Intuition on this definition of a poisson process

Question

I'm in a context of defaultable assets (stochastic process with a negative jump at some point). The paper I'm reading says that the arrival \textbf{rate} of default is $\psi_t$. What does it mean exactly? The paper uses the identity $\mathbb{E}_t[d N_t] = \psi_td t$, but I'm not familiar with this (abuse of?) notation. The paper can be found here (p. 9). Any help, please? I would appreciate as well books about stochastic integrals (in particular on poisson processes). Thanks.

Answer 1

In a credit default model the simplest way to define the default time $\tau$ is to draw a uniform $U\sim [0,1]$ and put it into the inverse of the CDF of the exponential distribution. With a time dependent default intensity $\psi(t)$ this CDF is $$ F(t)=1-\textstyle\exp(-\int_0^t\psi(s)\,ds)\,. $$ Then set $\tau=F^{-1}(U)\,.$ This ensures that the survival probability is what we expect: $$\mathbb P(\tau>t)=\mathbb P(U>F(t))=1-F(t)=\textstyle\underline{\underline{\exp(-\int_0^t\psi(s)\,ds)\,.}} $$ As far as credit default is concerned we could stop here because an entity defaults typically only once.

To define a Poisson process with time dependent intensity (an inhomogeneous PP) we draw independent uniforms $U_1,U_2,...$ and define independent exponential waiting times $\tau_1,\tau_2,\dots$ in the same way as above. The jump times are $$ \sigma_n=\sum_{i=1}^n\tau_i $$ and the Poisson process is $$ N_t=\sum_{i=1}^\infty 1_{\{\sigma_i\le t\}}\,. $$ The notation $\mathbb E[dN_t]=\psi(t)\,dt$ is nothing else than an abbreviation of the fact that $N_t-\int_0^t\psi(s)\,ds$ is a martingale which we can write also as: $$ \mathbb E[N_t|{\cal F}_s]=N_s+\textstyle\int_s^t\psi(u)\,du\,. $$