# Pseudorandom binary sequences: quality measures and number-theoretic
constructions

Arne Winterhof
Johann Radon Institute for Computational and Applied Mathematics
Austrian Academy of Sciences

###### Abstract

In this survey we summarize properties of pseudorandomness and non-randomness
of some number-theoretic sequences and present results on their behaviour
under the following measures of pseudorandomness: balance, linear complexity,
correlation measure of order $k$, expansion complexity and $2$-adic
complexity. The number-theoretic sequences are the Legendre sequence and the
two-prime generator, the Thue-Morse sequence and its sub-sequence along
squares, and the prime omega sequences for integers and polynomials.

Keywords. pseudorandom sequences, linear complexity, correlation measure,
expansion complexity, $2$-adic complexity, Legendre sequence, Thue-Morse
sequence, prime divisor function

##  1 Introduction

Let

${\cal S}=(s_{n})_{n=0}^{\infty},\quad
s_{n}\in{\mathbb{F}}_{2}=\\{0,1\\},\quad n=0,1,\ldots,$

be a binary sequence. We call it pseudorandom if it is deterministically
generated but cannot be distinguished from a truly random sequence.
Pseudorandom sequences are crucial for cryptographic applications such as
stream ciphers, see for example [12].

###  1.1 Measures of pseudorandomness

There are several measures of pseudorandomness which can be used to detect
cryptographically weak sequences including

  * •

balance,

  * •

linear complexity,

  * •

maximum-order complexity,

  * •

correlation measure of order $k$,

  * •

expansion complexity

  * •

and $2$-adic complexity.

These measures are partly not independent and partly complement each other. We
will discuss some of their relations.

###  1.2 Pseudorandom sequences

We summarize results on these measures for the following number-theoretic
sequences,

  * •

the Legendre sequence and the two-prime generator,

  * •

the Thue-Morse sequence and its sub-sequence along squares,

  * •

the prime omega sequence modulo $2$ for integers and for polynomials.

Each section will focus on one of the above measures of pseudorandomness.

It turns out that

  * •

the Legendre sequence has no obvious flaw (if the period is long enough),

  * •

the two-prime generator suffers a large correlation measure of order $4$ and
is not pseudorandom,

  * •

the Thue-Morse sequence has an undesirable deviation from the expected value
$N/2$ of the linear complexity, a large correlation measure of order $2$ and a
small expansion complexity, and is not suitable in cryptography,

  * •

the Thue-Morse sequence along squares seems to be an attractive candidate for
cryptography,

  * •

the $N$th linear complexity of the prime omega sequence for integers seems to
be too regular,

  * •

there is no obvious deficiency of the omega sequence for polynomials.

For earlier surveys on measures of pseudorandomness see [16, 39, 40, 42, 52,
54].

##  2 Balance and definitions of the sequences

###  2.1 Definition of balance and its expected value

The $N$th balance $B({\cal S},N)$ of a binary sequence ${\cal
S}=(s_{n})_{n=0}^{\infty}$ is

$B({\cal S},N)=\left|\sum_{n=0}^{N-1}(-1)^{s_{n}}\right|.$

The balance of a sequence which is not distinguishable from a random sequence
should be of order of magnitude $N^{1/2}$, see Alon et al. [1, Lemma 12], or
at least

$N^{o(1)}\ll B({\cal S},N)=o(N).$

Here we use the notation

$f(N)=O(g(N))\Longleftrightarrow|f(N)|\leq cg(N)$

for some absolute constant $c>0$,

$f(N)\ll g(N)\Longleftrightarrow f(N)=O(g(N))$

and

$f(N)=o(g(N))\Longleftrightarrow\lim_{N\rightarrow\infty}\frac{f(N)}{g(N)}=0.$

Now we give a list of some number-theoretic sequences with desirable balance.

###  2.2 Legendre sequence

For a prime $p>2$ the Legendre sequence ${\cal
L}_{p}=(\ell_{n})_{n=0}^{\infty}$ is the $p$-periodic sequence defined by

$\ell_{n}=\left\\{\begin{array}[]{cc}\frac{1}{2}\left(1-\left(\frac{n}{p}\right)\right),&\gcd(n,p)=1,\\\
0,&n\equiv 0\bmod p,\end{array}\right.$ (1)

where

$\left(\frac{n}{p}\right)=\left\\{\begin{array}[]{cc}1,&n\mbox{ a quadratic
residue modulo }p,\\\ -1,&n\mbox{ a quadratic non-residue modulo }p,\\\
0,&n\equiv 0\bmod p,\end{array}\right.$

is the Legendre symbol. Since there are $(p-1)/2$ quadratic residues and
$(p-1)/2$ quadratic non-residues ($0$ is neither a residue nor a non-residue),
we obviously have

$B({\cal L}_{p},p)=1.$

By the Burgess bound, see for example [26, $(12.58)$], we have

$B({\cal L}_{p},N)=O\left(N^{1-\frac{1}{r}}p^{\frac{r+1}{4r^{2}}}(\log
p)^{\frac{1}{r}}\right)$

for any $r\geq 1$. In particular, we have ($r=1$)

$B({\cal L}_{p},N)=O\left(p^{\frac{1}{2}}\log p\right)$

and

$B({\cal L}_{p},N)=o(N)\quad\mbox{for $N\geq p^{\frac{1}{4}+o(1)}$}.$

###  2.3 Two-prime generator

For two odd primes $p$ and $q$ with, say, $p<q<2p$ the two-prime generator
${\cal W}=(w_{n})_{n=0}^{\infty}$ of period $pq$ satisfies

$w_{n}=\frac{1}{2}\left(1-\left(\frac{n}{p}\right)\left(\frac{n}{q}\right)\right),\quad\gcd(n,pq)=1.$

For any choice of $w_{n}$ with $\gcd(n,pq)>1$, by [3, Lemma 4] we have

$B({\cal W},N)=O\left((pq)^{1/2}\log(pq)\right),\quad 1\leq N\leq pq,$

and again by the Burgess bound

$B({\cal W},N)=o(N)\quad\mbox{for }N\geq(pq)^{1/4+o(1)}.$

###  2.4 Thue-Morse sequence (along squares)

The Thue-Morse sequence ${\cal T}=(t_{n})_{n=0}^{\infty}$ over
${\mathbb{F}}_{2}$ is defined by

$t_{n}=\left\\{\begin{array}[]{cl}t_{n/2},&\mbox{$n$ even},\\\
t_{(n-1)/2}+1,&\mbox{$n$ odd},\end{array}\right.\quad n=1,2,\ldots$ (2)

with initial value $t_{0}=0$. Since $t_{2n}\neq t_{2n+1}$ we have

$B({\cal T},N)\leq 1,\quad N=1,2,\ldots$

This already points to some undesirable structure of the Thue-Morse sequence.
Further weaknesses of this sequence are mentioned below. We will also see that
the Thue-Morse sequence has some desirable features such as a large linear
complexity.

Certain sub-sequences, such as the sub-sequence of the Thue-Morse sequence
along squares, may keep the good properties of the original sequence but avoid
the bad ones. For the sub-sequence of the Thue-Morse sequence along squares
${\cal Q}=(t_{n^{2}})_{n=0}^{\infty}$ we have

$B({\cal Q},N)=o(N)$

by Mauduit and Rivat [32, Théorème 1].

###  2.5 Omega sequence (for integers and polynomials)

Let $n=p_{1}^{a_{1}}p_{2}^{a_{2}}\cdots p_{r}^{a_{r}}$ be the prime
factorization of a positive integer $n$. The $\Omega$ function is defined by

$\Omega(n)=a_{1}+a_{2}+\ldots+a_{r}.$

We consider the sequence ${\cal O}=(o_{n})_{n=0}^{\infty}$ with

$o_{0}=0,\quad o_{n}=\Omega(n)\bmod 2,\quad n=1,2,\ldots$

We have

$B({\cal O},N)=o(N)$

and the Riemann hypothesis is equivalent to, see Humphries [25],

$B({\cal O},N)=O\left(N^{\frac{1}{2}+\varepsilon}\right)\quad\mbox{for any
$\varepsilon>0$}.$

Similarly, for a polynomial $F(X)$ over the finite field ${\mathbb{F}}_{p}$ of
prime order $p$, $\Omega_{p}(F)$ denotes the total number of irreducible
factors over ${\mathbb{F}}_{p}$ of $F(X)$. For fixed degree $d\geq 3$ we order
the monic polynomials of degree $d$,

$F_{n}(X)=X^{d}+n_{d-1}X^{d-1}+\ldots+n_{1}X+n_{0},$

where

$n=n_{0}+n_{1}p+\ldots+n_{d-1}p^{d-1}\quad\mbox{with }0\leq
n_{0},n_{1},\ldots,n_{d-1}<p,$

and define the sequence ${\cal P}={\cal P}_{d,p}=(p_{n})_{n=0}^{p^{d}-1}$ of
length $p^{d}$ by

$p_{n}=\Omega_{p}(F_{n})\bmod 2,\quad n=0,1,\ldots,p^{d}-1.$

Carlitz [5] proved for $N=p^{d}$

$B({\cal P},p^{d})=p^{\left\lfloor\frac{d+1}{2}\right\rfloor}$

and for $p>2$ and $N<p^{d}$ we have, see [37],

$B({\cal P},N)=O\left(\frac{dN}{p^{1/2}}\right)\quad\mbox{for }N\geq p^{2}\log
p,$

that is, for fixed $d\geq 3$ and $p\rightarrow\infty$ we get

$B({\cal P},N)=o(N),\quad N\geq p^{2}\log p.$

A closely related measure of pseudorandomness, the well-distribution measure,
that is, roughly speaking, the balance of the sequence along arithmetic
progressions, was studied in the series of papers [1, 8, 33, 34, 37, 39, 46].

