Comparison and converse comparison theorems for backward stochastic   differential equations with Markov chain noise
  Comparison and converse comparison theorems are important parts of the research on backward stochastic differential equations. In this paper, we obtain comparison results for one dimensional backward stochastic differential equations with Markov chain noise, extending and generalizing previous work under natural and simplified hypotheses, and establish a converse comparison theorem for the same type of equation after giving the definition and properties of a type of nonlinear expectation: $f$-expectation. 
Zhe Yang, Dimbinirina Ramarimbahoaka, Robert J. Elliott
math.PR