T. Siegl, R.F. Tichy Insurance: Mathematics and Economics 26 2000 59–73 67
This operator is strictly contracting as we get for the maximum norm of the difference of two solutions U and V
kIU − IV k
∞
= max
≤x∞
|IUx − IV x|, by the linearity of I the relation
kIU − IV k
∞
= max
≤x∞
Z
∞
e
−λ+γ +ηt
λ Z
x +ct
U − V x + ct − y dF yγ U − V αx + ct + K
dt ≤ max
≤x∞
Z
∞
e
−λ+γ +ηt
λ Z
x +ct
dF y + γ
kU − V k
∞
dt ≤ max
≤x∞
Z
∞
e
−λ+γ +ηt
λ + γ dtkU − V k
∞
, ≤ 1kU − V k
∞
, where 1
= λ + γ λ + γ + η 1. Therefore, we have proved contraction with an explicit contraction constant.
6. Numerical solution
We consider the case with a finite upper barrier and α 1 to be the most interesting case in applications. Therefore, we will show some methods to solve this model with stochastic simulation. While the exponential distribution is very
convenient for analytical arguments, it can seldom be justified in practice. However, when we develop numerical methods, we often want such a benchmark model to test the accuracy and speed of the solution. The idea is, that
similar problems will have a comparable behavior.
6.1. Evaluation of the analytical solution We used the parameters in Table 1 for the numerical experiments. The series in Eq. 3 was computed by truncating
the series at a given index J = 80. We have computed the free coefficients under the restrictions Eqs. 8, 17 and
18. Furthermore, we have replaced the limit in Eq. 17 by the finite truncation point J . The symbolic computation package MAPLE was used to evaluate the function. Due to the comparatively simple operations needed to solve
our problem, and the fact that convergence is very fast for α ≪ 1 the analytic solution was much faster than any
simulation for comparable accuracy. 6.2. Discussion of the dependency on the parameters
Based on the parameters in Table 1 we have plotted the probability of survival at ˜x = x
max
2 in Fig. 2. We have assumed that we can buy a unit of securitization payout at the cost of a continuous payment of γ . It can be seen that
the optimal securitization size is K ≈ 0.5 with c ≈ 1.15.
From the definition of the process we can easily see that the probability of survival at a given initial capital x is an increasing function of K and c. Furthermore, we obtain that U
˜x
c, K is a concave function of c and K. Thus,
if we finance a securitization of size K by a continuous payment proportional to K, we obtain a concave function of K. Therefore, we have shown that U
˜x
˜c − γ K, K, which is depicted in Fig. 2, is a concave function of K. Furthermore, we can easily see that U
˜x
is a decreasing function of x
max
, λ and γ , and an increasing function of α.
6.3. Stochastic simulation We have compared the simulated values with the analytic solution using a crude standard simulation of the process
with N = 10 000 as explained in, e.g. Siegl and Tichy 1996 see Fig. 3. We used the pseudo-random sequence
68 T. Siegl, R.F. Tichy Insurance: Mathematics and Economics 26 2000 59–73
Fig. 2. Comparison of different securitization sizes K where c = ˜c − γ K.
Fig. 3. Comparison of analytical and numerical solution.
of Afflerbach 1990 based on the recursion x
i
= 532 393x
i −1
+ 1 mod 268 435 456 30
with x = 12 785. Here, we want to note that one can find a large number of alternative efficient pseudo-random
number generators which could be used instead of the Afflerbach sequence see e.g. Fishman, 1990. Our choice is motivated by our previous experience regarding applications of the sequence in similar problems. The sequence is
also used in the hybrid method presented later. The integral operator can also be used to obtain a numerical solution for our problem by means of a simulation.
