RECENT ADVANCES AND APPLICATIONS OF THE THEORY OF

STOCHASTIC CONVEXITY. APPLICATION TO COMPLEX

BIO-INSPIRED AND EVOLUTION MODELS

Eva Maria Ortega

and Jose Alonso

Centro de Investigacion Operativa, University Miguel Hernandez Elche, 03202 Elche, Spain

Hospital Universitario Virgen de Arrixaca, Servicio Radiologia, El Palmar 30012, Murcia, Spain

Keywords:

Stochastic directional convexity, Stochastic ordering, Increasing directionally convex function, Variabil-

ity, bounding, biologically inspired models, Evolution models, mixtures, Multiplicative process, Tree network.

Abstract:

The theory of stochastic convexity is widely recognised as a framework to analyze the stochastic behaviour

of parameterized models by different notions in both univariate and multivariate settings. These proper-

ties have been applied in areas as diverse as engineering, biotechnology, and actuarial science. Consider a

family of parameterized univariate or multivariate random variables {X(θ)|θ ∈ T} over a probability space

(Ω, ℑ, Pr), where the parameter θ usually represents some distribution moments. Regular, sample-path, and

strong stochastic convexity notions have been deﬁned to intuitively describe how the random objects X(θ)

grow convexly (or concavely) concerning their parameters. These notions were extended to the multivariate

case by means of directionally convex functions, yielding the concepts of stochastic directional convexity for

multivariate random vectors and multivariate parameters. We aim to explain some of the basic concepts of

stochastic convexity, to discuss how this theory has been used into the stochastic analysis, both theoretically

and in practice, and to provide some of the recent and of the historically relevant literature on the topic. Finally,

we describe some applications to computing/communication systems based on bio-inspired models.

1 INTRODUCTION

The theory of stochastic convexity is widely recog-

nised as a framework to analyze the stochastic be-

haviour of parameterized models by different notions

in both univariate and multivariate settings. These

properties have been applied in areas as diverse as

engineering, biotechnology, hydrology, actuarial sci-

ence, and medicine, and they have been useful to

analyze queueing systems, total times to complete

jobs, wireless communication networks, total claim

amount distributions of insurance portfolios, and resi-

dence times of a substance in the human body, among

others. Consider a family of parameterized univariate

or multivariate random variables {X(θ)|θ ∈ T} over a

probability space (Ω, ℑ, Pr), with T ⊂ R

with n ≥ 1,

being some sublattice, where the parameter θ usually

represents the mean or the variance, or both them for

the random variable X(θ), although it also may repre-

sent another rate inﬂuencing the probability distribu-

tion of X(θ). For practical instances, θ is sometimes

assumed to reﬂect some biological, physical, expo-

sure, environmental, or economical conditions, that

determine a scenario where the random object X(θ)

is analyzed. Let g(θ) = E[φ(X(θ))] be the expected

value of the functional φ for a univariate random vari-

able X(θ). Let ≤ denote the componentwise order-

ing for vectors of scalars. The stochastic increasing

monotonicity of X(θ) (or g(θ)) with respect to its pa-

rameter (in the sense that θ

≤ θ

implies X(θ

) ≤

X(θ

) or equivalently,Pr(X(θ

) > t) ≤ Pr(X(θ

) > t)

for all t) (see also Shaked and Shanthikumar, 2007)

constitutes a useful property in a wide range of ar-

eas as those aforementioned. Numerous examples of

functionals of random variables having such property

can be found in the literature, for instance, an increas-

ing net proﬁt (or an increasing expected net proﬁt)

is a common assumption in several ﬁnancial settings.

Additionally, different second-order properties, as the

convexity, the concavity and the supermodularity of

the parameterized random variable X(θ) with respect

to its parameter,have been described by different con-

cepts in the literature. Regular, sample-path, and

strong stochastic convexity notions have been deﬁned

to intuitively describe how the random objects X(θ)

grow convexly (or concavely) concerning their pa-

245

Ortega E. and Alonso J..

RECENT ADVANCES AND APPLICATIONS OF THE THEORY OF STOCHASTIC CONVEXITY. APPLICATION TO COMPLEX BIO-INSPIRED AND

EVOLUTION MODELS.

