Hazard Function And Hazard Process Of Random Time In Default Risk Model

There are a wide range of contract objects which are related to credit risk in the financial market,including bonds,loans and various derivatives.The most critical step in pricing those contract objects is about the characterization of the default.Two main methods are frequently used in research before.One is structural-form model,another is reduce-form model(intensity-based model).We focus on the sigma-field which represents the information about the diffusion process or stopping time and introduce hazard function and hazard process.Under several assumptions,we would compare different martingales and represent theorem in different sigma-field.