## survival probability

The

*survival probability***(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.**

*P*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

**(0,T), for any fixed T, is a known number that represents the probability that the event E will not occur until T.**

*P*It follows that

**(0,T) is always a decreasing function of T with**

*P***(0,T) -> 0 for T -> infinity.**

*P*Also

**(0,0) = 1 provided that E has not occurred so far, otherwise**

*P***(0,0) = 0.**

*P*Note that

**(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.**

*P*The following mathematical relations hold between the survival probability

**, default density**

*P(T)***and hazard rate**

*g(T)***, where**

*h(T)***stands for**

*P(T)***(t,T) with the first time parameter dropped for notational simplicity and similarly for**

*P***and**

*g(T)***:**

*h(T)***=**

*P(T)*

*e⁻ᴵ⁽ᵀ⁾***=**

*g(T)*

*h(T)*

*e⁻ᴵ⁽ᵀ⁾*where

**is the time integral of h(t,u) over the second parameter u from t to T.**

*I(T)*It turns out that the knowledge of any one of

**,**

*P***or**

*g***as a function of time suffices to compute the value of the other two as well.**

*h*