Nonlinearly perturbed stochastic processes
This paper is a survey of results presented in the recent book [25]). This book is devoted to studies of quasi-stationary phenomena in nonlinearly perturbed stochastic systems. New methods of asymptotic analysis for nonlinearly perturbed stochastic processes based on new types of asymptotic expansio...
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irk-123456789-45742009-12-08T12:00:36Z Nonlinearly perturbed stochastic processes Silvestrov, D. This paper is a survey of results presented in the recent book [25]). This book is devoted to studies of quasi-stationary phenomena in nonlinearly perturbed stochastic systems. New methods of asymptotic analysis for nonlinearly perturbed stochastic processes based on new types of asymptotic expansions for perturbed renewal equation and recurrence algorithms for construction of asymptotic expansions for Markov type processes with absorption are presented. Asymptotic expansions are given in mixed ergodic (for processes) and large deviation theorems (for absorption times) for nonlinearly perturbed regenerative processes, semi-Markov processes, and Markov chains. Applications to analysis of quasi-stationary phenomena in nonlinearly perturbed queueing systems, population dynamics and epidemic models, and risk processes are presented. The book also contains an extended bibliography of works in the area. 2008 Article Nonlinearly perturbed stochastic processes / D. Silvestrov // Theory of Stochastic Processes. — 2008. — Т. 14 (30), № 3-4. — С. 129-164. — Бібліогр.: 77 назв.— англ. 0321-3900 http://dspace.nbuv.gov.ua/handle/123456789/4574 en Інститут математики НАН України |
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This paper is a survey of results presented in the recent book [25]). This book is devoted to studies of quasi-stationary phenomena in nonlinearly perturbed stochastic systems. New methods of asymptotic analysis for nonlinearly perturbed stochastic processes based on new types of asymptotic expansions for perturbed renewal equation and recurrence algorithms for construction of asymptotic expansions for Markov type processes with absorption are presented. Asymptotic expansions are given in mixed ergodic (for processes) and large deviation theorems (for absorption times) for nonlinearly perturbed regenerative processes, semi-Markov processes, and Markov chains. Applications to analysis of quasi-stationary phenomena in nonlinearly perturbed queueing systems, population dynamics and epidemic models, and risk processes are presented. The book also contains an extended bibliography of works in the area. |
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Article |
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Silvestrov, D. |
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Silvestrov, D. Nonlinearly perturbed stochastic processes |
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Silvestrov, D. |
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Silvestrov, D. |
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Nonlinearly perturbed stochastic processes |
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Nonlinearly perturbed stochastic processes |
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Nonlinearly perturbed stochastic processes |
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Nonlinearly perturbed stochastic processes |
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Nonlinearly perturbed stochastic processes |
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nonlinearly perturbed stochastic processes |
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Інститут математики НАН України |
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2008 |
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Nonlinearly perturbed stochastic processes / D. Silvestrov // Theory of Stochastic Processes. — 2008. — Т. 14 (30), № 3-4. — С. 129-164. — Бібліогр.: 77 назв.— англ. |
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Theory of Stochastic Processes
Vol. 14 (30), no. 3-4, 2008, pp.129-164
DMITRII SILVESTROV
NONLINEARLY PERTURBED
STOCHASTIC PROCESSES
This paper is a survey of results presented in the recent book [25]1) .
This book is devoted to studies of quasi-stationary phenomena in
nonlinearly perturbed stochastic systems. New methods of asymp-
totic analysis for nonlinearly perturbed stochastic processes based
on new types of asymptotic expansions for perturbed renewal equa-
tion and recurrence algorithms for construction of asymptotic ex-
pansions for Markov type processes with absorption are presented.
Asymptotic expansions are given in mixed ergodic (for processes) and
large deviation theorems (for absorption times) for nonlinearly per-
turbed regenerative processes, semi-Markov processes, and Markov
chains. Applications to analysis of quasi-stationary phenomena in
nonlinearly perturbed queueing systems, population dynamics and
epidemic models, and risk processes are presented. The book also
contains an extended bibliography of works in the area.
1. Introduction
The book mentioned above presents new methods of asymptotic analy-
sis of nonlinearly perturbed stochastic processes and systems with random
lifetimes.