We employ a hybrid Monte Carlo method that makes use of a low discrepancy sequence. 6.4. Quasi-Monte Carlo sequences
By employing the linear congruential generator 30 we are already using a deterministic algorithm. Instead of a Monte Carlo algorithm we speak of a pseudo-Monte Carlo algorithm. In the beginning of the application of
simulation techniques it has been widely believed that truly random sequences would give the best simulation results. Nowadays, the situation has changed. For certain applications, such as the estimation of integrals by simulation,
nonrandom sequences have been proven to have superior performance. Such deterministic sequences are called quasi-Monte Carlo sequences. Among them the low discrepancy sequences are best suited for solving integral
T. Siegl, R.F. Tichy Insurance: Mathematics and Economics 26 2000 59–73 69
equations by simulation. One of the central definitions in this area is the discrepancy which is defined as the maximal difference between the empirical and the theoretical distribution function. We restrict ourselves to the
uniform distribution. Let
= I
s
be the s dimensional unit cube. Then the discrepancy of a sequence ξ = {ξ
n
}
∞ n
=1
with ξ
n
∈ is given by
D
∗ N
ξ = sup
x x
x ∈
{n : ξ
n
≤ x}
N −
s
Y
i =1
x
i
. where
{n : B
n
} denotes the number of n satisfying condition B
n
componentwise. Traditional Monte Carlo integration does not give an exact error bound. However, we can obtain a bound on the
probable error by the Chebyshev inequality Kalos, 1986
P Z
f t dt
− 1
N
N
X
n =1
f ξ
n
≥ r Var[f x]
εN ≤ ε,
where Var[f x] is the variance of f ξ . By selecting a suitable ε, we obtain that the probable error is proportional to 1
√ N
, which is independent of s. This is optimal for large dimensions s 20 and reasonable N because for large dimensions s 20 the error bounds for quasi-Monte Carlo integration are next to useless. A deterministic
error bound for such integrals is given by the Koksma–Hlawka inequality Z
f t dt
− 1
N
N
X
n =1
f ξ
n
≤ V f D
∗ N
ξ , where V f is the total variation of f in the sense of Hardy and Krause. For special sequences, which we will
discuss next, the discrepancy satisfies D
∗ N
ξ ≤ C
s
log N
s
N ,
31 with an explicitly computable constant C
s
Niederreiter, 1992. The bounds for the constant are usually pessimistic and often the actual error made by quasi-Monte Carlo integration is much lower than the error bound Caflisch, 1998.
This has been observed in many applications ranging from the pricing of mortage backed securities Papageorgiou and Traub, 1996; Paskov, 1994; Paskov and Traub, 1995 to the pricing of options Joy et al., 1996. These recent
results have one observation in common: certain quasi-Monte Carlo sequences have been shown to be superior to pseudo-Monte Carlo sequences even in large dimensions, although the error bounds for the quasi-Monte Carlo
methods are worse.
Sequences that satisfy 31 are often called low-discrepancy sequences as they have the currently best asymptotic discrepancy estimates as N
→ ∞. Note however, that the constant C
s
can be influenced by the construction of the quasi-Monte Carlo sequence.
• The constant C
s
for the Halton 1960 sequence tends to infinity super-exponentially for s → ∞. This sequence
is defined as a sequence of vectors based on the digit representation of n in base p
i
ξ
n
= b
p
1
n, b
p
2
n, . . . , b
p
s
n, where p
i
is the ith prime number and b
p
n is the digit reversal function for base p given by
b
p
n =
∞
X
k =0
n
k
p
−k−1
, n
=
∞
X
k =0
n
k
p
k
, where the n
k
are integers. It is also possible to use arbitrary pairwise-coprime base numbers, but the error estimate is best possible for prime bases p
n
.
