Titel_SS06

t

x(t),E(t)

x(t)

E(t)

Figure 8.3: Illustration of a realisation of a continuous stochastic process.

Consider the case where the probability of a first out-crossing is of interest for the scalar
process >X(),^33 0,T? with the failure domain bounded by E()t as illustrated in Figure

8.4.

x(t),E(t)

t

E(t)

N 1

N 0

2 N 0

(^3132)
2 4
N 0
2 N 1
x(t),E(t)
t
E(t)
N 1
N 0
2 N 0
(^3132)
2 4
N 0
2 N 1
Figure 8.4: Illustration of realisations of continuous stochastic processes.
The probability of the process being in the failure domain in the interval 0,T may be written
as:
0
0

() (0) (1 (0)) ()

T

PT PffPff d 33 (8.7)

where PPXf(0)( (0) E(0))

0 



, i.e. the probability that the process starts in the failure domain

and f 3 is the first excursion probability density function.

Equation (8.7) may easily be derived by consideration of Figure 8.4. From this figure it is seen
that all possible realisations N at time 3  0 may be divided up into two types of realisations,
i.e. N 0 realisations in the safe domain and N 1 realisations in the failure domain. For 3 & 0 tree

types of realisations are considered, namely the realisations 2 N 0 starting in the safe domain

and having a first excursion in the time interval  3312 , , 2 N 0 * the number of N 0 realisations