##  3 Linear complexity

###  3.1 Definition

The $N$th linear complexity $L({\cal S},N)$ of a binary sequence ${\cal S}$ is
the smallest positive integer $L$ such that there are constants
$c_{0},\ldots,c_{L-1}\in{\mathbb{F}}_{2}$ with

$s_{n+L}=c_{L-1}s_{n+L-1}+\ldots+c_{0}s_{n},~{}0\leq n<N-L.$

The linear complexity $L({\cal S})$ of ${\cal S}$ is

$L({\cal S})=\sup_{N\geq 1}L({\cal S},N).$

In particular, for a $T$-periodic sequence ${\cal S}_{T}$ we have

$L({\cal S}_{T})\leq T$

and $L({\cal S})<\infty$ if and only if ${\cal S}$ is ultimately periodic.

A sequence of small linear complexity is predictable and thus unsuitable in
cryptography. However, the converse is not true. There are many predictable
sequences of very large linear complexity, for example periodic sequences
containing only a single one in a period, and, in addition, finer quality
measures have to be studied.

###  3.2 Expected value

Let denote by $A(N,L)$ the number of
$(s_{0},\ldots,s_{N-1})\in{\mathbb{F}}_{2}^{N}$ which are the initial values
of a sequence ${\cal S}$ with $L({\cal S},N)=L$. The expected value

$E_{N}=\frac{1}{2^{N}}\sum_{L=0}^{N}A(N,L)L$

was analyzed in Gustavson [20].

###### Theorem 1

The expected value of $L({\cal S},N)$ is

$E_{N}=\frac{N}{2}+O(1).$

Niederreiter [41] showed that the $N$th linear complexity of a random sequence
follows closely but irregularly the $N/2$-line and deviations from $N/2$ of
the order of magnitude $\log N$ must appear for infinitely many $N$.

From a computational point of view to avoid an attack via the Berlekamp-Massey
algorithm, see [31], say $L({\cal S},N)\geq N^{o(1)}$ would be good enough.

For periodic sequences the expected value of the linear complexity depends on
the period [35]. For example, if the period $T$ is a prime and $2$ is a
primitive root modulo $p$, then any non-constant sequence ${\cal S}$ of period
$T$ is of linear complexity $T$ or $T-1$ and the expected value is very close
to $T$, see [12].

###  3.3 Legendre sequence and two-prime generator

The linear complexity of the Legendre sequence ${\cal L}_{p}$ defined by
$(\ref{legdef})$ was determined by Turyn [53], see also [14].

###### Theorem 2

For a prime $p>2$ the linear complexity $L({\cal L}_{p})$ of the $p$-periodic
Legendre sequence ${\cal L}_{p}$ is

$L({\cal L}_{p})=\left\\{\begin{array}[]{cl}(p-1)/2,&p\equiv 1\bmod 8,\\\
p,&p\equiv 3\bmod 8,\\\ p-1,&p\equiv-3\bmod 8,\\\ (p+1)/2,&p\equiv-1\bmod
8.\end{array}\right.$

For the $N$th linear complexity we have the following bound due to Chen et al.
[9].

###### Theorem 3

$L({\cal L}_{p},N)\geq\frac{\min\\{N,p\\}}{p^{1/2}}\quad\mbox{for
}N=0,1,\ldots$

It would be important to improve this lower bound getting closer to the
conjectured lower bound $N/2+o(N)$.

For the two-prime generator we get by [12, Theorem 8.2.9]

$L({\cal W})\geq\frac{(p-1)(q-1)}{2}$

and by [3]

$L({\cal W},N)\geq\frac{\min\\{N,pq\\}}{(pq)^{1/2}}.$

###  3.4 Thue-Morse sequence (along squares)

The following two results are due to [38] and [51].

###### Theorem 4

For the $N$th linear complexity of the Thue-Morse sequence ${\cal T}$ we have

$L({\cal T},N)=2\left\lfloor\frac{N+2}{4}\right\rfloor,\quad N=1,2,\ldots$

###### Theorem 5

For the $N$th linear complexity of the Thue-Morse sequence ${\cal Q}$ along
squares we have

$L({\cal Q},N)\geq\left(\frac{2N}{5}\right)^{1/2}\quad\mbox{for }N\geq 21.$

Note that the deviation of the $N$th linear complexity of the Thue-Morse
sequence from $N/2$ is $O(1)$ which is too regular. For the Thue-Morse
sequence along squares we conjecture the desirable $L({\cal
Q},N)=\frac{N}{2}+o(N)$.

###  3.5 Omega sequences

Up to our knowledge there is no lower bound on $L({\cal O},N)$ in the
literature. However, our numerical data leads to the following conjecture.

###### Conjecture 1

$L({\cal O},N)=\frac{N}{2}+O(1).$

If this conjecture is true, then the integer omega sequence can be
distinguished from a random sequence by the deviation of the $N$th linear
complexity from $N/2$.

For the polynomial omega sequence, combining [9, Corollary 4] and [37] we get
a lower bound on $L({\cal P},N)$ of order of magnitude
$\min\\{N,p^{d}\\}^{1/2}p^{\frac{1}{4}-\frac{d}{2}}$.

###  3.6 Balance and linear complexity

Balance and linear complexity are independent measures of pseudorandomness in
the following sense:

  1. 1.

Both measures detect the non-randomness of constant sequences.

  2. 2.

The non-randomness of the sequence $s_{n}=0$, $n=0,1,\ldots,N-2$, $s_{N-1}=1$,
is detected by the balance but not by the $N$th linear complexity.

  3. 3.

The balance of the Thue-Morse sequence is too small but its $N$th linear
complexity is large enough.

  4. 4.

We have seen several examples, for example the Legendre sequence, with both a
high $N$th linear complexity and a desirable balance.

##  4 Correlation measure

###  4.1 Definition and expected value

The $N$th correlation measure of order $k$ of ${\cal S}$ introduced by Mauduit
and Sárközy [33] is

$C_{k}({\cal
S},N)=\max_{M,D}\left|\sum^{M-1}_{n=0}(-1)^{s_{n+d_{1}}}\cdots(-1)^{s_{n+d_{k}}}\right|,\quad
k\geq 1,$

where the maximum is taken over all $D=(d_{1},d_{2},\ldots,d_{k})$ with
integers satisfying $0\leq d_{1}<d_{2}<\cdots<d_{k}$ and $1\leq M\leq
N-d_{k}$.

The correlation measure of order $k$ provides information about the
independence of parts of the sequence and their shifts. For a random sequence
this similarity and thus the correlation measure of order $k$ is expected to
be small. More precisely, by [1] we have the following result.

###### Theorem 6

For any $\varepsilon>0$ there exist an $N_{0}$ such that for all $N\geq N_{0}$
we have for a randomly chosen sequence ${\cal S}$

$\frac{2}{5}\sqrt{N\log{N\choose k}}<C_{k}({\cal
S},N)<\frac{7}{4}\sqrt{N\log{N\choose k}}$ (3)

with probability at least $1-\varepsilon$.

Hence, $C_{k}({\cal S},N)$ should be up to some logarithmic factor of order of
magnitude $\sqrt{N}$ or at least $o(N)$.

###  4.2 Correlation measure and linear complexity

The following lower bound on the linear complexity profile in terms of the
correlation measure was proved in [4].

###### Theorem 7

Let ${\cal S}$ be a $T$-periodic binary sequence. For $2\leq N\leq T$ we have

$L({\cal S},N)\geq N-\max_{1\leq k\leq L({\cal S},N)+1}C_{k}(s_{n},T).$

For a recent improvement which saves typically a factor $\log N$ see Chen et
al. [9, Corollary 4]. For example, combining this relation between linear
complexity and correlation measure with the bound on the correlation measure
in Theorem 8 below we immediately get the lower bound on the linear complexity
of Theorem 3 above. In this sense we may say that the correlation measure of
order $k$ is a finer measure of pseudorandomness than the linear complexity.
However, from an algorithmic point of view the $N$th correlation measure of
order $k$ is much more difficult to analyze than the $N$th linear complexity.
Still, for some special number-theoretic sequences such as the Legendre
sequence one can estimate it theoretically.

###  4.3 Legendre sequence and two-prime generator

Although almost all sequences satisfy $(\ref{correxp})$, it is difficult to
find concrete examples. Roughly speaking, if you can describe a sequence, it
does not behave like a randomly chosen sequence anymore. However, for fixed
$k$ and sufficiently large $p$, the correlation measure of order $k$ of the
Legendre sequence essentially behaves like the one for a randomly chosen
sequence up to logarithmic terms, see [33].