70 T. Siegl, R.F. Tichy Insurance: Mathematics and Economics 26 2000 59–73
• The first improvement of this situation is given by the Sobol’ 1967, 1976 sequence. While C
s
still tends to infinity super-exponentially Morokoff, 1994 the constant is much lower than that of the Halton sequence. The
general definition of the Sobol’ sequence would be beyond the scope of this paper, thus we only present an elegant definition that can be found in Beck and Chen 1987 for a Sobol type sequence
{x
1
, y
1
, x
2
, y
2
, . . . }
in dimension 2, where a
ν
is the binary representation of n n
= P
∞ ν
=1
a
ν
2
ν −1
, b
ν
= Ma
ν
mod 2, x
n
= P
∞ ν
=1
a
ν
2
−ν
, y
n
= P
∞ ν
=1
b
ν
2
−ν
. Here the matrix
M
i,j
= i
− 1 j
− 1
mod 2 consists of the binomial coefficients in the finite field F
2
. • The Faure 1981, 1982 sequence was the first low discrepancy sequence for which the constant C
s
does not grow, but tends to zero as s
→ ∞. • Further classes of low-discrepancy sequences are due to Niederreiter 1987. Among them, the Niederreiter
sequence with base 2 is best suited for computation as the arithmetic can be performed modulo 2. The Sobol’, Faure, and Niederreiter sequences are examples of so-called t, m, s-nets or nets for short. These nets
are based on the b-adic representation of vectors in the s ≥ 1 dimensional unit cube . Instead of optimizing the
discrepancy itself, an easier task is set: to control the discrepancy over a specified set of rectangles J . Let J ; N
denote the number of n, 1 + ≤ n ≤ N with x
n
∈ J, J ∈ J , and VolJ the volume of J . Now, the idea is that we try to obtain a low discrepancy, not for arbitrary intervals, but for a family of intervals that is convenient for mathematical
analysis, such as the family of elementary intervals in base b. They have the form J =
Q
s i
=1
[a
i
b
−d
i
, a
i
+ 1b
−d
i
, with integers d
i
≥ 0 and integers 0 ≤ a
i
b
d
i
for 1 ≤ i ≤ s.
The next idea is to look at the discrepancy of s-dimensional intervals, not at an arbitrary length of the sequence, but rather at a power m of the base b. We say that a point set P with cardP
= b
m
is a t, m, s-net, if J ; b
m
= b
t
for every elementary interval J with VolJ = b
t −m
. The parameter t is a quality parameter. If t = 0, then we have
minimal discrepancy of the point set P with respect to the family of elementary intervals. Based on this definition we now consider sequences.
Definition 1 t, s-sequence. Let t
≥ 0 be an integer. A sequence ξ
1
, ξ
2
, . . . of points in I
s
is called a t, s
-sequence in base b, if for all integers k ≥ 0 and m t the point set consisting of the ξ
n
with kb
m
n ≤ k + 1b
m
is a t, m, s-net in base b. The theory of t, s-sequences is given in detail by Niederreiter 1992 and Drmota and Tichy 1997. The Sobol’
sequence is a t, s-sequence in base 2 with values t that depend on s. The Faure sequence is a 0, s-sequence in base p
s
, where p
s
is the smallest prime number s. The Niederreiter sequences yield t, s-sequences in arbitrary base; among them there are 0, s-sequences in prime power bases b
≥ s. 6.5. Numerical simulation of the integral equation
In the simulation of the integral equation we use hybrid Monte Carlo sequences for the generation of uniformly distributed sequences. The Faure 1982, Halton 1960 and Niederreiter 1987 in base 2, and Sobol’ 1976
sequences are used for the initial 48 dimensions. The pseudo-random sequence of Afflerbach 1990 is used for the remaining dimensions s
≤ 3000. We need three dimensions per recursion step in order to avoid branching into two different results into U x
+ ct
− y and Uαx + ct. Thus, the variable in the first dimension is transformed into an exponentially distributed variable t cut off at T
= x
max
− xc with parameter λ + δ. The second dimension is used to construct a Bernoulli random variable B with p
= λλ + γ and the third dimension for B = 1 is used to obtain the exponentially distributed random claim cut off at x
+ct with parameter 1. The next step of the recursion proceeds with the following three dimensions in the same way until all recursion steps are completed. The details are given in the following
T. Siegl, R.F. Tichy Insurance: Mathematics and Economics 26 2000 59–73 71
Fig. 4. Mean square error for hybrid dim 48 and pseudo-Monte Carlo.
