Filtration-Consistent Dynamic Operator with a Floor and Associated Reflected Backward Stochastic Differential Equations

This paper introduces the notion of an {Ft, 0 ≤ t ≤ T}-consistent dynamic operator with a floor, by suitably formulating four axioms. It is shown that an {Ft, 0 ≤ t ≤ T}consistent dynamic operator {Es,t, 0 ≤ s ≤ t ≤ T} with a continuous upper-bounded floor S is necessarily represented by the solution of a backward stochastic differential equation (BSDE) reflected upwards on the floor, if it is Eμ-super-dominated for some μ > 0 and if it has the non-increasing and floor-above-invariant property of forward translation. We make full use of the two assumptions to extend the underlying {Ft, 0 ≤ t ≤ T}-consistent dynamic operator from the subset L (FT ;ST ) of floor-dominating square-integrable random variables to the whole space L(FT ) of square-integrable random variables. The extended dynamic operators are shown to be identified to an {Ft, 0 ≤ t ≤ T}-consistent expectation Ẽ . The generator g of its BSDE representation given by Coquet, Hu, Memin, and Peng [1, Probability Theory and This work is partially supported by the NSFC under grants 10325101 (distinguished youth foundation) and 101310310 (key project), and the Science Foundation of Chinese Ministry of Education under grant 20030246004. Institute of Mathematics, School of Mathematical Sciences, Fudan University, Shanghai 200433, China, & Key Laboratory of Mathematics for Nonlinear Sciences (Fudan University), Ministry of Education. Department of Finance and Control Sciences, School of Mathematical Sciences, Fudan University, Shanghai 200433, China, & Key Laboratory of Mathematics for Nonlinear Sciences (Fudan University), Ministry of Education. E-mail: [email protected].