Usually the behaviour of a stochastic system can be described in terms
of some Markov type stochastic process η(ε)(t) and its lifetime defined to be
Ivited lecture.
2000 Mathematics Subject Classifications: Primary: 60F05, 60F10; 60K05, 60K15;
Secondary: 60J22, 60J27, 60K20, 60K25, 65C40.
Key words: nonlinear perturbation, quasi-stationary phenomenon, pseudo-stationary
phenomenon, stochastic system, renewal equation, asymptotic expansion, ergodic the-
orem, limit theorem, large deviation, regenerative process, regenerative stopping time,
semi-Markov process, Markov chain, absorption time, queueing system, population dy-
namics, epidemic model, lifetime, risk process, ruin probability, Cramér-Lundberg ap-
proximation, diffusion approximation.
1)Gyllenberg, M., Silvestrov, D.S. (2008). Quasi-Stationary Phenomena in Nonlin-
early Perturbed Stochastic Systems, De Gruyter Expositions in Mathematics 44, Walter
de Gruyter, Berlin, XII + 579 pp.
129
130 DMITRII SILVESTROV
the time μ(ε) at which the process η(ε)(t) hits a special absorption subset
of the phase space of this process for the first time. The object of interest
is the joint distribution of the process η(ε)(t) subject to a condition of non-
absorption of the process up to a moment t, i.e., the probabilities P{η(ε)(t) ∈
A, μ(ε) > t}, and the asymptotic behaviour of these probabilities for t→ ∞.
A typical situation is when the process η(ε)(t) and the absorption time
μ(ε) depend on a small parameter ε ≥ 0 in the sense that some of their local
“transition” characteristics depend on the parameter ε. The parameter ε is
involved in the model in such a way that the corresponding local character-
istics are continuous at the point ε = 0, if regarded as functions of ε. These
continuity conditions permit to consider the process η(ε)(t), for ε > 0, as a
perturbed version of the process η(0)(t).
In models with perturbations, it is natural to study the asymptotic be-
haviour of the probabilities P{η(ε)(t) ∈ A, μ(ε) > t} when the time t → ∞
and the perturbation parameter ε → 0 simultaneously. Without loss of
generality it can be assumed that the time t = t(ε) is a function of the
parameter ε such that t(ε) → 0 as ε→ 0.
The corresponding studies obviously relate to two classical asymptotic
problems.
The first problem is connected with limit theorems for lifetimes of sto-
chastic systems, i.e., propositions about convergence of probabilities
P{μ(ε) > t(ε)} to some non-zero limits as ε → 0, as well as with large devi-
ation theorems for lifetimes, i.e., propositions about asymptotic behaviour
of these probabilities in the case where they tend to zero as ε→ 0.
The second problem is connected with a mathematical description of
quasi-stationary phenomena for the corresponding processes. These phe-
nomena describe the behaviour of stochastic systems with random life-
times. The core of the quasi-stationary phenomenon is that one can ob-
serve something that resembles a stationary behaviour of the system before
the lifetime goes to the end, i.e., stabilisation of conditional probabilities
P{η(ε)(t(ε)) ∈ A/μ(ε) > t(ε)} as ε → 0.
The principal novelty of results presented in book [25] is that the models
with nonlinear perturbations are studied. Local transition characteristics
that were mentioned above are usually some scalar or vector moment func-
tionals p(ε) of local transition probabilities for the corresponding processes.
By a nonlinear perturbation we mean that these characteristics are non-
linear functions of the perturbation parameter ε and that the assumptions
made imply that the characteristics can be expanded in an asymptotic power
series with respect to ε up to and including some order k,
p(ε) = p(0) + p[1]ε+ · · · + p[k]εk + o(εk). (1)
The case k = 1 corresponds to models with usual linear perturbations
while the cases k > 1 correspond to models with nonlinear perturbations.
