Reflected Quadratic BSDEs Driven by G -Brownian Motions
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Chinese Annals of Mathematics, Series B c The Editorial Office of CAM and
Springer-Verlag Berlin Heidelberg 2020
Reflected Quadratic BSDEs Driven by G-Brownian Motions∗ Dong CAO1
Shanjian TANG2
Abstract In this paper, the authors consider a reflected backward stochastic differential equation driven by a G-Brownian motion (G-BSDE for short), with the generator growing quadratically in the second unknown. The authors obtain the existence by the penalty method, and some a priori estimates which imply the uniqueness, for solutions of the G-BSDE. Moreover, focusing their discussion at the Markovian setting, the authors give a nonlinear Feynman-Kac formula for solutions of a fully nonlinear partial differential equation. Keywords G-Brownian motion, G-Martingale, Quandratic growth, G-BSDEs, Probabilistic representation 2000 MR Subject Classification 60H10, 60H30
1 Introduction A general backward stochastic differential equation (BSDE for short) takes the following form: Z T Z T Yt = ξ + f (s, Ys , Zs )ds − Zs dWs , t ∈ [0, T ]. t
t
The function f is conventionally called the generator and the random variable ξ is called the terminal value. Bismut [2–3] initially gave a complete linear theory, where the generator is linear in both unknown variables, and derived the stochastic Riccati equation as a particular nonlinear BSDE where the generator is quadratic in the second unkown variable. Pardoux and Peng [29] established the existence and uniqueness result when the generator f is uniformly Lipschitz continuous in both unknown variables and the terminal value ξ is square integrable. Subsequently, an intensive attention has been given to relax the assumption of the uniformly Lipschitz continuity on the generator. In particular, the one-dimensional BSDE with a quadratic generator (i.e., the so-called quadratic BSDE) was studied by Kobylanski [18] for a bounded terminal value ξ, and by Briand and Hu [5–6] for an unbounded terminal value ξ of some suitable exponential moments. The multi-dimensional quadratic BSDE was discussed by Tang [39] and Hu and Tang [16]. Manuscript received March 13, 2019. of Mathematical Sciences, Fudan University, Shanghai 200433, China. E-mail: [email protected] 2 Department of Finance and Control Sciences, School of Mathematical Sciences, Fudan University, Shanghai 200433, China. E-mail: [email protected] ∗ This work was supported by the National Science Foundation of China (No. 11631004) and the Science and Technology Commission of Shanghai Municipality (No. 14XD1400400). 1 School
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D. Cao and S. J. Tang
As a constrained BSDE, a reflected backward stochastic differential equation (RBSDE for short) was formulated and studied by El Karoui et al. [11], where the first unknown Y is required to stay up a given continuous process S and an additional increasing process which satisfies the Skorohod condition, is thus introduced into the equation. Subsequently, much efforts have been made to relax the Lipschitz assumption on the generator. For the quadratic case, see Kobylanski et al. [19] with
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