The Study On Financial Problems Under Asymmetric Credit Risk Information

It is know that additional information, including insider information, the default information impact the investment decision-making, risk measurement and product pricing. Therefore, these phenomena need deeply analysis and it is interesting to set up the mathematical model. We will study the progressively enlarged ?ltration with theoretical analysis and discuss its ?nancial applications.Assume that there exist two kinds of investors in the market, the F-investors and the G-investors. The F-investors have the market information F( also called initial ?ltration), which is given by a d-dimensional Brownian motion W =(W1, · · ·, Wd)′as well as an integer-valued random measure μ(du, dy). The market might has default at time τ, modeled by the so called the generalized Cox model, and the information of the G-investors is the by the default τ progressively enlarged ?ltration of F. We characterize this model with a triplet and describe main properties such as the survival process and the conditional density of τ.Then the thesis derives the CRRA-utility indi?erence value(UIV) Ctof the F-investors with respect to the G-investors and describe the dynamics of Ctby two BSDEs. In the end, we give an explicit expression of Ctin an example.For the generalized Cox model we typically have that Ct≥ 1 in contrast to the standard Cox model.Next, the random variables τ is the default time and L is default lose. Let G = {Gt; t ≥ 0} be the progressively enlargement of F by( τ, L), i.e, G is the smallest ?ltration including F such that τ is a G-stopping time and L is G τ-measurable. Under the density hypothesis, we consider the G-decomposition of a(P, F) martingale and the representation of a G-martingale. We characterize the conditional density process by ps(s, l), θ1(u; s, l)Iu>sand θ2(u, y; s, l)Iu>s,which makes us to describe the survival process G explicitly. Then we give the G-decomposition of a F martingale explicitly and obtain the predictable representation theorem for a(P, G)-martingale. Formula parametrization in the enlarged?ltration is a useful quality in ?nancial modeling.As the applications, the thesis mainly considers the forward CDS with loss in the framework of stochastic interest rates whose term structures are modeled in the sense of the Heath-Jarrow-Morton model with jumps under the general conditional density hypothesis. We describe the dynamics of the defaultable bond in G and the forward CDS with random loss explicitly by the BSDEs method.