NONLINEARLY PERTURBED STOCHASTIC PROCESSES 131
It turns out that the relation between the velocities with which ε tends
to zero and the time t = t(ε) tends to infinity has a delicate influence on the
quasi-stationary asymptotics. The balance between the rate of the pertur-
bation and the rate of growth of time can be characterized by the following
condition which is assumed to hold for some 1 ≤ r ≤ k:
εrt(ε) → λr <∞ as ε→ 0. (2)
The main results are represented by so-called mixed ergodic and limit/
large deviation theorems given in a form of exponential expansions for the
probabilities P{η(ε)(t(ε)) ∈ A, μ(ε) > t(ε)}. Under the mentioned above
assumptions on the perturbations and some additional natural conditions
of Cramér type, the following exponential asymptotic relations are obtained:
P{η(ε)(t(ε)) ∈ A, μ(ε) > t(ε)}
exp{−(ρ(0) + a1ε+ · · ·+ ar−1εr−1)t(ε)} → π(0)(A)e−λrar as ε → 0. (3)
With these relations, explicit algorithms for calculating the coefficients
ρ(0) and a1, . . . , ak, as functions of the coefficients in the expansions of the
local transition characteristics that appear in the initial nonlinear pertur-
bation conditions of type (1), are given. It is a non-trivial problem due to
the nonlinear character of the perturbations involved.
The asymptotic behaviour of P{η(ε)(t) = j, μ(ε) > t} is very much dif-
ferent in the two alternative cases: (a) ρ(0) = 0 and (b) ρ(0) > 0. In the
first case, the absorption times μ(ε) are stochastically unbounded random
variables, that is, they tend to ∞ in probability as ε → 0. In the second
case, the absorption times μ
(ε)
0 are stochastically bounded as ε → 0.
In the literature, the asymptotics related to the second model is known
as quasi-stationary asymptotics. To distinguish between the asymptotics
(3) in the first and the second cases, the term pseudo-stationary was coined
for the first one.
Another class of asymptotic expansions systematically studied in the
book concerns the so-called quasi-stationary distributions. Under some
natural moment, communication and aperiodicity conditions imposed on
the local transition characteristics there exists a so-called quasi-stationary
distribution for the process η(ε)(t), which is given by the formula,
π(ε)(A) = lim
t→∞
P{η(ε)(t) ∈ A / μ(ε) > t}. (4)
Asymptotics (3) let one also obtain asymptotic expansions for the quasi-
stationary distributions of the nonlinearly perturbed processes η(ε)(t),
π(ε)(A) = π(0)(A) + g1(A)ε+ · · · + gk(A)εk + o(εk). (5)
As in the case of asymptotics (3), an explicit recurrence algorithm for
calculating the coefficients g1(A), . . . , gk(A), as functions of the coefficients
132 DMITRII SILVESTROV
in the expansions for local transition characteristics of the processes η(ε)(t)
in the corresponding initial perturbation conditions, is given.
The classes of processes for which this program is realised include non-
linearly perturbed regenerative processes, semi-Markov processes, and con-
tinuous time Markov chains with absorption.
The approach is based on advanced techniques, developed in the book,
of nonlinearly perturbed renewal equations.
Applications to the analysis of quasi-stationary phenomena in models
of nonlinearly perturbed stochastic systems considered in the book pertain
to models of highly reliable queueing systems, M/G queueing systems with
quick service, stochastic systems of birth-death type, including epidemic
and population dynamics models, metapopulation dynamic models, and
perturbed risk processes.
As was mentioned above, quasi-stationary phenomena were a subject
of intensive studies during several decades. An extensive survey of the
literature and comments can be found in the bibliographical remarks given
in book [25].
The results related to the asymptotics given in (3) and (5) and known
in the literature mainly cover the case k = 1, which corresponds to the
model of linearly perturbed processes. Some known results also relate to
the case where the perturbation condition (1) has the special form of relation
(c) p(ε) = p(0) + p[1]ε. In context of nonlinearly perturbed models, this is
a particular case of nonlinear perturbation condition (1) which holds for
every k ≥ 1 with all higher order terms p[2], p[3], . . . ≡ 0. It should be
noted that the asymptotic expansions given in (3) and (5) obviously cover
this case but not vise versa. Indeed, the asymptotic expansions obtained
for the models with perturbation condition of the form (c) do not give
any information about contribution of the terms p[2], p[3], . . . in nonlinear
asymptotic expansions given in (3) and (5) for the models where these higher
order terms take non-zero values.
