On absolutely continuous compensators and nonlinear filtering equations in default risk models (Q454855)

This paper deals with the financial instrument with default risk, which is a corporate bond with maturity \(T\) that pays the owner \(F\) units of a currency if the firm does not default until time \(T\). If the firm defaults before, usually there is a nonzero rebate \(R\) paid to the bond holder. In the case of a zero-coupon defaultable bond, \(R=0\) and \(F=1\), the time \(t\) price of this bond on the event \([\tau>t]\) is given by \(J_{t}-E[{\mathbf 1}_{[t<\tau\leq T]}\Delta J_{\tau}|{\mathcal G}_{t}]\), \(t\in[0,T]\), where \(J_{t}=E\left[\left.\exp\left(-\int_{t}^{T\wedge \tau}\lambda_{s}ds\right) \right|{\mathcal G}_{t}\right]\), where \(\tau\) is a default time and \(\lambda\) is the default intensity. The author shows that, under natural regularity conditions, \(({\mathbf 1}_{[\tau\leq t]})_{t\geq0}\) has an absolutely continuous \({\mathcal G}\)-compensator when \(\tau\) is the first hitting time of 0 for the diffusion \(X_{t}=X_0+W_{t}+\int_{0}^{t}a(X_{s})ds\) and the observation process is given by \(Y_{t}=B_{t}+\int_{0}^{t}b(s,X_{s})ds\), where \(B\) and \(W\) are independent standard Brownian motions. An alternative formula for the price of a zero-coupon defaultable bond before default is given by \(Z^{-1}_{t}E[Z_{T}|{\mathcal F}_{t}^{Y}]\), where \(Z\) is the so-called Azéma supermartingale defined by \(Z_{t}:=P[\tau>t| {\mathcal F}_{t}^{Y}]\). It is shown that the finite variation part of the Doob-Meyer decomposition of \(Z\) is absolutely continuous, which will in turn imply the existence of \(\lambda\). An explicit representation for \(\lambda\) is presented. The author solves the nonlinear filtering problem corresponding to the \({\mathcal G}\)-optional projection of semimartingales and obtains the Kushner-Stratonovich equations for the \({\mathcal G}\)-conditional distribution of \(X\). As an application of the filtering equations, the Doob-Meyer decomposition for the value process of the rebate is obtained and the equation of extrapolation is presented.