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The price for a simple credit bond, where a credit event is modeled as the first jump of a Poisson process $N$, with stochastic hazard rate $\lambda$, is given by

$$P_t = P(t, \lambda, N)$$

such that,

$$\mathrm{d}P_t = \frac{\partial P}{\partial t}\mathrm{d}t + \frac{\partial P}{\partial \lambda }\mathrm{d}\lambda + \frac{\partial P}{\partial N}\mathrm{d}N + \frac{1}{2}\frac{\partial^2 P}{\partial \lambda^2}\mathrm{d}\lambda ^2$$

This is the general formulation of the reduced form of the model. Since $\lambda$ simply describes $N$, why do we need them both in the equation? Also, why are we keeping the $\lambda^2$ term but not the $N^2$ term? Does $N^2$ vanish somehow under Ito's lemma?

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  • $\begingroup$ This seems to follow from Ito's lemma but because $N$ is a finite variation process it enters similarly to $t$. $\endgroup$
    – fes
    Dec 17, 2020 at 17:47
  • $\begingroup$ @Erik Kjellgren Are you sure $\delta$ instead of $\partial$ didn't have a meaning? $\endgroup$
    – fes
    Dec 17, 2020 at 19:33
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    $\begingroup$ @fesman $\delta$ would imply it is a functional derivative. Meaning that $t$, $\lambda$, and $N$ are not variables, but functions themself. If we look at Wikipedia for Itô's lemma, en.wikipedia.org/wiki/It%C3%B4%27s_lemma. We can see that it is for partial derivatives and not functional derivatives. $\endgroup$
    – Erik Kjellgren
    Dec 17, 2020 at 19:38
  • $\begingroup$ thanks guys, yeah i meant to write partial $\endgroup$
    – dayum
    Dec 17, 2020 at 19:44
  • $\begingroup$ @Erik Kjellgren Ok thanks I thought it might have a different symbol if the multiplier on $dN$ is not a partial derivative but a discrete change in the function when the jump occurs, (see Ito's lemma for jump processes). $\endgroup$
    – fes
    Dec 17, 2020 at 19:47

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