DOI: 10.5220/0003724102450251

In Proceedings of the International Conference on Evolutionary Computation Theory and Applications (ECTA-2011), pages 245-251

ISBN: 978-989-8425-83-6

 2011 SCITEPRESS (Science and Technology Publications, Lda.)

rameters, and they have played a prominent role into

the stochastic analysis in different ﬁelds. The purpose

of this paper is to explain some of the basic concepts

of stochastic convexity, and to discuss how this theory

has been used into the stochastic analysis, both theo-

retically and in practice. We will omit the technical

details of the results that the reader can check in the

corresponding articles. A second purpose is to pro-

vide the reader to some of the recent and historically

relevant literature on this topic. Finally, we focus on

several applications to bio-inspired models.

For a univariate random variable X(θ) with re-

spect to one real or integer-valued scalar θ, next, we

recall several stochastic convexity notions, where we

will use {X(θ)} as a shorthand of {X(θ)|θ ∈ T}.

First, {X(θ)} is stochastic increasing and convex in

the sample-path sense, denoted by SICX − sp, if for

any four parameter values θ

, i = 1, 2, 3, 4, that fulﬁl

+ θ

= θ

+ θ

and θ

≥ max{θ

, θ

}, there exists

four random variables

, i = 1, 2, 3, 4 on a common

probability space (Ω, ℑ, Pr), such that

X(θ

) for

any i; and ∀w ∈ Ω

(w) ≥ max{

(w),

(w)} (1)

(w) +

(w) ≥

(w) +

(w). (2)

This concept was developed in Shaked and Shan-

thikumar, 1988a, 1990a, 1990b. From (Shanthiku-

mar and Yao, 1991), we have the following stronger

notion. {X(θ)} is strong stochastic increasing and

convex, denoted by SSICX, if X(θ) =

φ(ε, θ), for

any φ being an increasing function on θ and ε being

a random variable whose distribution function is in-

dependent of θ. The regular notion, also called the

functional notion (Shaked and Shanthikumar, 1990a)

is deﬁned as follows. {X(θ)} is stochastic increas-

ing and convex, denoted by SICX, if E[φ(X(θ))] is

increasing and convex in θ for any increasing and

convex function φ. Another related concept intro-

duced by (Shaked and Shanthikumar, 1990b) is given

next. {X(θ)} is stochastic increasing and convex in

stochastic sense, denoted by SICX −st, if E[φ(X(θ))]

is increasing and convex in θ for any increasing func-

tion φ. Notice that the convexity of φ is not required

for this deﬁnition. The SICX − st property is equiva-

lent to the fact that the distribution function of X(θ),

denoted by F(x, θ), is decreasing and concave in θ for

any x. The following implications hold for the previ-

ous notions:

SSICX ⇒ SICX −sp ⇒ SICX and SICX −st ⇒ SICX −sp.

(3)

However, SICX − st and SSICX do not imply each

other. Many well known parametric families of distri-

butions, as the Normal, Gamma, Lognormal, Poisson,

Binomial and Geometric, fulﬁl some of these proper-

ties. We refer to the earlier references for the deﬁ-

nitions of the stochastic concavity notions. Nice re-

views of these concepts, their basic properties and ap-

plications can be found in (Shaked and Shanthikumar,

1991); (Chang et al., 1994) and (Shaked and Shan-

thikumar, 2007). The earlier notions are extended to

the multivariate case by means of directionally con-

vex functions (see Marinacci and Montrucchio, 2005,

for the deﬁnition, main properties and background of

these functions, that were deﬁned by Wright, 1954).