###### Theorem 8

The correlation measure of order $k$ of the Legendre sequence satisfies

$C_{k}({\cal L}_{p},N)=O(kp^{1/2}\log p),\quad 1\leq N\leq p.$

The situation is different for the two-prime generator. On the one hand, by
[46] we still have

$C_{2}({\cal T},N)=O((pq)^{3/4}),\quad 1\leq N\leq pq.$

On the other hand, taking the lags

$d_{0}=0,d_{1}=p,d_{2}=q,d_{3}=p+q$

we get

$C_{4}({\cal T},N)=N+O(N^{1/2}),\quad 1\leq N\leq pq,$

showing that the two-prime generator is not a good candidate for cryptography.

###  4.4 Thue-Morse sequence (along squares)

By [34] we have

$C_{2}({\cal T},N)>\frac{N}{12},\quad N\geq 5.$

We believe that this feature of non-randomness is destroyed by taking the sub-
sequence along squares.

###### Conjecture 2

$C_{k}({\cal Q},N)=o(N)\quad\mbox{for }k=2,3,\ldots$

If we assume that the lags are bounded by a constant $C$, that is, $d_{k}\leq
C$ and $N\rightarrow\infty$, the analog of the correlation measure of order
$k$ with bounded lags is $o(N)$ by [15].

##  5 Omega sequences

The following is essentially Chowla’s conjecture, see [10].

###### Conjecture 3

$C_{k}({\cal O},N)=o(N).$

For recent progress on Chowla’s conjecture see Tao and Teräväinen [48, 49] and
references therein. The correlation measure of order $k$ of a modified omega
sequence was studied by Cassaigne et al. [8].

The Chowla conjecture for polynomials was settled by Carmon and Rudnick [7]
for $p>2$ and Carmon [6] for $p=2$ in the case that $p$ is fixed and the
degree $d$ goes to infinity. In particular, we have the following bound, see
[37, Theorem 3].

###### Theorem 9

$C_{k}({\cal P}_{d},p^{d})=O(k^{2}dp^{d-1/2}\log p).$

However, nothing is known for fixed $p$ and $d\rightarrow\infty$. For
polynomials over finite fields $\mathbb{F}_{p^{r}}$ with $r\geq 3$ there has
been a recent breakthrough by Sawin and Shusterman [47]. However, it seems
that the case $r=1$ is out of reach.

##  6 Maximum-order complexity

###  6.1 Definition, expected value and relation to other measures

The $N$th maximum order complexity $M({\cal S},N)$ is the smallest positive
integer $M$ with

$s_{n+M}=f(s_{n+M-1},\ldots,s_{n}),\quad 0\leq n\leq N-M-1,$

for some mapping $f:{\mathbb{F}}_{2}^{M}\mapsto{\mathbb{F}}_{2}$.

The maximum order complexity was introduced by Jansen in [27, Chapter 3], see
also [28]. The typical value for the $N$th maximum order complexity is of
order of magnitude $\log N$, see [27, 28].

Obviously, we have

$M({\cal S},N)\leq L({\cal S},N)$

and we may consider the maximum-order complexity a finer measure than the
linear complexity. However, from an algorithmic point of view the linear
complexity can be much easier determined via the Berlekamp-Massey algorithm
than the maximum-order complexity. An algorithm for calculating the maximum
order complexity profile of linear time and memory was presented by Jansen
[27, 28] using the graph algorithm introduced by Blumer et al. [2].

Although a large $M({\cal S},N)$ is desired it should not be too large since
otherwise the correlation measure of order $2$ is large, see [39, (5.6)].
Combining this inequality with [9, Theorem 5] we get:

###### Theorem 10

We have

$C_{2}({\cal S},N)\geq\max\left\\{M({\cal S},N)-1,N+1-2^{M({\cal
S},N)}\right\\}.$

In Subsection 6.3 we study the Thue-Morse sequence ${\cal
T}=(t_{n})_{n=0}^{\infty}$ defined by $(\ref{tmdef})$. It turns out that
$M({\cal T},N)$ is of order of magnitude $N$. However, this implies that the
correlation measure $C_{2}({\cal T},N)$ of order $2$ is also of order of
magnitude $N$ and concerning this measure the Thue-Morse sequence does not
behave like a random sequence.

However, for the Thue-Morse sequence along squares ${\cal
Q}=(t_{n^{2}})_{n=0}^{\infty}$ we mention that $M({\cal Q},N)$ is at least of
order of magnitude $N^{1/2}$.

We can also define and study

$M({\cal S})=\sup_{N\geq 1}M({\cal S},N),$

see for example [30].

###  6.2 Legendre sequence

Combining Theorem 10 and Theorem 8 we get the following bound on the maximum-
order complexity of the Legendre sequence.

###### Corollary 1

For $1\leq N\leq p$ we have

$M({\cal L}_{p},N)=O(p^{1/2}\log p)$

and

$M({\cal L}_{p},N)\geq\frac{\log N-\frac{1}{2}\log p+O(\log\log p)}{\log 2}.$

It is not difficult to obtain a similar bound for the two-prime generator.
However, because of its large correlation measure of order $4$ there is no
need of further studies of this sequence.

###  6.3 Thue-Morse sequence (along squares)

The following result is due to [50].

###### Theorem 11

For $N\geq 4$, the $N$th maximum order complexity of the Thue-Morse sequence
${\mathcal{T}}$ satisfies

$\displaystyle M({\mathcal{T}},N)=2^{\ell}+1,$
$\displaystyle\ell=\left\lceil\frac{\log(N/5)}{\log 2}\right\rceil.$

It is easy to see that

$\displaystyle\frac{N}{5}+1\leq M({\mathcal{T}},N)\leq
2\frac{N-1}{5}+1\quad\text{for}\;\;N\geq 4$ $\displaystyle
M({\mathcal{T}},1)=0,\quad M({\mathcal{T}},2)=M({\mathcal{T}},3)=1.$

For the Thue-Morse sequence along squares see [51].

###### Theorem 12

$M({\cal Q},N)\geq\sqrt{\frac{2N}{5}},\quad N\geq 21.$

For an extension to sub-sequences of the Thue-Morse sequence along polynomial
values see [44].

###  6.4 Omega sequences

Our numerical data leads to the following conjecture.

###### Conjecture 4

$M({\cal O},N)$ is of order of magnitude $\log N$.

For the polynomial analog we get from Theorems 9 and 10 the following result.

###### Corollary 2

$M({\cal P}_{d},p^{d})=O(dp^{d-1/2}\log p)$

and

$M({\cal P}_{d},p^{d})\geq\frac{(d-1/2)\log p}{\log 2}+o(\log p).$

##  7 Expansion complexity

###  7.1 Definition and Thue-Morse sequence

Let

$G(x)=\sum_{n=0}^{\infty}s_{n}x^{n}$

be the generating function of the sequence ${\cal S}$ with
$(s_{0},\ldots,s_{N-1})\neq(0,\ldots,0)$. The smallest degree $E({\cal S},N)$
of a polynomial $h(x,y)\neq 0$ with

$h(x,G(x))\equiv 0\bmod x^{N}$

is called $N$th expansion complexity of ${\cal S}$.
For $(s_{0},\ldots,s_{N-1})=(0,\ldots,0)$ we define $E({\cal S},N)=0$.
The expansion complexity $E({\cal S})$ of ${\cal S}$ is

$E({\cal S})=\sup_{N\geq 1}E({\cal S},N).$

The expansion complexity was introduced by Diem [13] and we have

$E({\cal S},N)<\sqrt{2N}$

by [19, Theorem 4]. By Christol’s theorem [11] automatic sequences are
characterized by $E({\cal S})<\infty$.

From the well-known equation

$(1+x)^{3}G(x)^{2}+(1+x)^{2}G(x)+x=0$

we immediately get the following bound.

###### Corollary 3

For $N=1,2,\ldots$, the $N$th expansion complexity $E({\cal T},N)$ of the
Thue-Morse sequence is at most $5$.

The expansion complexity is another measure for the predictability of a
sequence. Despite of its very large $N$th linear complexity, the Thue-Morse
sequence is very predictable because of its extremely small $N$th expansion
complexity. Hence, the $N$th expansion complexity can be substantially smaller
than the $N$th linear complexity. However, we will see in the next section
that in the periodic case expansion complexity and linear complexity are
essentially the same.

The expected value of $E({\cal S},N)$ of a random sequence is of order of
magnitude $N^{1/2}$, see [18, Theorem 2].

Diem showed [13] that if a sequence has small expansion complexity, then long
parts of such sequences can be computed efficiently from short ones. An
algorithm based on Gröbner basis is given in [18].

###  7.2 Expansion complexity and linear complexity

The following results are from [36].

###### Theorem 13

Let ${\cal S}$ be a (purely) periodic sequence. Then we have

$E({\cal S})=L({\cal S})+1.$

In the aperiodic case we get the following.

###### Theorem 14

$E({\cal S},N)\leq\min\\{L({\cal S},N)+1,N+2-L({\cal S},N)\\}.$

This result and the results on the Thue-Morse sequence imply that the $N$th
expansion complexity is a strictly finer measure than the linear complexity,
more precisely, than the deviation of $L({\cal S},N)$ from the expected value
$\frac{N}{2}$.