Fig. 5. Regression analysis of the mean square error.
algorithm, where Exp ·|z denotes a random variable cut off at z, and M gives the recursion depth M = 1000 in
our example. Solution of the integral equation
: Compute ˜
U =
1 N
P
N n
=1
RM, x Function Rm, x:
If m = 0: Return0;
Generate t ∼ Expλ + γ |
x
max
−x c
Generate B ∼ Bernoulli
λ λ
+γ
If B =
1 : Generate y
∼ Exp1|x + ct
0 : y
= 1 − αx + ct + K Return
λ +γ
1 −B e
−x−ct
1 −e
−λ+γ T
Rm − 1, x + ct − y + e
−λ+γ T
End Function In the following experiments we compare the simulation results of different hybrid sequences with the result of
the exact solution that is computed according to Section 6.1. As the initial part of the hybrid sequence is always the same due to the construction of the quasi-Monte Carlo sequence, no average case error over the sequence can be
computed meaningfully. We address this problem by computing a mean error over the simulation results ˜ U x
p
for a given argument set P
= {x
p
} consisting of |P | = 20 equally spaced points x
p
in the interval [0, 10]. We compute
72 T. Siegl, R.F. Tichy Insurance: Mathematics and Economics 26 2000 59–73
Fig. 6. Gain for the hybrid sequences compared to the pseudo-Monte Carlo sequence.
the mean square error S
= v
u u
t 1
|P | X
x
p
∈P
˜Ux
p
− Ux
p 2
for the simulated values ˜ U
and record the results of the simulation at N = 2
i
i = 1, . . . , 17 samples. The evolution
of the mean square error S is shown in Fig. 4. The pseudo-Monte Carlo sequence is included for comparison. We have also considered the inversive congruential generator Hellekalek, 1995
x
n +1
= 858 993 221x
−1 n
+ 1 mod 2 147 483 053 instead of the Afflerbach sequence. However, the resulting picture does not differ significantly from Fig. 4 where
the Afflerbach sequence is used. To quantify the effect of using a low discrepancy part in the sequence we perform a regression analysis on the
sample results by fitting log
2
S = F
a
1
,a
2
,a
3
N = a
+ a
1
log
2
N + a
2
log
2
log
2
N + ε
32 to the data of the log–log plot using least square optimization of ε. Note that the Koksma–Hlawka inequality could
be interpreted as implying a
2
= s and a
1
= −1. However, we do not set a
2
= s as the effective dimension of the simulation may be different to the theoretical dimension and we do not set a
2
= −1 as we are using a hybrid sequence. The fitted curves are given in Fig. 5.
We measure the efficiency gain by comparing the regression results. For a given error bound S = S
∗
we compute the minimal sample size N
= N
∗
sequence such that the regression error estimate Eq. 32 is lower than S
∗
. The required sample size using the hybrid sequences is compared to the required sample size using the pseudo-Monte
Carlo sequence. The gain in using the hybrid sequence is expressed as a factor gain = N
∗
pseudoN
∗
hybrid of reduction of the required sample size for a given error. Therefore, the pseudo-Monte Carlo sequence takes gain
times the number of samples to reach the same error as the given hybrid sequence. Fig. 6 shows gain as a function of the error bound.
The Sobol’ sequence outperforms the pseudo-Monte Carlo sequence by a factor of 2–3, while the other sequences do not show advantages. Good results of the Sobol’ sequence even in problems with an extremely high dimension
are also reported in, e.g. Paskov 1994.
T. Siegl, R.F. Tichy Insurance: Mathematics and Economics 26 2000 59–73 73
7. Conclusion