A useful characterization of the increasing direction-

ally convex functions was given in (Shaked and Shan-

thikumar, 1990a). A real-valued function φ deﬁned

on R

is increasing and directionally convex if and

only if φ is increasing, componentwise convex, and

supermodular. Also, φ is said to be supermodular (see

Marshall and Olkin, 1979) if φ(x ∨ y) + φ(x ∧ y) ≥

φ(x) + φ(y), for all x, y ∈ R

with the operators +,

∨ and ∧ denoting, respectively, the componentwise

sum, maximum and minimum. Stochastic directional

convexity for multivariate random vectors with re-

spect to multivariate parameters was introduced by

(Shaked and Shanthikumar, 1990a), when the param-

eters have on values over convex subsets of the real

line. These concepts were also studied by (Meester

and Shanthikumar,1993), who used this theory to ob-

tain the regularity of some stochastic processes, in-

cluding Markov chains, and to derive applications in

queueing systems; and extended to a general space by

(Meester and Shanthikumar, 1999). Stochastic con-

vexity in the sense of the usual stochastic order, intro-

duced by (Shaked and Shanthikumar, 1990b) in the

univariate case, was extended to the multivariate case

by (Shanthikumar and Yao, 1991), where is called

strong stochastic convexity. The functional notion of

stochastic increasingness and directional convexity is

denoted by SI − DCX. Other notions of stochastic

convexity in a generalized sense are deﬁned and stud-

ied in (Denuit et al., 1999), and notions involving par-

tial sums in (Chao and Luh, 2004).

2 DISCUSSION

In this section we discuss how the theory of the

stochastic convexity has been used into the stochas-

tic analysis, both theoretically and in practice, and

we provide some historically relevant and recent bib-

liographic references and remarks on their main de-

velopments and applications. Generally, the sample-

path notions of stochastic convexity are useful in the

mathematical proofs to state stochastic convexity of

random objects, while the functional deﬁnitions are

ECTA 2011 - International Conference on Evolutionary Computation Theory and Applications

246

good for applications, for example, when φ is the

functional of a performance measure of a system in

engineering to be bounded or the objective function

of a mathematical model to be optimized. In addi-

tion, the functional notions allow one to apply the

compositional rules in (Meester and Shanthikumar,

1999) to stand the stochastic increasingness and di-

rectional convexity (concavity) by using some trans-

formation of the parameters. The fact that this the-

ory does not rely on closed-form distribution func-

tions allows one to analyze stochastic models which

are unwieldy analytically. Several techniques of prob-

abilistic risk assessment, statistics and operational re-

search have been connected with the stochastic con-

vexity and the stochastic directional convexity. They

lead to solve some decision-making problems in sev-

eral contexts. First, looking to a dynamical perspec-

tive, these structural properties allow one to charac-

terize the spatio and the temporal behaviour of some

stochastic processes, being relevant for the probabilis-

tic risk asessment concerning some maintenance poli-

cies in reliability (see e.g., (Shanthikumar and Yao,

1992, and Meester and Shanthikumar, 1993). Sec-

ondly, in many optimization problems, the mono-

tonicity and the convexity assumptions play an im-

portant role. In deterministic dynamic programming

problems, for example, if the reward function g(s, x)

is strictly concave in s and x and increasing in s, the

feasible action correspondence is an increasing and

convex graph; and the transition function is contin-

uous, bounded and concave, then the value function

is continuous, bounded, increasing and concave (see

Stokey et al., 1989). In stochastic dynamic program-

ming, the problem is more difﬁcult due to the mono-

tonicity and the convexity for a transition probabil-

ity, and some conditions for the differentiability and

the concavity of the value function are stated by using

notions similar to the stochastic convexity in stochas-

tic sense in (Atakan, 2003). The stochastic convexity

also arises to solve optimization problems involving

other operational research models, for allocation or

manufacturing as in (Gallego et al., 1993) and (Kim

and Park, 1999); and other problems involving pair-

wise interchange arguments based on the interplay of

this theory with the stochastic majorization of vec-

tors, for scheduling as in (Shanthikumar, 1987), and

(Chang and Yao, 1990) (we refer to the book Marshall

and Olkin, 1979, for majorization concepts). Thirdly,

the stochastic convexity has been a bridge property

into the analysis of the variability of mixture models

with uncertain parameters. The statistical modelling

of the uncertainty constitutes one of the major goals

in experimental studies. Theoretically, a distinction

between the random uncertainty (due to natural con-

ditions) and the epistemic uncertainty (due to the lack

of knowledge of the data) is given in the literature.