###  7.3 Thue-Morse along squares, Legendre sequence and omega sequences

Our numerical data leads to the conjecture that all four sequences have
expansion complexity of order of magnitude $N^{1/2}$ with the restriction
$N\leq p^{2}$ for the Legendre sequence.

##  8 $2$-adic complexity

Besides the linear complexity, that is the length of a shortest linear feed
shift register which generates the sequence, the $2$-adic complexity has been
studied, which is closely related to the length of a shortest feedback with
carry shift registers which generates the sequence and was introduced by
Goresky and Klapper, see [17] and references therein. Although the theory of
$2$-adic complexity has been very well-developed for the periodic case, almost
nothing is known for the aperiodic case.

More precisely, the $2$-adic complexity $C({\cal S})$ of a $T$-periodic
sequence is

$C({\cal S})=\frac{\log\left(\frac{2^{T}-1}{\gcd(2^{T}-1,S(2)}\right)}{\log
2},$

where

$S(X)=\sum_{n=0}^{T-1}s_{n}X^{n}.$

The expected value of the $2$-adic complexity of $T$-periodic sequences is
$T-O(\log T)$, see [17, Corollary 18.2.2].

Since the linear complexity satisfies

$L({\cal S})=T-\deg(\gcd(S(X),X^{T}-1))$

it is easy to see that linear complexity and $2$-adic complexity complement
each other. For example, let $2^{T}-1$ be a Mersenne prime. Then any non-
constant sequence has maximum $2$-adic complexity. However, $X^{T}-1$ may
still have a nontrivial divisor $S(X)$ of large degree and the linear
complexity can be small. Conversely, if $T$ is a prime and
$1+X+\ldots+X^{T-1}$ is irreducible, that is, $2$ is a primitive root modulo
$T$, then any non-constant sequence has maximal linear complexity. However,
$2^{T}-1$ may have a large nontrivial divisor and the $2$-adic complexity can
be small.

Moreover, an $m$-sequence of period $2^{r}-1$ has linear complexity only $r$
but maximal $2$-adic complexity [55]. Conversely, $\ell$-sequences have
minimal $2$-adic complexity but can have very large linear complexity [45].

The Legendre sequence has maximal $2$-adic complexity and the $2$-adic
complexity of the two-prime generator is very large as well if $p$ and $q$ are
essentially of the same size, see [21, 24, 55].

It would be very important to study also the aperiodic case, in particular, to
get results for the Thue-Morse sequence along squares and the omega sequences.
More precisely, the $N$th $2$-adic complexity $C({\cal S},N)$ of ${\cal S}$ is
the binary logarithm of

$\min\\{\max\\{|f|,|g|\\}:f,g\in\mathbb{Z},g\mbox{ odd, }gS(2)\equiv f\bmod
2^{N}\\},$

where

$S(2)=\sum_{n=0}^{N-1}s_{n}2^{n}.$

The question about the expected value is open, see [29, Section 5.1]. Our
numerical data obtained using the rational approximation algorithm, see [17,
Chapter 17], leads, for example, to the following conjecture for the Legendre
sequence.

###### Conjecture 5

$C({\cal L}_{p},N)=\frac{\min\\{N,2p\\}}{2}+O(1).$

Similar conjectures can be stated for the Thue-Morse sequence along squares
and the omega-sequences.

Another very promising balanced number-theoretic sequence ${\cal X}$ defined
by

$x_{n}=(g^{n}\bmod p)\bmod 2,\quad n=0,1,\ldots$

for some $g\in{\mathbb{F}}_{p}^{*}$ has been introduced very recently in [43].
It is natural to ask for lower bounds on linear complexity, maximum-order
complexity etc. for this sequence as well. Note that in the case that $g=2$ is
a primitive root modulo $p$ we get an $\ell$-sequence by [17, Theorem 4.5.2].

There is also a $2$-adic analog of the correlation measure of order $2$
(aperiodic autocorrelation) called arithmetic autocorrelation which can be
estimated in terms of the correlation measure, see [22]. In particular, the
arithmetic autocorrelation of the Legendre sequence was estimated in [23].

## Acknowledgment

The author wishes to thank Zhixiong Chen and László Mérai for useful comments.

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# Optimal Reinsurance: A Ruin-Related Uncertain Programming Approach

Wrya Vakili and Alireza Ghaffari-Hadigheh
Dept. of Applied Math.
Azarbaijan Shahid Madani University, Tabriz, Iran
<EMAIL_ADDRESS>
<EMAIL_ADDRESS>
###### Abstract

We investigate the role of reinsurance in maximizing the wealth of an
insurance company. We use Liu’s uncertainty theory (B. Liu, 2007) for the
problem modeling and follow-up computations. The uncertainty measure of ruin
for the insurance company is considered as the optimization criterion. Since
calculating the ruin index is very difficult, we introduce a simple
computational method to identify the uncertain measure of ruin for an
insurance company. Finally, a generalized model is presented, granting the
model be more practical.

##  1 Introduction

Risk is an inevitable feature of a business. Analyzing the behavior of
involved risk and it’s consequences have always been subjected to different
theories that deal with unpredictable environments. Risk theory investigates
the effect of potential deviation of the outcome from their expected value,
prevent it from occurring, or being prepared to take action when it is
inescapable. It mostly practices the modeling of cash flow as a surplus
process, e.g., the business’s wealth during the activity period [10].

Using insurance, people aim to hedge their property against diverse sorts of
risks they might encounter. Insurance companies take their customers’ risks in
exchange for a specified amount of money referred to as the premium. The
collective risk model, also called the Cramer-Lundberg model, was initiated by
Lundberg and latter developed by Cramer [5]. It is usually counted as a
standard configuration of an insurance company’s risk process concerning the
risks it takes by selling insurance underwriting.

In the non-life insurance industry, consider an insurance company that sells
many finite fire insurance contracts. To compete against other companies, it
may offer a lower premium, leading to enduring a high risk in each contract.
Notice that in such contracts, the chance of damage of property by the fire is
significantly low. However, the policyholder’s loss in an incident is
extremely higher than the premium amount. Exercising a mass of such a risk,
the company would prefer to hedge itself by purchasing reinsurance to
transform some of the risks to another company. Here, we model such a
situation as an optimization problem at which the optimal decision is the
amount of risk it should share.

Consider this classical model as appeared in most of the standard actuarial
literature [5]

$U_{t}=u+ct-R_{t},$ (1)

where $u$ is the initial capital of the company, $c>0$ is a constant that
represents insurance premium indicating that company receives $c$ units of
money per unit time. The process $R_{t}=\sum_{i=1}^{N_{t}}\eta_{i}$ is so-
called compound process, where $N_{t}$ is the process of counting the number
of claims that the company might have to pay during $(0,t]$. Further,
$\eta_{i}$’s are random variables denoting the severity of claims. Finally,
$U_{t}$ is the wealth of the company regarding its contracts over $(0,t]$.

Observe that the interval $(0,t]$ can be partitioned into separated points
$s_{1},s_{2},\ldots$ in which losses take place. We refer to
$\xi_{i}=s_{i+1}-s_{i}$ as interarrival times. In the Lundberg-Cramer model,
these random variables are iid with exponential distribution. It is also
assumed that $N_{t}$ and $\eta_{i}$’s are independent, and $\eta_{i}$’s are
iid random variables. Anderson et al.[3] generalized the classic model by
letting $N_{t}$ to be any renewal process, not just the Poisson process. In
these models, the only non-deterministic component is $R_{t}$, while
calculating its distribution is a very costly task, and most of the time, it
is almost impossible to do. However, an acceptable distribution can be
approximated by discretizing the variables to reduce them to be more
tractable.

Ruin is an essential and primary concept of mathematical approaches in the
risk management of insurance activities. It is defined as the company’s
surplus being negative in some future, concerning active contracts of some
policies [5]. Observe that ruin happens when the collected premiums don’t
satisfy the losses claimed by policyholders. Just by restricting the model to
some simplified considerations, it is possible to obtain some results. For
example, Panjer [38] proved that when severity only takes positive integers,
and $N_{t}$ is of some special cases, calculation of the distribution of
$R_{t}$ and consequently the ruin, is possible through some recursive
algorithm. In another study, diffusion approximation was proposed [24] and
followed by Grandell [22], as another approach to make the computation of ruin
possible. Here, the main idea is using the central limit theorem in favor of
approximating the risk process by a Wiener process. Another approximation was
proposed by De Vydler et al. [17], where the first three moments of the
$R_{t}$ were used to mimic its behavior and find a recursive algorithm for
calculating the probability distribution of the ruin. Thus, direct calculating
of the probability of ruin is far from being a practical approach.

Some generalized versions of classical models were developed, where more
details of the real-world problem are taken into account. Dufresne et al. [18]
used a diffusion process to perturb the risk process, allowing more details to
the model, such as the effect of interest rate or dependency between the
number and severity of the claims. More assumptions, such as taxes and
dividends, have been added to models enriching the literature and making the
models more practical. We refer to [2, 42], as well as to the masterpiece
textbooks in the field [5, 19] for more details.

