survival probability

The survival probability P(t,T) for some predefined stochastic event E (often thought as a "default" event) and times t, T such that t <= T is defined as the probability that the event E will not have occurred by time T.
In Finance one usually sets the time origin t = 0 to mean today's date, or more precisely the current instant, when all market information is assumed to be known with certainty.
Then P(0,T), for any fixed T, is a known number that represents the probability that the event E will not occur until T.
It follows that P(0,T) is always a decreasing function of T with P(0,T) -> 0 for T -> infinity.
Also P(0,0) = 1 provided that E has not occurred so far, otherwise P(0,0) = 0.
Note that P(t,T), for 0 < t < T is not a simple number, but rather a random variable, which may assume any possible number between 0 and 1 at the future time t.
The following mathematical relations hold between the
survival probability P(T), default density g(T) and hazard rate h(T), where P(T) stands for P(t,T) with the first time parameter dropped for notational simplicity and similarly for g(T) and h(T):
P(T) = e⁻ᴵ⁽ᵀ⁾
g(T) = h(T)e⁻ᴵ⁽ᵀ⁾
where I(T) is the time integral of h(t,u) over the second parameter u from t to T.
It turns out that the knowledge of any one of P, g or h as a function of time suffices to compute the value of the other two as well.