This distinction is not so easy in practice. Parame-

ter uncertainty involves both issues, since when fore-

castings are performed by a model often there is a

random component in which some of the conditions

given in the inputs are not controlled and speciﬁed

(see e.g., Ang and Tang, 2007). Mixture models with

random parameters, having arbitrary probability dis-

tributions of the marginals, and an arbitrary joint dis-

tribution function to model the dependence structure

of the random vector, provide a non-parametric set-

ting to account for the uncertainty in a broad sense.

We notice that the SI − DCX property of the mix-

ture model X(θ) when the parameters are held ﬁxed

implies the increasingness and directional convexity

of the expected value of any increasing and convex

function of the parameterized random variable X(θ).

This property leads to variability comparisons of the

mixture models X(Θ) for two random vectors of pa-

rameters Θ and Θ

′

describing two scenarios, whose

mixing distributions are ordered by the increasing

directionally convex order, see (Shaked and Shan-

thikumar, 2007) for details on the directionally con-

vex orders. Recall that the variability ordering, also

known as increasing convexordering, (see Muller and

Stoyan, 2002), denoted by X(Θ) ≤

icx

X(Θ

′

) means

that E[φ(X(Θ))] ≤ E[φ(X(Θ

′

))], for every increasing

convex real-valued function φ for which the expecta-

tions exist. In particular, if the mixture model X(Θ)

fulﬁlls the SI − DCX property, then the following re-

lationship holds:

Θ ≤

idcx

′

⇒ X(Θ) ≤

icx

X(Θ

′

) (4)

where Θ ≤

idcx

′

denotes the increasing direction-

ally convex ordering, that allows one to compare

the marginal distributions in increasing convex sense,

jointly with the increasingness in positive dependence

of the joint distribution of the random vector (see

Muller and Scarsini, 2001). The decreasing direction-

ally convex order, denoted by Θ ≤

ddcx

′

, is used to

combine less valued with more volatility components.

They belong to the class of multivariate integral or-

ders that are deﬁned by the corresponding property

of the expected value of a functional of the random

vector. Other properties and results on the connec-

tion of the directionally convex orders and the vari-

ability of mixture models can be found in (Ruschen-

dorf, 2005). A ﬁrst consequence of the relationship

(4) is the construction of distributional bounds for the

mixture X(Θ) by ﬁxing some values of the parameter

′

or of the random variable X(Θ

′

). We notice that

the distribution moments are increasing convex func-

tionals of the random variables and can be bounded

by the previous stochastic comparison. Accordingly,

RECENT ADVANCES AND APPLICATIONS OF THE THEORY OF STOCHASTIC CONVEXITY. APPLICATION

TO COMPLEX BIO-INSPIRED AND EVOLUTION MODELS

247

when X ≤

icx

Y and the random variables have equal

means, then it is written X ≤

Y and it implies that

Var(X) ≤ Var(Y), where Var denotes the variance of

the random variable. Hence, the convex ordering pro-

vides an instrument to compare and to quantitatively

evaluate the magnitude and the dispersion of a perfor-

mance measure of a system, under different mixing

distributions of the parameters. Convex risk measures

of the random variable are preserved by this order-

ing. Other convex functions, as the maximal value,

for a sequence of independent and identically dis-

tributed random variables can be bounded by using

the previous relationship (4). Bounds for the mix-

ture models X(Θ) from Equation (4) can be derived

also by ﬁxing the dependence structure of the param-

eter vector Θ. Some dependence concepts allow one

to understand the relationship between the stochastic

convexity and the stochastic dependence of random

vectors of parameters by means of directionally con-

vex orders (see (Joe, 1997) for a general treatment

of the stochastic dependence). Positive quadrant de-

pendence (denoted by PQD) and positive supermodu-

lar dependence notions involve integral comparisons

with a given random vector with ﬁxed marginals and

independent components. Some examples of bivari-

ate distributions fulﬁlling the PQD property for some

values of their parameters are given by the Kibbles bi-

variate Gamma, the Moran-Dowton exponential, the

Marshall-Olkin, the F-G-M bivariate exponential and

the Lomax distributions. Nonparametric statistical

tests to check the earlier dependence notion can be

found in (Denuit and Scaillet, 2004) and (Scaillet,

2006).

