# A model to stochastic hazard rate and CDS spread term structure

I'm interested in the term structure of CDS spread.

It's known that the Market CDS rate (fair CDS spread or T-maturity spread) of a CDS contract initiated at $s$, maturity $T$ and recovery function $R$ is given by

$$S(s,T) = \frac{-\int_s^T R(u) dG(u)}{\int_s^T G(u)du}$$

where $G(t)=\mathbb P[\tau>t]$ is the survival probability associated to default with default time $\tau$. Also one has that $G(t)= \exp({-\int_0^t h_u du})$ where $h$ is the hazard rate (HR).

My first question is what is the easy and market practice to get the hazard ratio from CDS spread.

Assume that the CDS spread is given by a know stochastic process. Let's say we have even a analytical tractability for the stochastic CDS spread as in the SRMR model.

How to obtain the survival probability $G$ ?

Another doubt I have is concerning the SRMR model and the role of maturity. Unless I missed something he gives us model to the dynamic of the market spread (for the log return so for the rate itself) for a fixed maturity $T$ which is not explicated anywhere in this model. Assuming we how to solve the integral equation $\int_s^T R(u) dG(u)+ S(t, T)\int_s^T G(u)du =0$ I would be treating every T-maturity as they had the same dynamic.

That may be ok and the distinction between their behaviors must comes from the different parameter obtained by proper calibration for each different maturity.

$$dH_t= -\mu_t H_t ~dt+\sigma_tdW_t$$ with $H_t := \ln(H_t)$.