2013/01/07 by Kun Tian, Tian, Kun, Dewen Xiong +3
Economics, Econometrics and Finance · Mathematics · #FOS: Mathematics #Financial Risk and Volatility Modeling #Probability (math.PR) #Stochastic processes and financial applications #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.1301.1119
openalex publication_date 2013/01/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper, we assume that the filtration \bb F is generated by a d-dimensional Brownian motion W=(W1,⋯,Wd)' as well as an integer-valued random measure μ(du,dy). The random variable \ttau is the default time and L is the default loss. Let \mathbb G=\\scr Gt;t≥ 0\ be the progressive enlargement of \bb F by (\ttau,L), i.e, \bb G is the smallest filtration including \bb F such that \ttau is a \bb G-stopping time and L is \scr G_\ttau-measurable. We parameterize the conditional density process, which allows us to describe the survival process G explicitly. We also obtain the explicit \bb G-decomposition of a \bb F martingale and the predictable representation theorem for a (P,\bb G)-martingale by all known parameters. Formula parametrization in the enlarged filtration is a useful quality in financial modeling.