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In this chapter, we consider the stochastic differential equations and backward stochastic differential equations driven by G-Brownian motion. The conditions and proofs of existence and uniqueness of a stochastic differential equation is similar to the classical situation. However the corresponding problems for backward stochastic differential equations are not that easy, many are still open. We only give partial results to this direction.
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The material in this chapter is mainly from Peng . There are many excellent books on Itô’s stochastic calculus and stochastic differential equations based by Itô’s original paper . The ideas of that notes were further developed to build the nonlinear martingale theory. For the corresponding classical Brownian motion framework under a probability measure space, readers are referred to Chung and Williams  , Dellacherie and Meyer , He, Wang and Yan , Itô and McKean , Ikeda and Watanabe , Kallenberg , Karatzas and Shreve , Øksendal , Protter , Revuz and Yor  and Yong and Zhou . Linear backward stochastic differential equations (BSDEs) were first introduced by Bismut in [17, 19]. Bensoussan developed this approach in [12, 13]. The existence and uniqueness theorem of a general nonlinear BSDE, was obtained in 1990 in Pardoux and Peng . Here we present a version of a proof based on El Karoui, Peng and Quenez , which is an excellent survey paper on BSDE theory and its applications, especially in finance. Comparison theorem of BSDEs was obtained in Peng  for the case when g is a \(C^1\)-function and then in  when g is Lipschitz. Nonlinear Feynman-Kac formula for BSDE was introduced by Peng [127, 129]. Here we obtain the corresponding Feynman-Kac formula for a fully nonlinear PDE, within the framework of G-expectation. We also refer to Yong and Zhou , as well as Peng  (in 1997, in Chinese) and  and more resent monographs of Crepey , Pardoux and Rascanu  and Zhang  for systematic presentations of BSDE theory and its applications. In the framework of fully nonlinear expectation, typically G-expectation, a challenging problem is to prove the well-posedness of a BSDE which is general enough to contain the above ‘classical’ BSDE as a special case. By applying and developing methods of quasi-surely analysis and aggregations, Soner et al. [156–158], introduced a weak formulation and then proved the existence and uniqueness of weak solution 2nd order BSDE (2BSDE). We also refer to Zhang  a systematic presentation. Then, by using a totally different approach of G-martingale representation and a type of Galerkin approximation, Hu et al.  proved the existence and uniqueness of solution of BSDE driven by G-Brownian motions ( G-BSDE). As in the classical situation, G-BSDE is a natural generalization of representation of G-martingale. The assumption for the well-posedness of 2BSDEs is weaker than that of G-BSDE, whereas the solution ( Y, Z, K) obtained by GBSDE is quasi-surely continuous which is in general smoother than that of 2BSDE. A very interesting problem is how to combine the advantages of both methods. Then Hu and Wang  considered ergodic G-BSDEs, see also . In , Hu, Lin and Soumana Hima studied G-BSDEs under quadratic assumptions on coefficients. In , Li, Peng and Soumana Hima investigated the existence and uniqueness theorem for reflected G-BSDEs. Furthermore, Cao and Tang  dealed with reflected Quadratic BSDEs driven by G-Brownian Motions.
- Stochastic Differential Equations
- Springer Berlin Heidelberg
- Chapter 5
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