Hindawi Publishing Corporation Journal of Applied Mathematics Volume 2013, Article ID 729636, 7 pages http://dx.doi.org/10.1155/2013/729636
Research Article Reflected Backward Stochastic Differential Equations Driven by Countable Brownian Motions Pengju Duan,1,2 Min Ren,2 and Shilong Fei2 1 2
Laboratory of Intelligent Information Processing, Suzhou University, Anhui 234000, China Department of Mathematics, Suzhou University, Anhui 234000, China
Correspondence should be addressed to Pengju Duan;
[email protected] Received 16 June 2013; Accepted 15 September 2013 Academic Editor: Lotfollah Najjar Copyright © 2013 Pengju Duan et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. This paper deals with a new class of reflected backward stochastic differential equations driven by countable Brownian motions. The existence and uniqueness of the RBSDEs are obtained via Snell envelope and fixed point theorem.
1. Introduction The nonlinear backward stochastic differential equations (BSDEs in short) were introduced by Pardoux and Peng [1], who proved the existence and uniqueness of the solution under the Lipschitz conditions for giving the probabilistic interpretation of semilinear parabolic partial differential equations. Since then, many authors were devoted to studying the BSDEs (see, e.g., [2–8] and the references therein). At present, the theory of BSDEs becomes a powerful tool to solve practical matters. In 1994, Pardoux and Peng [9] firstly studied the backward doubly stochastic differential equations (BDSDEs in short), which are driven by two kinds of Brownian motions. Later, Boufoussi et al. [10] established the connection between a class of generalized BDSDEs and semilinear stochastic partial differential equations with a Neumann boundary condition. Reflected backward differential equations (RBSDEs in short) were introduced by El Karoui et al. [11]. Later, many researchers discussed various kinds of RBSDEs for their deep application in mathematical finance and partial differential equations. Ren and Hu [12] proposed the RBSDEs, driven by Teugels martingales and Brownian motion, and derived the existence and uniqueness of the solution by means of the Snell envelope and the fixed point theorem when the barrier was right continuous with left limits. Ren and El Otmani [13] discussed the generalized reflected BSDEs driven by L´evy process. Recently, Ren et al. [14] studied a new class
of reflected backward doubly stochastic differential equations driven by L´evy process and Brownian motion. As in all the previous works, the equations are driven by finite Brownian motions. To the best of our knowledge, there are no papers on the reflected backward stochastic differential equations driven by countable Brownian motions. In this paper, we aim to derive the existence and uniqueness of the solution for the RBSDEs driven by countable Brownian motions. The structure of the paper is organized as follows. In Section 2, we give some notations. Section 3 is devoted to the main result.
2. Notations Let 𝑇 be a positive constant. Throughout the paper (Ω, F, P) is a complete probability space equipped with the natural filtration {F𝑡 }𝑡≥0 satisfying the usual conditions. {𝛽𝑗 (𝑡)}∞ 𝑗=1 are mutual independent one-dimensional standard Brownian motions on the probability space. 𝑊(𝑡) is a standard Brownian motion on R𝑑 which is independent of 𝛽𝑗 (𝑡). Assume that ∞
𝛽
F𝑡 = (⋁F𝑡,𝑇𝑗 ) ⋁ F𝑊 𝑡 ⋁ N,
(1)
𝑗=1
𝜂
where for any process {𝜂𝑡 }, F𝑠,𝑡 = 𝜎{𝜂𝑟 − 𝜂𝑠 : 𝑠 ≤ 𝑟 ≤ 𝑡}, 𝜂 𝜂 F𝑡 = F0,𝑡 , and N denotes the class of P-null sets of F.