Reinsurance is regarded as a framework of protecting the insurance companies
against potential catastrophic high losses [1]. As an insurance company
charges its policyholders with premiums, the reinsurer also defines a premium
adapted to the amount of preassumed uncertain compensation cover that offers
the cedent party. Reinsurance’s role is to limit the primary insurance
liability with a predefined risk and increase its risk-taking capacity when
the likely loss is predicted to be overwhelming.

There are two basic forms of reinsurance contracts; proportional and non-
proportional. In the former, both parties admit a predetermined share of all
premiums and their potential losses. In the latter, reinsurance’s liability is
not determined already but depends on the number or severity of the claims
incurred before the reinsurance contract’s expiration [1].

One of the most important concerns about the reinsurance contracts is
determining optimal retention, i.e., the optimum amount of the risk that
primary insurance holds. Since the pioneering works by Borch [9], Kahn [26],
and Arrow [4], there have been many contributions to the subject. Followed by
these fundamental works, other researchers added a great deal of knowledge to
the literature, e.g., [10, 11, 20, 23, 40, 41].

The special case of proportional reinsurance was the subject of some other
researches [12, 13, 14, 27]. Common objectives include minimizing the
probability of ruin [6, 16, 39], maximizing the expected utility of the
terminal wealth [7, 21, 37, 29], and mean variance portfolio optimization [8,
27]. For a comprehensive review on optimal reinsurance approaches see [15].

Probability theory based models are assumed to be practical and useful, not to
be considered the perfect and the only way to model certain problems. As noted
by some people [35, 45, 46], probability theory has some shortcomings in
dealing with those problems with no sufficient data, or basic information is
based on some experts’ opinions.

Liu [30] introduced the uncertainty theory to deal with problems related to
belief degree in expert-based systems. Analogous to the stochastic point of
view, Liu [31] introduced the concept of renewal and renewal reward process
for initiated systems based on sudden arrival events. To deal with the ruin
theory problems, the concept of ruin index has been proposed in the
uncertainty theory framework as its substitution in probability theory [36].
Li et al. [28] proposed a premium calculation principle in an uncertain
environment. Later, Yao et al. [43, 44] modified the results on ruin index in
some extensions and generalized over different assumptions.

Uncertainty theory has two crucial features that make it a reliable tool
dealing with expert-based models. Firstly, it enjoys a well-defined axiomatic
structure similar to probability theory. Secondly, this theory is designed in
a way to be tractable for subjective modeling in the fields like actuarial
science that distributions are highly based on belief and not data. Here we
investigate the impact of proportional reinsurance on an insurance company’s
surplus process and its survival that is defined as leaving the uncertain
measure of ruin (UMR) under a fair bound. We assume a quota share reinsurance
contract, through which the primary insurance cedes premiums and losses with
an unknown cession rate to the reinsurer. We need to consider a safety loading
for both reinsurance and insurance premiums so that the UMR does not occur in
an almost certain way in the long run. We study the optimal retention of
proportional reinsurance under ruin-related optimization restrictions. The
model determines parameters that identify the best possible scenario to get
involved in a reinsurance contract. Based on a proportional type of
reinsurance contract, we only study the problem from an insurance point of
view, assuming that reinsurers are willing and capable of taking the whole
committed risk if required.

The rest of the paper is organized as follows. Since uncertainty theory is
relatively new, in the next section we present some basics on the theory which
will be used in the main parts of the paper. Section 3 includes three
subsections. First, we introduce a new way of calculating the uncertain
measure of ruin followed by a detailed example. We then discuss optimal
reinsurance as an uncertain optimization problem and describe the results
through an example in non-life insurance. The section will be ended with a
generalization of the proposed model. The conclusion suggests some possible
extensions that might make the model more reliable.

##  2 Preliminaries

Let $\Gamma$ be a nonempty set and $\mathscr{L}$ be a $\sigma$\- algebra over
$\Gamma$. Each element $\Lambda\in\mathscr{L}$ is called an event. A set
function $\mathscr{M}$ from $\mathscr{L}$ to $[0,1]$ that satisfies the
following axioms is called an uncertain measure [30]:

######  Axiom 1.

(Normality Axiom) $\mathscr{M}\\{\Gamma\\}=1$.

######  Axiom 2.

(Duality Axiom) $\mathscr{M}\\{\Lambda\\}+\mathscr{M}\\{\Lambda^{c}\\}=1$ for
any event $\Lambda$.

######  Axiom 3.

(Subadditivity Axiom) For every countable sequence of events
$\Lambda_{1},\Lambda_{2},\ldots$,

$\mathscr{M}\\{\bigcup_{i=1}^{\infty}\Lambda_{i}\\}\leq\sum_{i=1}^{\infty}\mathscr{M}\\{\Lambda_{i}\\}.$
(2)

The triplet $(\Gamma,\mathscr{L},\mathscr{M})$ is called an uncertainty space.
The Product axiom, distinguishing the probability theory from the uncertainty
theory, is defined as follows [32].

######  Axiom 4.

(Product Axiom) Let $(\Gamma_{k},\mathscr{L}_{k},\mathscr{M}_{k})$ be
uncertainty spaces for $k=1,2,\ldots$. The product uncertain measure
$\mathscr{M}$ is a measure on product $\sigma$-algebra
$\mathscr{L}_{1}\times\mathscr{L}_{2}\times\cdots\times\mathscr{L}_{n}$
satisfying

$\mathscr{M}\\{\prod_{k=1}^{\infty}\Lambda_{k}\\}=\bigwedge_{k=1}^{\infty}\mathscr{M}_{k}\\{\Lambda_{k}\\}.$
(3)

Uncertain variable has been defined to ease the quantitative modeling of
phenomena in uncertainty theory [30]. It is a measurable function from an
uncertain space $(\Gamma,\mathscr{L},\mathscr{M})$ to the set of real numbers,
in which for any Borel set $B$, the set $\\{\xi\in
B\\}=\\{\gamma\in\Gamma|\xi(\gamma)\in B\\}$ is an event. Uncertainty
distribution of an uncertain variable $\xi$ is defined as [30]

$\Phi(x)=\mathscr{M}\\{\xi\leq x\\},\qquad\forall x\in\Re.$ (4)

There are several uncertain variables in the literature of the uncertainty
theory. The simplest one is the linear uncertain variable with the following
distribution,

$\displaystyle\Phi(x)=\begin{cases}0&~{}{\rm if}~{}x\leq a\\\
\displaystyle\frac{x-a}{b-a}&~{}{\rm if}~{}a\leq x\leq b\\\ 0&~{}{\rm
if}~{}x\geq b,\end{cases}$ (5)

denoted by $\mathscr{L}(a,b)$ where $a$ and $b$ are real numbers with $a<b$.

An uncertain variable $\xi$ is called normal and denoted by
$\mathscr{N}(e,\sigma)$ [30], if it has the normal uncertain distribution

$\Phi(x)=\left(1+exp\left(\frac{\pi(e-x)}{\sqrt{3}\sigma}\right)\right)^{-1},x\in\Re,$
(6)

where $e$ and $\sigma$ are real numbers with $\sigma>0$. Moreover, an
uncertain variable $\xi$ is called lognormal if $\ln\xi$ is a normal uncertain
variable. Its distribution reads as

$\Phi(x)=\left(1+exp\left(\frac{\pi(e-\ln
x)}{\sqrt{3}\sigma}\right)\right)^{-1},x\geq 0.$ (7)

Uncertain variables $\xi_{1},\xi_{2},\dots,\xi_{n}$ are independent if [32]

$\mathscr{M}\left\\{\bigcap_{i=1}^{n}(\xi_{i}\in
B_{i})\right\\}=\bigwedge_{i=1}^{n}\mathscr{M}\left\\{\xi_{i}\in
B_{i}\right\\},$ (8)

for arbitrary Borel sets $B_{1},B_{2},\dots,B_{n}$. An uncertain distribution
$\Phi(x)$ is called regular if it is a continuous and strictly increasing
function w.r.t. $x$ for all $0<\Phi(x)<1$, and

$\lim_{x\rightarrow-\infty}\Phi(x)=0,\qquad\lim_{x\rightarrow+\infty}\Phi(x)=1.$
(9)

Let $\xi_{1},\xi_{2},\ldots,\xi_{n}$ be independent uncertain variables with
regular uncertainty distributions $\Phi_{1},\Phi_{2},\ldots,\Phi_{n}$,
respectively. If $f(\xi_{1},\xi_{2},\ldots,\xi_{n})$ is strictly increasing
w.r.t. $\xi_{1},\xi_{2},\ldots,\xi_{m}$ and strictly decreasing w.r.t.
$\xi_{m+1},\xi_{m+2},\ldots,\xi_{n}$, then $f$ is an uncertain variable with
the uncertainty distribution [34, Theorem 1.26]

$\Psi(x)=\sup_{f(x_{1},x_{2},\ldots,x_{n})=x}\left(\min_{1\leq i\leq
m}\Phi_{i}(x_{i})\wedge\min_{m+1\leq i\leq n}(1-\Phi(x_{i}))\right).$ (10)

######  Definition 1.

Let $\xi$ be an uncertain variable with regular uncertainty distribution
$\Phi(x)$. The inverse function $\Phi^{-1}(\alpha)$ is called the inverse
uncertainty distribution of $\xi$.