The ﬁrst applications of this topic can be

found in the queueing and the stochastic pro-

cesses theories: (Shaked and Shanthikumar

1988a,1988b,1990a,1990b), (Chang et al., 1991),

(Meester and Shanthikumar, 1990,1993). Recent ap-

plications in queueing systems are given in (Chao and

Luh, 2004), (Miyoshi and Rolski, 2004), (Shioda and

Ishi, 2004), (Rolski, 2005), and (Ortega, 2011). The

last article considers routing in queueing networks,

to develop some applications in communication

networks, that we will discuss later. The issue of

stochastic convexity arises recently into the study

of stochastic processes, especially point processes.

At the best of our knowledge, recent results can be

seen in (Miyoshi, 2004), and (Fernandez-Ponce, et

al, 2008a,b). They are involved in the multivariate

stochastic comparisons of some stochastic processes

(see Kulik and Szekli, 2005). The following mono-

graphs include an exhaustive development of the

theory, and/or some detailed applications for selected

operational research problems. Chang et al., 1994

provided applications to the analysis of a random

yield model, a joint setup problem, a manufacturing

process with trial runs, a production network with

constant work-in-process, and a scheduling model

in tandem production lines and parallel assembly

systems. The convexity and the concavity in the

stochastic sense of other measures associated with

production-inventory systems are studied in (Yao and

Zheng, 2002), that includes applications of this the-

ory for quality control with warranty, process con-

trol with inspection of machines for batch production,

and inventory control with substitution, among oth-

ers. Yao (1994,1996) provided surveys with manu-

facturing and production applications. Recent devel-

opments to operational research models can be found

in (Ahn et al., 2005), (Ott and Shanthikumar, 2006),

and (Ortega and Alonso, 2011). Stochastic directional

convexity of random sums of random variables has

been dealt with by (Fernandez-Ponce, et al., 2008a),

(Escudero et al., 2010), and (Ortega and Escudero,

2010), among others. Observe that general results in

the direction of Equation (4), for random sums, can

be found in the last articles. This is a classic problem

for sums and univariateparameters, as can be check in

Makowski and Philips, 1992. Applications have been

depicted for the sums and the random sums in dif-

ferent ﬁelds, as for instance, the total claim amounts

under the Sparre-Andersen risk model in actuarial sci-

ence, and the times of systems under shock models

in reliability, etc. The stochastic convexity of some

special linear combinations deﬁned by indicator func-

tions is applied in (Escudero and Ortega, 2008) for the

analysis of reinsurance models for large claims based

on truncation with random retention levels. Other lin-

ear combinations of random variables are studied in

(Ortega, 2010).

3 APPLICATION BASED ON

BIO-INSPIRED MODELS

Current research trends are exploiting the similar fea-

tures between biological systems and complex com-

munication systems to develop techniques in these

ﬁelds based on biological schemes. Nature has been

shown to be able to adapt rapidly to environmental

changes, and assemble simple structures into complex

operations. Next, we discuss some models that are

biologically inspired, where the stochastic convexity

properties play an important role.

Multiplicative processes arise in biology and ecol-

ogy to describe the growth of organisms and popula-

tions of species, as protein families, that evolve in a

multiplicative manner, see e.g., (Reed and Hughes,

ECTA 2011 - International Conference on Evolutionary Computation Theory and Applications

