how to value Credit Default Swap default leg and default probabilities

how to value CDS (credit default swap) default leg with following time structure:

0—-t1—–t2—-t3—–t4—–….—T

Suppose that default (at time \(\tau \)) can only occur at discrete times t1,t2,t3,..
and Qi=survival probability until time $$t_i$$

then

$$ \tau – time of default {=t_1,=t_2,=t_3 …} $$

$$ Q_i=P(\tau>t_i) – survival proba for counterparty $$

$$ P(\tau=t_3)=1-P(\tau\neq t_3)=1-( P(\tau<3 | \tau>=4) ) $$

$$ = 1 – ( 1-Q_2 + Q_3)= Q_2-Q_3 $$

default leg of Credit Default Swap pays:

$$ discounted payoff = (1-R) ( df_1 E{\bf1}(\tau=t_1) + df_2 E{\bf1}(\tau=t_2)+ …) $$
$$ =(1-R) ( df_1 P(\tau=t_1)+df_2 P(\tau=t_2)+ …) $$
$$ =(1-R) ( df_1(1-Q_1)+df_2(Q_1-Q_2)+df_3(Q_2-Q_3) … ) $$

where $$df_1$$ is discount factor for $$t_1$$ and R – recovery rate (normally assumed ot be 40%)

to use continuous time one normally use intensity:
$$ P(\tau>t+dt | \tau>t) = \lambda(t)dt $$
where we ssupose that default is the first jump of Poisson process
$$ Q(t)=exp( -\int_{0}^{T}\lambda(s)ds) $$
in practice $$\lambda$$ is assumed to be piecewise-constant chaning value on market CDS maturities

useful approximations:
Credit Default Swap spread $$\approx p(1-R) \approx \lambda (1-R) $$
where $$\lambda$$ is intensity and approx. equal to probability of default in first year.