2
Journal of Applied Mathematics
3. Main Result
For the convenience, let us introduce some spaces: 2
(i) H = {(𝜑𝑡 )0≤𝑡≤𝑇 : an F𝑡 -progressively measurable, R𝑇
valued process such that 𝐸 ∫0 |𝜑𝑡 |2 d𝑡 < ∞};
(ii) S2 = {(𝜓𝑡 )0≤𝑡≤𝑇 : an F𝑡 -progressively measurable, continuous process such that R𝑑 -valued 𝐸(sup0≤𝑡≤𝑇 |𝜓𝑡 |2 ) < ∞};
In order to get the solution of (2), we consider the following RBSDEs driven by finite Brownian motions: 𝑇
𝑌𝑡 = 𝜉 + ∫ 𝑓 (𝑠, 𝑌𝑠 , 𝑍𝑠 ) d𝑠 𝑡
𝑛
(iii) A2 = {(𝐾𝑡 )0≤𝑡≤𝑇 : an F𝑡 -adapted, continuous, increasing process such that 𝐾0 = 0, 𝐸|𝐾𝑡 |2 < ∞}.
𝑇
+ ∑ ∫ 𝑔𝑗 (𝑠, 𝑌𝑠 , 𝑍𝑠 ) d𝛽𝑗 (𝑠)
(3)
𝑗=1 𝑡 𝑇
With the previous preparations, we consider the following RBSDEs:
− ∫ 𝑍𝑠 d𝑊 (𝑠) + 𝐾𝑇 − 𝐾𝑡 ,
0 ≤ 𝑡 ≤ 𝑇.
𝑡
𝑇
𝑌𝑡 = 𝜉 + ∫ 𝑓 (𝑠, 𝑌𝑠 , 𝑍𝑠 ) d𝑠
Firstly, we consider a special case of (3); that is, the functions 𝑓 and 𝑔 do not depend on (𝑌, 𝑍):
𝑡
∞
𝑇
+ ∑ ∫ 𝑔𝑗 (𝑠, 𝑌𝑠 , 𝑍𝑠 ) d𝛽𝑗 (𝑠)
(2)
𝑗=1 𝑡
𝑇
𝑌𝑡 = 𝜉 + ∫ 𝑓 (𝑠) d𝑠 𝑡
𝑇
− ∫ 𝑍𝑠 d𝑊 (𝑠) + 𝐾𝑇 − 𝐾𝑡 , 𝑡
𝑛
0 ≤ 𝑡 ≤ 𝑇,
Definition 1. A solution of (2) is a triple of R × R𝑑 × R+ value process (𝑌𝑡 , 𝑍𝑡 , 𝐾𝑡 )0≤𝑡≤𝑇 , which satisfies (2), and (i) (𝑌𝑡 , 𝑍𝑡 , 𝐾𝑡 )0≤𝑡≤𝑇 ∈ S2 × H2 × A2 ; (ii) 𝑌𝑡 ≥ 𝑆𝑡 ; (iii) 𝐾𝑡 is a continuous and increasing process with 𝐾0 = 0 𝑇 and ∫0 (𝑌𝑡 − 𝑆𝑡 )d𝐾𝑡 = 0. In order to get the solution of (2), we propose the following assumptions: (H1) 𝜉 is an F𝑇 measurable square integrable random variable; (H2) the obstacle {𝑆𝑡 : 0 ≤ 𝑡 ≤ 𝑇} is an F𝑡 -progressive measurable continuous real valued process which satisfies 𝐸 sup0≤𝑡≤𝑇 (𝑆𝑡 )2 < ∞. We always assume that 𝑆𝑇 ≤ 𝜉, a.s.; (H3) 𝑓(⋅, 𝑦, 𝑧) and 𝑔𝑗 (⋅, 𝑦, 𝑧) are two progressive measurable functions such that, for any 𝑡 ∈ [0, 𝑇], 𝑦1 , 𝑦2 ∈ R, 𝑧1 , 𝑧2 ∈ R𝑑 , (3a) 𝑓(𝑠, ⋅, ⋅) is continuous and |𝑓(𝑠, 𝑦, 𝑧)| ≤ 𝑀(1 + |𝑦| + |𝑧|); 𝑇
∑∞ 𝑗=1
𝑇
