Backward Stochastic Differential Equations Driven by G -Brownian Motion with Double Reflections
- PDF / 465,440 Bytes
- 30 Pages / 439.37 x 666.142 pts Page_size
- 107 Downloads / 168 Views
Backward Stochastic Differential Equations Driven by G-Brownian Motion with Double Reflections Hanwu Li1
· Yongsheng Song2,3
Received: 22 April 2020 / Revised: 28 August 2020 © The Author(s) 2020
Abstract In this paper, we study the reflected backward stochastic differential equations driven by G-Brownian motion with two reflecting obstacles, which means that the solution lies between two prescribed processes. A new kind of approximate Skorohod condition is proposed to derive the uniqueness and existence of the solutions. The uniqueness can be proved by a priori estimates and the existence is obtained via a penalization method. Keywords G-expectation · Reflected backward SDE · Approximate Skorohod condition Mathematics Subject Classification (2010) 60H10
Li’s research was supported by the German Research Foundation (DFG) via CRC 1283. Song’s research was supported by National Key R&D Program of China (No. 2018YFA0703901); NCMIS; NSFCs (Nos. 11871458 & 11688101), and Key Research Program of Frontier Sciences, CAS (No. QYZDB-SSW-SYS017).
B
Hanwu Li [email protected] Yongsheng Song [email protected]
1
Center for Mathematical Economics (IMW), Bielefeld University, Universitätsstrasse 25, 33615 Bielefeld, Germany
2
RCSDS, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China
3
School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China
123
Journal of Theoretical Probability
1 Introduction Given a filtered probability space (, F, (Ft )t∈[0,T ] , P), Pardoux and Peng [20] first introduced the following type of nonlinear backward stochastic differential equations (BSDEs for short):
T
Yt = ξ +
T
f (s, Ys , Z s )ds −
t
Z s dBs ,
t
where the generator f (·, y, z) is progressively measurable and Lipschitz continuous with respect to (y, z), ξ is an FT -measurable and square integrable terminal value. They proved that there exists a unique pair of progressively measurable processes (Y , Z ) satisfying this equation. The BSDE theory attracts a great deal of attention due to its wide applications in mathematical finance, stochastic control and quasilinear partial differential equations (see [8,21], etc). One of the most important extensions is the reflected BSDE initiated by El Karoui et al. [7]. In addition to the generator f and the terminal value ξ , there is an additional continuous process S, called the obstacle, prescribed in this problem. The reflection means that the solution is forced to be above this given process S. More precisely, the solution of the reflected BSDE with parameters (ξ, f , S) is a triple of processes (Y , Z , L) such that Yt = ξ +
T
T
f (s, Ys , Z s )ds + L T − L t −
t
Z s dBs ,
t
T
Yt ≥ St , t ∈ [0, T ], and
(Ys − Ss )dL s = 0, P-a.s.,
0
where L is an increasing process to push the solution upward. Besides, it should behave in a minimal way, which means that L only acts when the solution Y reaches T the obstacle S. This requirement corresponds to the mathematical expr
Data Loading...
