Numerical solutions of doubly perturbed stochastic delay differential ...

Arab J Math (2012) 1:251–265 DOI 10.1007/s40065-012-0026-1

R E S E A R C H A RT I C L E

Xiaotai Wu · Litan Yan

Numerical solutions of doubly perturbed stochastic delay differential equations driven by Lèvy process

Received: 6 May 2011 / Accepted: 3 October 2011 / Published online: 11 April 2012 © The Author(s) 2012. This article is published with open access at Springerlink.com

Abstract In this paper, the numerical solutions of doubly perturbed stochastic delay differential equations driven by Lèvy process are investigated. Using the Euler–Maruyama method, we define the numerical solutions, and show that the numerical solutions converge to the true solutions under the local Lipschitz condition. As a corollary, we give the order of convergence under the global Lipschtiz condition. Mathematics Subject Classification (2010) 60H10 · 65C50 · 60J75

1 Introduction The purpose of this paper is to study the numerical solutions of doubly perturbed stochastic delay differential equations (DPSDDEs) driven by Lèvy process. Such DPSDDEs take the form: t x(t) = x(0) +

t f (x(s), x(δ(s)))ds +

0

t  +

g(x(s), x(δ(s)))d B(s) 0

(ds, dl) + α sup x(s) + β inf x(s), H (x(s−), x(δ(s)−), l) N

0 |l| 0 and R+ = [0, +∞). Let C([−τ, 0]; Rd ) be the family p of continuous functions from [−τ, 0] to Rd with norm ||ϕ|| = sup−τ ≤θ ≤0 ϕ(θ ). Denote by L Ft ([−τ, 0]; Rd ) the family of Ft -measurable, C([−τ, 0]; Rd )-valued random variables ξ = {ξ(s), −τ ≤ s ≤ 0} such that E||ξ || p = sup−τ ≤s≤0 E|ξ(s)| p < +∞. Let C(a) denote a constant, whose value depends only on a. For simplicity, we denote by a ∨ b = max{a, b} and a ∧ b = min{a, b}. Let B = (B(t), t ≥ 0) be an m-dimensional standard Ft -adapted Brownian motion and N be an indepen and intensity meadent Ft -adapted Poisson random measure defined on R+ × (Rd − {0}) with compensator N   sure ν, where ν is a Lèvy measure so that N (dt, dy) := N (dt, dy)−ν(dy)dt and Rd −{0} (|y|2 ∧1)ν(dy) < ∞. p For Eq. (1.1), the initial data x(0) = ξ(0) ∈ L Ft ([−τ, 0]; Rd ), and δ : [0, ∞) → R is a Lipschitz continuous function which satisfies − τ ≤ δ(t) ≤ t and |δ(t) − δ(s)| ≤ ρ|t − s|, ∀t, s ≥ 0, for some positive constant ρ.

123

(2.1)

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In this paper, we make the following assumptions: (H1) There exists a constant K 1 > 0 such that for all −τ ≤ s < t ≤ 0 E|ξ(t) − ξ(s)|2 ≤ K 1 (t − s).

(2.2)

(H2) For each n and each 2 ≤ η ≤ p, there exists a positive constant K 2 (n), such that | f (x1 , y1 ) − f (x2 , y2 )|2 ∨ |g(x1 , y1 ) − g(x2 , y2 )|2 ≤ K 2 (n)(|x1 − x2 |2 + |y1 − y2 |2 ) and



|H (x1 , y1 , l) − H (x2 , y2 , l)|η ν(dl) ≤ K 2 (n)(|x1 − x2 |η + |y1 − y2 |η ),

|l|T 2

0≤t≤T

and σn >T }

" +E

# sup |x(t) − y(t)| I{τn ≤T 2

0≤t≤T

or σn ≤T }

.

(4.1)

By the Young inequality p

p p − 2 p−2 a2 b , + ab ≤ 2h 2 p ph p−2

where a, b, h > 0 and p > 2, we obtain " # sup |x(t) − y(t)| I{τn ≤T 2

E

0≤t≤T

or σn ≤T }

Consequently, "

#

sup |x(t) − y(t)|2 I{τn ≤T

E

0≤t≤T

or σn ≤T }

"

# " 2h p−2 1 p ≤E I{τn ≤T sup |x(t) − y(t)| + E 2 p 0≤t≤T p h p−2

2h ≤ E p

#

"

sup |x(t) − y(t)| p + 0≤t≤T

Set C5 = C1 ∨ C2 . Notice that

|x(τn )| p P(τn ≤ T ) ≤ E I{τn ≤T } np



1 ≤ pE n

# or σn ≤T }

p−2 1 P(τn ≤ T or σn ≤ T ). 2 p h p−2 #

" sup |x(t)|

−τ ≤t≤T

p

≤

C5 np

and P(σn ≤ T ) ≤

C5 . np

We get P(τn ≤ T or σn ≤ T ) ≤ P(τn ≤ T ) + P(σn ≤ T ) ≤

2C5 . np

Moreover, " E

# sup |x(t) − y(t)| p ≤ 2 p−1 E 0≤t≤T

#

"

sup |x(t)| p + sup |y(t)| p ≤ 2 p C5 . 0≤t≤T

0≤t≤T

Substituting these inequalities above into Eq. (4.1) leads to " # " E

sup |x(t) − y(t)|

2

0≤t≤T

123

≤E

sup |x(t) − y(t)| I{τn >T 2

0≤t≤T

# and σn >T }

+

.

2 p+1 hC5 2( p − 2)C5 . (4.2) + 2 p ph p−2 n p

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Let us now estimate the first term on the right-hand side of Eq. (4.2). Clearly, # " sup |x(t) − y(t)|2 I{τn >T

E

0≤t≤T

and σn >T }

#

"

"

sup |x(t) − y(t)|2 1{vn >T } ≤ E

=E

# sup |x(t ∧ vn ) − y(t ∧ vn )|2 .

0≤t≤T

(4.3)

0≤t≤T

Thanks to Eqs. (1.1) and (2.6), we derive |x(t ∧ vn ) − y(t ∧ vn )| t∧v t∧v  n  n ≤| [ f (x(s), x(δ(s))) − f (z 1 (s), z 2 (s))]ds + [g(x(s), x(δ(s))) − g(z 1 (s), z 2 (s))]d B(s) 0 t∧v  n

0



(ds, dl)| + β| [H (x(s−), x(δ(s)−), l) − H (z 1 (s), z 2 (s), l)] N

+ 0 |l|