Page 468 - A First Course In Stochastic Models

F. NUMERICAL LAPLACE INVERSION 463

2. f (t) is continuous for t ≥ 0.
3. For any b > 0 the function e −bt f (t) is monotone for t ≥ t 0 (b) for some number
t 0 (b).
∞ −bt
4. e |f (t)| dt < ∞ for any b > 0.
0
In probability applications the function f (t) is often the complementary cumula-
tive probability distribution function of a continuous random variable. In this case
the conditions 1 to 4 are automatically satisfied. A basic result from analysis is that
a real-valued function f (t) is of bounded variation if and only if it can be writ-
ten as the difference of two monotone functions. Under the above conditions the
following version of the Poisson summation formula from Fourier analysis holds:

∞ ∞
2πn −b(t+2πn/h) h inht
f t + e = f (b + inh)e
∗
h 2π
n=−∞ n=−∞
for any constants h, b > 0. This Poisson summation formula is the basis for the
following algorithm of Abate and Whitt (1992).

Inversion algorithm of Abate and Whitt

In Abate and Whitt (1992) it was shown that

1 1
a ∞
e 2 a # a $ e 2 k a + 2kπi
f (t) = f ∗ + (−1) Re f ∗ − ǫ(a, t) (F.1)
2t 2t t 2t
k=1
for any constant a > 0, where the error term ǫ(a, t) is given by
∞
−na
ǫ(a, t) = e f ((2n + 1)t).
n=1

To calculate f (t) from (F.1) for a given value of t, we need f (s) for the sequence
∗
{(a + 2kπ)/2t, k = 0, 1, . . . } of complex numbers. In calculating f (t) through the
representation (F.1) there are three possible sources of error. First the discretization
error associated with ǫ(a, t). Second, the truncation error associated with approxi-
mately calculating the infinite series in (F.1). Third, the round-off error associated
with subtracting positive numbers that are close to each other. The discretization
error can be controlled by choosing the constant a sufficiently large. Assuming that
the function f (t) is bounded by 1, as typically holds in probability applications, it
follows from the inequality
e −a
|ǫ(a, t)| ≤ −a
1 − e