非Lipschitz条件下基于g-期望的Jensen不等式

Jensen's Inequality for g-Expectation with Non-Lipschitz Generator

研究了一类具有非Lipschitz生成元的g-期望的Jensen不等式,利用g-期望的定义,严格单调性以及一个生成元表示定理,证明了一类非Lipschitz生成元的惟一性.对于线性凸函数φba(x)=ax+b,x∈R,定义了一个新的生成元gc(t,y,z)∶=g(t,y-c,z),利用生成元惟一性,BSDEs解的存在惟一性,证明了基于g-期望的Jensen不等式对单调增加的凸函数成立的充分必要条件是g不依赖于变量y并且g关于变量z是正齐次的.

Jensen;s inequality for g-expectation with non-Lipschitz generator was studied. Based on the definition of g-expectation, strict monotonicity and a general representation theorem for generators of BSDEs, a uniqueness theorem for g-expectation with non-Lipschitz generator was proved. For a linear convex ruction φa^b （x） = ax ＋ b, A↓ x ∈ R, a new generator g^c（t,y,z） ：= g（t,y-c,z） was defined. According to the uniqueness theorem for g-expectation and the uniqueness of the solution of the BSDEs, it is proved that Jensen＇s inequality for g-expectation with non-Lipschitz generator holds for monotonic, increasing and convex function if and only if the generator g is independent of y and positively homogeneous with respect to z.