ds

p +

Z T 0

E

h

b s, Xs(n)
− bs, Xη(n)

n (s)



pi ds



+ Cpn−pλ

Z T

0



1 + E



Xη(n)

n (s)

p(2l+2)

ds



+ Cp



ε2pα

(log δ)p +

Z T 0

δp

εp(log δ)pE

h

Us(n)

p(1+2α)i

ds



+ CpE



sup

0≤u≤t

Z u 0

φ0δε Ys(n)
hσ(s, Xs) − σs, Xη(n)

n (s)

i

dWs

p (4.6)

Applying Burkholder-Davis-Gundy’s inequality, H¨older’s inequality and Lemma 4.1, the last expectation in (4.6) is less than

E

"

Z t 0

φ0

δε Ys(n)
2h

σ(s, Xs) − σs, Xη(n)

n (s)

i2

ds

p/2#

≤ Cp

( E

Z t 0

Ys(n)

1+2α

ds

p/2

+ 1

np(1+2α)/4

)

This estimate together with Lemmas 4.1 and 4.2 implies

E

h

sup

0≤u≤t

Yu(n)

pi

≤ Cp



εp+ E

Z t 0

sup

0≤u≤s

Yu(n)ds

p

+ 1

npλ + ε

2pα

(log δ)p + δ

p

εp(log δ)p 1

np(1+2α)/2+ +E

Z t

0

Ys(n)

1+2α

ds

p/2

+ 1

np(1+2α)/4 + 1

np/2

)

By choosing ε = √1

n and δ = 2 when α ∈ (0,12]; and ε = log n1 and δ = n1/3 when α = 0 and applying Lemma 4.3 together with Theorem 3.2, we get the

desired result

4.5 Proof of Theorem 3.5

For each R > 0, we denote τR = inf{t ≥ 0 : |Xt| ≥ R}, ρnR = inf{t ≥ 0 :

|Xt(n)| ≥ R} and νnR = τR∧ ρnR, and χ(n)(s) = Ys∧ν(n)nR = Xs∧ν nR − Xs∧ν(n)nR

We recall the following result form Sabanis, S (2013)

Lemma 4.4 Suppose that A1 holds Then for any R > 0, q > p > 2 and

η > 0, one has

E



sup

0≤t≤T

Yt(n)

p

≤ ηp

q Cq+

q − p

qηp/(q−p)RpCp + E

 sup

0≤t≤T

χ(n)t

p

Furthermore, suppose that A4 holds Then for any R > 0, there is a Cp(R) >

0 such that

Z T

0

E

h

b s, Xs(n)
− bs, Xη(n)

n (s)



p

1(s≤νnR)ids ≤ CpL

p R

np/2 The proof of Theorem 3.5 is divided into three steps

Step 1: We will show that sup0≤t≤TE[|χ(n)(t)|] → 0 as n → ∞ Indeed,

from (4.2) we have

|χ(n)t | ≤ ε +

Z t 0

φ0δε(Ys(n))hb(s, Xs) − bns, Xη(n)

n (s)

i

1[s≤νnR]ds

+ 1 2

Z t 0

φ00δε(Ys(n))hσ(s, Xs) − σs, Xη(n)

n (s)

i2

1[s≤νnR]ds

+

Z t 0

φ0δε(Ys(n))hσ(s, Xs) − σs, Xη(n)

n (s)

i

1[s≤νnR]dWs (4.8)

A similar argument as in (4.3) and Condition A5 imply that

Z t

0

φ0δε(Ys(n))hb(s, Xs) − bns, Xη(n)

n (s)

i

1[s≤νnR]ds

≤ LR

Z t

0

χ(n)s ds +

Z T 0

b(s, Xs(n)) − bs, Xη(n)

n (s)



1[s≤νnR]ds + CR

nλ (4.9) Thanks to condition A6, we have

Z t

φ00δε Ys(n)
hσ(s, Xs) − σs, Xη(n)

n (s)

i2

1[s≤νnR]ds ≤ CRε

2α

log δ +

Z T CRδ

ε log δ

Us(n)

1+2α

ds

to SDEs with Hăolder continuous diffusion coefficient and linear growth drift (see Gyăongy, I and Rasonyi, M (2011))

Finally we show the convergence of tamed Euler-Maruyama... proof of Theorem 3.1 will be given at Sections 4.6 and 4.7

Throughout this section C denotes some positive constants which may depend on L, l, T, α and x0 but independent of