248

2002) from a probabilistic point of view. The main

idea is that the random growth of an organism is ex-

pressed as a percentage of its current actual size. As-

suming that we start with an organism of size X

, then

at each step j, the organism may grow or shrink, ac-

cording to a random variable Y

, such that the size of

an organism at step j is given by X

(θ) = X

j−1

(θ)Y

In ﬁnancial mathematics, the Black-Scholes option

pricing model, which is a speciﬁc application of Itos

lemma, describes the price of a security that moves

in discrete time steps according to the earlier equality,

where Y

is lognormally distributed usually (see e.g.,

Hull, 2002). Some bio-inspired models and networks

have been used to model evolution in computing and

communication systems. Some of these systems are

characterized by strongly dynamic environments and

heterogeneous nodes, among others. Huberman and

Adamic have applied the multiplicative processes to

describe the growth of sites on the web, as well as

the growth of user trafﬁc on web sites (see Huberman

and Adamic, 1999,2000). Mitzenmacher,2004 used a

similar model to explain the behaviour of both tails of

the distribution of the size of computer ﬁles. There is

a vast recent literature on the use of the random prod-

ucts in the study of computer systems. Other applica-

tions of multiplicative processes can be found in hy-

drology, geology, and chemistry (Crow and Shimizu,

1988 and Escudero et al., 2010).

Ortega, 2010 has provided the conditions under

which the multiplicative processes are SI−DCX. Ap-

plications are given there to the analysis of the size of

trafﬁc on web sites. The stochastic convexity of the

size of the trafﬁc X

(θ) with respect to its environ-

mental parameter means that the expected value of

any convex increasing function of the size increases

and behaves componentwise convex and supermod-

ular with respect to its parameter. The main sta-

tistical consequences of this property are described

above from the relationship (4), especially the con-

struction of distributional bounds. For communica-

tion systems, bounds of these measures provide valu-

able insight into the primary factors affecting the per-

formance of systems, as those inﬂuencing the sys-

tem bottleneck; and they become useful to elimi-

nate non appropriate options at an early stage of the

analysis. Similar conclusions may be reached for

other bio-inspired adaptive models. Speciﬁcally, the

tree networks constitute one of the most useful net-

work topologies, with applications in computer sci-

ence, taxonomy, location, molecular biology, evolu-

tion, and ecology. Ortega, 2011 considers rooted tree

networks and rooted butterﬂy networks that are mod-

elled by queueing systems, where the items enter at

the root and proceed away until they reach their desti-

nation and exit the system, the arcs of the network are

represented by FIFO servers, and the service times

have arbitrary probability distributions and they cor-

respond to the times needed for an item to cross edges.

The SI − DCX of the exit time of the item k from the

queue j in the network is the key property to develop

variability comparisons of departure times and delay

times of the system with a discrete probability distri-

bution for the routing policy and with correlated inter-

arrival times and service times. Bounds for the previ-

ous performance measures are derived as above.

4 CONCLUSIONS

To ﬁnish, we emphasize on different advantages of

the concepts of stochastic convexity. Many mea-

sures of the performance of a system are modelled as

mixtures deﬁned by composition of arithmetic func-

tionals of parameterized non-negative random vari-

ables, and these functionals fulﬁl monotonic direc-

tional convexity properties. Nevertheless, by appro-

priate assumptions for the random components in the

model for ﬁxed parameters; and by using Theorem 3.2

in (Meester and Shanthikumar, 1999), we can state

that the parameterized model is SI -DCX. We recall

that the stochastic directional convexity allows one to

analyze the mixtures without requiring an analytical

expression of the distribution function of the random

variable, and to obtain bounds from the relationship

(4). This makes this theory to become an attractive

alternative to evaluate or to explore the probabilistic

behaviour of the performance measure, when there is

lack of information about the marginal distribution of

the parameters and the correlations among them (pro-

vided that modelling the dependence structure from

empirical data is a rather complex task; and the same

happens for selection of prior distributions to account

for covariates). Also, the deﬁnitions of the stochas-

tic convexity properties are set for parameters taking

on values over general spaces in (Meester and Shan-

thikumar, 1999), and this allows versatile scenarios

of parameters. Some related optimization problems,

e.g., to ﬁnd the maximal value of a performance mea-

sure, can be addressed directly from stochastic con-

vexity properties, by using notions of dependence as

the comonotonicity (see Escudero et al., 2010).

ACKNOWLEDGEMENTS

Prof. Laureano Escudero, and the research project

MTM2009-13433 from Spanish Ministry of Science

and Innovation are gratefully acknowledged.

RECENT ADVANCES AND APPLICATIONS OF THE THEORY OF STOCHASTIC CONVEXITY. APPLICATION

TO COMPLEX BIO-INSPIRED AND EVOLUTION MODELS

249

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