######  Theorem 2.

If $f(\xi_{1},\xi_{2},\ldots,\xi_{n})$ is strictly increasing w.r.t.
$\xi_{1},\xi_{2},\ldots,\xi_{m}$ and strictly decreasing w.r.t.
$\xi_{m+1},\xi_{m+2},\ldots,\xi_{n}$, then $f$ is an uncertain variable with
the inverse uncertainty distribution

$\Psi^{-1}(\alpha)=f(\Phi^{-1}(\alpha),\ldots,\Phi^{-1}(\alpha),\Phi_{m+1}^{-1}(1-\alpha),\ldots,\Phi_{n}^{-1}(1-\alpha)).$
(11)

######  Theorem 3.

[33] Let $\xi_{1},\xi_{2},\ldots,\xi_{n}$ be independent uncertain variables
with regular uncertainty distributions $\Phi_{1},\ldots,\Phi_{n}$,
respectively. If $f(\xi_{1},\ldots,\xi_{n})$ is strictly increasing with
respect to $\xi_{1},\xi_{2},\ldots,\xi_{m}$ and strictly decreasing with
respect to $\xi_{m+1},\xi_{m+2},\ldots,\xi_{n}$, then

$\mathscr{M}\left\\{f(\xi_{1},\xi_{2},\ldots,\xi_{n})\leq 0\right\\},$ (12)

is the root $\alpha$ of the equation

$f(\Phi_{1}^{-1}(\alpha),\ldots,\Phi_{m}^{-1}(\alpha),\Phi_{m+1}^{-1}(1-\alpha),\ldots,\Phi_{n}^{-1}(1-\alpha))=0.$
(13)

If
$f(\Phi_{1}^{-1}(\alpha),\ldots,\Phi_{m}^{-1}(\alpha),\Phi_{m+1}^{-1}(1-\alpha),\ldots,\Phi_{n}^{-1}(1-\alpha))<0,$
for all $\alpha$, then we set the root $\alpha=1$; and if
$f(\Phi_{1}^{-1}(\alpha),\ldots,\Phi_{m}^{-1}(\alpha),\Phi_{m+1}^{-1}(1-\alpha),\ldots,\Phi_{n}^{-1}(1-\alpha))>0$
for all $\alpha$, then we set the root $\alpha=0$.

The expected value of an uncertain variable $\xi$ is defined as [30]

$E[\xi]=\int_{0}^{+\infty}\mathscr{M}\\{\xi>r\\}dr-\int_{-\infty}^{0}\mathscr{M}\\{\xi\leq
r\\}dr,$ (14)

provided that at least one of the two integrals is finite. Having an uncertain
distribution $\Phi(x)$ of an uncertain variable $\xi$, one can calculate its
expected value by [30]

$E[\xi]=\int_{0}^{\infty}(1-\Phi(x))dx-\int_{-\infty}^{0}\Phi(x)dx,$ (15)

provided that at least one of these integrals is finite. It is proved that for
an uncertain variable $\xi$ with regular uncertain distribution $\Phi(x)$, it
holds

$E[\xi]=\int_{0}^{1}\Phi^{-1}(\alpha)d\alpha.$ (16)

Variance of an uncertain variable $\xi$ with finite expected value $E[\xi]$ is
also defined as [30]

$V[\xi]=E\left[(\xi-E[\xi])^{2}\right].$ (17)

Let $\xi$ and $\eta$ be independent uncertain variables with finite expected
values. Then, for any real numbers $a$ and $b$ [34]

$E\left[a\xi+b\eta\right]=aE\left[\xi\right]+bE\left[\eta\right].$ (18)

### Uncertain process

Let $(\Gamma_{k},\mathscr{L}_{k},\mathscr{M}_{k})$ be uncertainty spaces for
$k=1,2,\ldots$, and $T$ be a totally ordered set (e.g. time). For the sake of
simplicity, we use the term “time” for each member of this set. An uncertain
process is a function $X_{t}(\gamma)$ from
$T\times(\Gamma_{k},\mathscr{L}_{k},\mathscr{M}_{k})$ to the set of real
numbers such that $\\{X_{t}\in B\\}$ is an event for any Borel set $B$ of real
numbers at each time $t$. An uncertain process $X_{t}$ has independent
increments if

$X_{t_{0}},X_{t_{1}}-X_{t_{0}},X_{t_{2}}-X_{t_{1}},\ldots,X_{t_{k}}-X_{t_{k-1}},$
(19)

are independent uncertain variables where $t_{0}$ is the initial time and
$t_{1},t_{2},\ldots,t_{k}$ are any times with
$t_{0}<t_{1}<t_{2}<\cdots<t_{k}$.

An uncertain process has stationary increments if its increments are
independent identically distributed (iid) uncertain variables whenever the
time intervals have equal lengths. For uncertain process $X_{t}$, fixing the
event $\omega$, the resulted function $X_{t}(\omega)$ is called a sample path
of $X_{t}$.

######  Definition 4.

[32] An uncertain process $C(t)$ is called Liu process if
(a) $C(0)=0$ and almost all sample paths are Lipschitz continuous,
(b) $C(t)$ has stationary and independent increments,
(c) Every increment $C(t+s)-C(s)$ is a normal uncertain variable with expected
value zero and variance $t^{2}$.

######  Definition 5.

[31] Let $\xi_{1},\xi_{2},\ldots$ be iid uncertain interarrival times. Define
$S_{0}=0$ and $S_{n}=\xi_{1}+\xi_{2}+\cdots+\xi_{n}$ for $n\geq 1$. The
uncertain process

$N_{t}=\max_{n\geq 0}\\{n|S_{n}\leq t\\}$ (20)

is called an uncertain renewal process.

######  Theorem 6.

[34] Let $N_{t}$ be a renewal process with iid uncertain interarrival times
$\xi_{1},\xi_{2},\ldots$. Further, let $\Phi$ denote their common uncertainty
distribution. Then, $N_{t}$ has an uncertainty distribution

$\Upsilon_{t}(x)=1-\Phi\left(\frac{t}{\lfloor x\rfloor+1}\right),\qquad\forall
x\geq 0,$ (21)

where $\lfloor x\rfloor$ represents the floor function.

######  Theorem 7.

[34] Let $N_{t}$ be a renewal process with iid uncertain interarrival times
$\xi_{1},\xi_{2},\ldots$ Then the average renewal number

$\frac{N_{t}}{t}\rightarrow\frac{1}{\xi_{1}}$ (22)

in the sense of convergence in distribution as $t\rightarrow\infty$.

######  Theorem 8.

[34] Let $N_{t}$ be a renewal process with iid uncertain interarrival times
$\xi_{1},\xi_{2},\ldots$ Then

$\lim_{t\rightarrow\infty}\frac{E[N_{t}]}{t}=E\left[\frac{1}{\xi_{1}}\right].$
(23)

If $\Phi$ is regular, then

$\lim_{t\rightarrow\infty}\frac{E[N_{t}]}{t}=\int_{0}^{1}\frac{1}{\Phi^{-1}(\alpha)}d\alpha.$
(24)

######  Definition 9.

[34] Let $\xi_{1},\xi_{2},\ldots$ be iid uncertain interarrival times.
Further, let $\eta_{1},\eta_{2},\ldots$ be iid uncertain rewards or costs (
losses in insurance case) associated with $i$-th interarrival times $\xi_{i}$
for $i=1,2,\cdots$Then

$R_{t}=\sum_{i=1}^{N_{t}}\eta_{i},$ (25)

is called a renewal reward process, where $N_{t}$ is the renewal process with
uncertain interarrival times $\xi_{1},\xi_{2},\ldots$

######  Theorem 10.

[34] Let $R_{t}$ be a renewal reward process with iid uncertain interarrival
times $\xi_{1},\xi_{2},\ldots$ and iid uncertain rewards
$\eta_{1},\eta_{2},\ldots$ Assume $(\xi_{1},\xi_{2},\ldots)$ and
$(\eta_{1},\eta_{2},\ldots)$ are independent uncertain vectors, and those
interarrival times and rewards have uncertainty distributions $\Phi$ and
$\Psi$, respectively. Then, $R_{t}$ has uncertainty distribution

$\Upsilon_{t}(x)=\max_{k\geq
0}\left\\{\left(1-\Phi\left(\frac{t}{k+1}\right)\right)\bigwedge\Psi(\frac{x}{k})\right\\}.$
(26)

######  Theorem 11.

Let $R_{t}$ be a renewal reward process with iid uncertain interarrival times
$\xi_{1},\xi_{2},\ldots$ and iid uncertain rewards $\eta_{1},\eta_{2},\ldots$
Assume $(\xi_{1},\xi_{2},\ldots)$ and $(\eta_{1},\eta_{2},\ldots)$ are
independent uncertain vectors. Then the reward rate

$\frac{R_{t}}{t}\rightarrow\frac{\eta_{1}}{\xi_{1}},$ (27)

in the sense of convergence in distribution as $t\rightarrow\infty$.

######  Definition 12.

[36] Let $U_{t}$ be an insurance risk process. Then the ruin index is defined
as the uncertain measure that $U_{t}$ eventually becomes negative, i.e.,

${\rm Ruin}=\mathscr{M}\left\\{\inf_{t\geq 0}U_{t}<0\right\\}.$ (28)

##  3 The Optimization Model

Here, we consider the process of variation in the wealth of the primary
insurance as

$U_{t}^{x}=u+\left[x(1+\rho)-(\rho-\theta)\right]ct-xR_{t},$ (29)

where, $x\in(0,1]$ is a control parameter; and $\rho$ and $\theta$ are
reinsurance and insurance safety loads, respectively. It is worthy to remind
that term $c$, as the insurance premium, should be chosen in a way to ensure
that ruin does not happen almost surely for any amount of the initial capital
or reinsurance program.

When a company considers taking a reinsurance contract, solving the following
optimization problem identifies the ruin index.

${\rm Ruin}=\max_{k\geq 1}\sup_{z\geq
0}\bigg{\\{}\Phi\Big{(}\frac{z}{k+1}\Big{)}\bigwedge
1-\Psi\Big{(}\frac{u+\big{[}x(1+\rho)-(\rho-\theta)\big{]}cz}{xk}\Big{)}\bigg{\\}}.$
(30)

Note that solving this problem is very costly; it is almost impossible for
most uncertain distributions, and just its approximations would be identified.
To overcome this difficulty, we propose a simpler method for the UMR and
construct a constraint to prevent ruin. For the ultimate ruin, the ruin at the
infinite time, it is sensible to model an optimization problem dependent on
$t$ and taking the objective function as the asymptotic uncertain expected
value of the insurer’s risk process. Formally, we consider the following
constrained optimization problem.

$\begin{array}[]{rrclcl}\displaystyle\max_{0\leq x\leq
1}&\lx<EMAIL_ADDRESS>\textrm{s.t.}&{\rm Ruin}\leq\varepsilon,\end{array}$ (31)

where $\varepsilon$ is a small positive number chosen by the administration as
maximum tolerable ruin.

###  3.1 Calculating the Uncertain Measure of Ruin

First, we need to compute the UMR for primary insurance. In the contract
duration, arrival times coincide with some point in the interval $(0,t]$.
Since ruin might happen in only one of these arrivals, we regard
$t=\sum_{i=1}^{N_{t}}\xi_{i}$, where $\xi_{i}$’s are uncertain interarrival
times. Therefore, the risk process becomes

$U_{t}=u+\sum_{i=1}^{N_{t}}(c\xi_{i}-\eta_{i}).$ (32)

Let $\xi_{i}$ and $\eta_{i}$ be independent uncertain variables. Then by
Theorem, the inverse uncertainty distribution of $\eta_{1}-c\xi_{1}$ is

$\Upsilon^{-1}_{1}(\alpha)=\Psi^{-1}_{1}(\alpha)-c\Phi^{-1}_{1}(1-\alpha)=\Psi^{-1}(\alpha)-c\Phi^{-1}(1-\alpha).$
(33)

Let $\Theta_{i}=\eta_{i}-c\xi_{i}$ and $M_{k}=\Theta_{1}+\cdots+\Theta_{k}$.
It can be easily understood that the inverse uncertainty distribution of
$M_{k}$ for all $k$ is

$\displaystyle L^{-1}_{k}(\alpha)=k[\Psi^{-1}(\alpha)-c\Phi^{-1}(1-\alpha)],$
(34)

where $L_{k}$ is the uncertainty distribution of the uncertain variable
$M_{k}$.

Notice that ruin happens when $\max_{k\geq 1}\left\\{M_{k}\right\\}\geq u.$
Recall that direct calculation of this quantity is a very costly task using
its uncertainty distribution, especially when uncertain distributions like
lognormal are involved in the model. Taking
$f=\max_{k}\left\\{x_{1},\ldots,x_{k}\right\\}-u$, $M_{k}$’s and $f$ satisfy
the requirements of Theorem 3. Therefore, the UMR is

$\mathscr{M}\left\\{f(M_{1},M_{2},\ldots,M_{k})>0\right\\}.$ (35)

For $k$ big enough, the root $\alpha$ of

$\displaystyle\max_{k}\left\\{L^{-1}_{1}(1-\alpha),L^{-1}_{2}(1-\alpha),\ldots,L^{-1}_{k}(1-\alpha)\right\\}-u=0$
(36)

identifies the UMR. Equivalently, we have to solve

$\displaystyle\max\bigg{\\{}\Psi^{-1}(1-\alpha)-c\Phi^{-1}(\alpha),\ldots,k(\Psi^{-1}(1-\alpha)-c\Phi^{-1}(\alpha))\bigg{\\}}=u.$
(37)

We need the assumption

${\rm Premium}>E[\eta_{i}]E[\xi_{i}],$ (38)

in the selection of value $c$, because after each interarrival time, one
expects $E[\xi_{i}]$ as the amount of loss, and so the premium must be bigger
than this amount. Observe that $\eta_{i}$’s are iid, means that for all $i$
these expectation amounts are identical, as for the $\xi_{i}$s. This criterion
on premiums prevents the model from the obvious event of ruin being determined
in a long run.

To illustrate the practicality of this approach in calculating the ruin,
consider the following concrete example.

######  Example 1.

Consider an insurance company takes some risk in signing a particular class of
underwriting. They invite some experts to build two uncertain distributions
for the potential number of claims and their severity. Suppose their opinion
on the severity of claims and the interarrival times are $\mathscr{LOGN}(2,1)$
and $\mathscr{L}(1,3)$, respectively.

A way to know the measure of possible bankruptcy caused by these contracts is
to identify the company’s UMR. As it was said, it is the root $\alpha$ of the
following equation. To calculate the UMR, we used online Mathematica [25].

$\displaystyle\max_{k\geq 1}$
$\displaystyle\bigg{\\{}\exp\Big{(}2+\frac{\sqrt{3}}{\pi}\ln\frac{1-\alpha}{\alpha}\Big{)}-26(2\alpha+1),\ldots,$
(40) $\displaystyle\hskip
28.45274ptk\bigg{(}\exp\Big{(}2+\frac{\sqrt{3}}{\pi}\ln\frac{1-\alpha}{\alpha}\Big{)}-26(2\alpha+1)\bigg{)}\bigg{\\}}=u.$

Fig.1(a) represents the UMR concerning the number of claims $k$ with fixed
$u=10000$. Notice that this measure gets stable after $k$ gets big enough (
for $k=10000$ and higher), and then the measure does not change considerably.
This observation is a completely rational outcome because we expect the
ultimate ruin to lift by increasing the number of claims. After a point on,
there is no significant fluctuation in the number UMR, which is mostly because
of the premiums’ growth. We also illustrated the UMR for different amounts of
the initial capital $u$, for fixed $k=100$, Fig.1(b). As it is depicting,
increasing the initial capital reduces the measure of ruin substantially to a
point where it is almost zero.

(a) Number of Claims $k$ for $u=10000$.

(b) Different amounts of $u$ for $k=100$.

Figure 1: The sample UMR in Example 1.

###  3.2 Optimal Retention

Applying these results for the case of quota share reinsurance contract, we
devise a model to explain the primary insurance situation. Solving the
optimization problem provides us the optimal retention value $x$. First, we
need to calculate the UMR for the case of reinsurance involvement. After
sharing its risk with reinsurance, the primary insurance has the following
wealth process

$U_{t}^{x}=u+\sum_{i=1}^{N_{t}}\left[x(1+\rho)-(\rho-\theta)\right]c\xi_{i}-x\eta_{i},$
(41)

with the assumption $\rho>\theta$, meaning that the risk in reinsurance is
more significant than the one in primary insurance. Notice that if we take
$x=1$, the wealth process (41) describes the primary insurance without
reinsurance partnership.

Let $\left[x(1+\rho)-(\rho-\theta)\right]c$ be denoted by $\beta$. Similar to
Sec. 3.1, each $x\eta_{i}-\beta\xi_{i}$ has the inverse uncertainty
distribution

$\hat{\Upsilon}^{-1}(\alpha)=x\Psi^{-1}(\alpha)-\beta\Phi^{-1}(1-\alpha).$
(42)

Moreover, for $\hat{\Theta}_{i}=x\eta_{i}-\beta\xi_{i}$, uncertain variables
$\hat{M}_{k}=\hat{\Theta}_{1}+\cdots+\hat{\Theta}_{k}$ has the following
inverse uncertainty distributions

$\hat{L}_{k}^{-1}(\alpha)=k\left(x\Psi^{-1}(\alpha)-\beta\Phi^{-1}(1-\alpha)\right),$
(43)

where $\hat{L}_{k}$ is the uncertain distribution of $\hat{M}_{k}$. To reflect
the profitability of the contracts for the primary insurance company, we
consider

$\Bigg{[}x(1+\rho)-(\rho-\theta)\Bigg{]}c-x\int_{0}^{1}\left[\frac{\Psi^{-1}(\alpha)}{\Phi^{-1}(1-\alpha)}\right]d\alpha,$
(44)

as the objective function, that is the expected value of the wealth with
respect to $t$. It means that the company makes a profit equal to the
objective function at each unit of time in the long run. Our purpose is to
make this amount positive and as large as possible by an optimal value of $x$.
Considering the profitablity of the company in small time intervals influences
the idea of optimizing the expected value of the primary insurance company’s
wealth asymptotically.

The problem identifies the optimal retention rate that maximizes the objective
function, the company’s wealth, constrained by keeping its UMR under a
predefined value. Thus, we have the following optimization problem.

$\begin{array}[]{l}\displaystyle\max_{0\leq x\leq
1}\Big{[}x(1+\rho)-(\rho-\theta)\Big{]}c-x\int_{0}^{1}\frac{\Psi^{-1}(\alpha)}{\Phi^{-1}(1-\alpha)}d\alpha\\\\[8.53581pt]
\textrm{s.t.}\quad\textrm{Root}\Big{\\{}\max_{k\geq
1}\big{\\{}\hat{L}^{-1}_{1}(1-\alpha),\ldots,\hat{L}^{-1}_{k}(1-\alpha)\big{\\}}-u=0\Big{\\}}\leq\varepsilon.\end{array}$
(45)

######  Remark 1.

Observe that in this optimization problem, for $x=0$, the insurance company
transfers the whole risk to the reinsurance and leaves with no premium. On the
other hand, $x=1$ means that the primary insurance takes the whole risk by
itself and does not get into the reinsurance contract. The value of $x$ is
completely dependant on the distributions of severity and number of claims.
Recall that an expert provides these uncertainty distributions. While, initial
capital, parameters such as $\rho$ and $\theta$ depend on the amount of
premium $c$. The value of $c$ is also provided based on the management policy.

######  Example 2.

We assume an insurance company is deciding on arranging a great amount of non-
life insurance contracts. Let it aims to underwrite many houses and businesses
in a new area. Since it has no enough experience in this region, the
management may ask some experts to contribute their opinions via uncertain
distributions on the number and severity of claims based on their experiences.
Consider the case where the interarrival times of the renewal process have a
linear distribution, $\mathscr{L}(1,3)$, and claim variables follow the
lognormal distribution $\mathscr{LOGN}(2,1)$. This means that experts believe,
overall, that for each policyholder there would be at least one claim in the
long run, and the number of claims is not bigger than 3. Also, they believe
that the amount of loss that each claim produces is in a way that makes a log-
normal distribution, meaning that there is always a considerable uncertain
measure devoted to rare events.

Now suppose $\rho=0.9,\theta=0.8,c=26$, and $\varepsilon=0.005$. With these
assumptions, the optimization problem reads as

$\begin{array}[]{l l}\displaystyle\max_{0\leq x\leq 1}&39.6x-2.6\\\
\textrm{s.t.}&\textrm{Root}\Big{\\{}\max_{k\geq
1}k\big{[}x\exp(2+\frac{\sqrt{3}}{\pi}\ln\frac{1-\alpha}{\alpha})\\\ &\hskip
85.35826pt-(49.4x-2.6)(2\alpha+1)\big{]}-u\Big{\\}}\leq 0.005.\end{array}$
(46)

The optimal retentions of the company for $k=100$, a fixed number of possible
claims, and for different initial capital amounts are represented in Fig.2(a),
which show the behavior of retention rate with varying this value. To analysis
the effect of claims’ number on the retention rate, we fix the initial capital
at $u=100000$.

When $u$ is small the UMR is significantly high, consequently, the retention
needs to be small to keep the UMR under the assumed limit. Raising the initial
capital certainly gives a more chance to the insurance company to take more
risks and at the same time keeping the UMR under the limit, Graph (2(a)).
Investigating Graph (2(b)) demonstrates that, growing the number $k$,
increases the UMR, which makes the retention rate to decrease in a way that
the problem’s constraint stays satisfied. Almost by $k=7000$ (the fixed
initial capital), and the numbers higher than that, the amount of retention
gets stabilized.

(a) Different Amount of the Initial Capital.

(b) Different Possible Amount of the Claims.

Figure 2: Optimal Retention Rate for the Insurance Company Constrained to Keep
UMR Under Preassumed Number in Example 2.

Notice that the number of sold contracts by the primary insurance company can
be considered as the potential claims’ number. With this assumption, one can
assume that a catastrophe situation would occur when all these contracts have
a loss and confirm a claim.

This optimization problem can also be more practically used by incorporating
other optimal control parameters to design contracts and reduce the risk while
raising their potential gain. It should also be noticed that increasing the
initial capital $u$ lessens the risk of ruin, too. However, we don’t use it as
a control parameter as it is absent in the objective function.

###  3.3 A Perturbed Extension

Following the idea of perturbing the risk process by a Brownian motion [18],
one can perturb the UMR by the Liu process. By perturbation, we mean adding an
uncertain process that can take positive and negative numbers. This
generalized model is used to include more practical considerations in the
model. We can interpret it as the imperfect human reckoning in the expert
opinion’s contribution when only a few data is available. In this way, adding
a potential fluctuation to the model allows us to mimic this incompetence. It
can also be recognized as the variation caused by dependency between uncertain
variables used in the independent model, recalling that they are mostly
dependent in real life.

The perturbed problem would be modeled as

$U_{t}=u+ct-R_{t}+C_{t},$ (47)

where $C_{t}$ is a Liu process having the following inverse uncertainty
distribution.

$\tilde{\Phi}_{t}^{-1}(\alpha)=t\frac{\sqrt{3}}{\pi}\ln(\frac{\alpha}{1-\alpha}).$
(48)

We assume that the Liu process, at each point of time, is independent of other
uncertain variables in the model. Following the inference for the non-
perturbed case, we consider the uncertain variables

$\tilde{\Theta}_{i}=\eta_{i}-c\xi_{i}-C_{i},$ (49)

as a perturbed version of $\Theta$. The inverse uncertainty distribution of
this uncertain variable is

$\tilde{\Upsilon}^{-1}_{i}(\alpha)=\Psi^{-1}_{i}(\alpha)-c\Phi^{-1}_{i}(1-\alpha)-\tilde{\Phi}_{i}^{-1}(1-\alpha).$
(50)

Defining
$\tilde{M}_{k}=\tilde{\Theta}_{1}+\tilde{\Theta}_{2}+\cdots+\tilde{\Theta}_{k},$
the UMR becomes the uncertain measure of the following event

$\max_{k\geq 1}\left\\{\tilde{M}_{k}\right\\}\geq u,$ (51)

with the inverse uncertainty distribution

$\displaystyle\tilde{L}^{-1}_{k}(\alpha)$ $\displaystyle=$ $\displaystyle
k(\Psi^{-1}(\alpha)-c\Phi^{-1}(1-\alpha)-\tilde{\Phi}_{k}^{-1}(1-\alpha)),\qquad
k\geq 1.$ (52)

Using Theorem 3, for $k$ big enough, the UMR is the root $\alpha$ of

$\displaystyle\max_{k}\left\\{\tilde{L}^{-1}_{1}(1-\alpha),\tilde{L}^{-1}_{2}(1-\alpha),\ldots,\tilde{L}^{-1}_{k}(1-\alpha)\right\\}-u=0,$
(53)

or equivalently the root of

$\begin{array}[]{l}\max_{k\geq
1}\Big{\\{}\Psi^{-1}(1-\alpha)-c\Phi^{-1}(\alpha)-\tilde{\Phi}_{k}^{-1}(\alpha),\ldots,\\\
\hskip
42.67912ptk(\Psi^{-1}(1-\alpha)-c\Phi^{-1}(\alpha)-\tilde{\Phi}_{k}^{-1}(\alpha))\Big{\\}}=u.\end{array}$
(54)

Similarly, when reinsurance is involved in the model, we have

$U_{t}^{x}=u+\left[x(1+\rho)-(\rho-\theta)\right]ct-xR_{t}+C_{t}.$ (55)

Defining $\tilde{\Theta}_{i}=x\eta_{i}-\beta\xi_{i}-C_{i}$, and
$\tilde{M}_{k}=\tilde{\Theta}_{1}+\tilde{\Theta}_{2}+\cdots+\tilde{\Theta}_{k},$
with the inverse uncertainty distribution

$\tilde{L}_{k}^{-1}(\alpha)=k\left(x\Psi^{-1}(\alpha)-\beta\Phi^{-1}(1-\alpha)-\tilde{\Phi}_{k}^{-1}(1-\alpha))\right),$
(56)

the UMR is the root $\alpha$ of

$\max_{k\geq
1}\left\\{\tilde{L}^{-1}_{1}(1-\alpha),\tilde{L}^{-1}_{2}(1-\alpha),\ldots,\tilde{L}^{-1}_{k}(1-\alpha)\right\\}-u=0.$
(57)

##  4 Conclusions

In this paper, we investigated a model designed for an insurance company that
wants to share a part of its risk with a reinsurer. Uncertainty theory was
applied, for the first time, to model the involved uncertainty of the problem.
Some comparisons between initial capital, the maximum number of possible
claims, and the optimal retention were achieved. Further research can model
other kinds of reinsurance contracts as an uncertain optimization problem with
similar or different sorts of constraints.

## Conflict of interest

The authors declare that they have no conflict of interest.

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