The book [25] contains an extended introduction, where the main prob-
lems, methods, and algorithms that constitute the content of the book are
presented in informal form. In Chapters 1 and 2, results which deal with a
generalisation of the classical renewal theorem to a model of the perturbed
renewal equation are presented. These results are interesting by their own
and, as we think, can find various applications beyond the areas mentioned
in the book. In Chapters 3, 4, and 5 asymptotics of the types (3) and (5)
for nonlinearly perturbed regenerative processes, semi-Markov processes,
and continuous time Markov chains with absorption are studied. Chap-
ters 6 and 7 are devoted to applications of the theoretical results to studies
of quasi-stationary phenomena for various nonlinearly perturbed models of
stochastic systems. In Chapter 6, quasi-stationary phenomena are stud-
ied for highly reliable queueing systems, M/G queueing systems with quick
NONLINEARLY PERTURBED STOCHASTIC PROCESSES 133
service, stochastic systems of birth-death type, including epidemic and pop-
ulation dynamics models, and metapopulation dynamic models; Chapter 7
deals with perturbed risk processes. The last Chapter 8 contains three
supplements. The first one gives some basic operation formulas for scalar
and matrix asymptotic expansions. In the second supplement some new
prospective directions for future research in the are discussed. In the last
supplement, bibliographical remarks to the bibliography that includes more
than 1000 references are given.
2. Nonlinearly perturbed renewal equation
2.1. Renewal theorem for perturbed renewal equation. Let us
consider the family of renewal equations,
x(ε)(t) = q(ε)(t) +
∫ t
0
x(ε)(t− s)F (ε)(ds), t ≥ 0, (6)
where, for every ε ≥ 0, we have the following: (a) q(ε)(t) is a real-valued func-
tion on [0,∞) that is Borel measurable and locally bounded, i.e., bounded
on every finite interval, and (b) F (ε)(s) is a distribution function on [0,∞)
which is not concentrated at 0 but can be improper, i.e. F (ε)(∞) ≤ 1.
As well known, there exists the unique Borel measurable and bounded
on every finite interval solution x(ε)(t) of equation (6).
In the model of perturbed renewal, the forcing function q(ε)(t) and distri-
bution F (ε)(s) depend on some perturbation parameter ε ≥ 0 and converge
in some sense to q(0)(t) and F (0)(s) as ε→ 0.
The fundamental fact of the renewal theory connected with this equation
is the renewal theorem given in its final form by Feller (1966). This theorem
describes the asymptotic behavior of solution in the form of asymptotic
relation x(0)(t) → x(0)(∞) as t→ ∞ for non-perturbed renewal equation.
The renewal theorem is a very powerful tool for proving ergodic theo-
rems for regenerative stochastic processes. This class of processes is very
broad. It includes Markov processes with discrete phase space. Moreover,
Markov processes with a general phase space can be included, under some
minor conditions, in a model of regenerative processes with the help of the
procedure of artificial regeneration.
Applying the renewal theorem to ergodic theorems for regenerative type
processes is based on the well known fact that the distribution of a regen-
erative process at a moment t satisfies a renewal equation. This makes
it possible to apply the renewal theorem and to describe the asymptotic
behaviour of the distribution of the regenerative process as t→ ∞.
Theorems that generalise the classical renewal theorem to a model of the
perturbed renewal equation was proved in papers Silvestrov (1976, 1978,
1979). These results are presented in Chapter 1 of book [25].
134 DMITRII SILVESTROV
As usual the symbol F (ε)(·) ⇒ F (0)(·) as ε→ 0 means weak convergence
of the distribution functions that is, the pointwise convergence in each point
of continuity of the limiting distribution function.
Further, the following notations are used,
f (ε) = 1 − F (ε)(∞), m(ε)
r =
∫ ∞
0
srF (ε)(ds), r ≥ 1.
We assume that the functions q(ε)(t) and the distributions F (ε)(s) sat-
isfy the following continuity conditions at the point ε = 0, if regarded as
functions of ε:
D1: F
(ε)(·) ⇒ F (0)(·) as ε → 0, where F (0)(s) is a proper non-arithmetic
distribution function;
M1: m
(ε)
1 → m
(0)
1 <∞ as ε→ 0;
and
F1: (a) limu→0 lim0≤ε→0 sup|v|≤u |q(ε)(t + v) − q(0)(t)| = 0 almost every-
where with respect to the Lebesgue measure on [0,∞);
(b) lim0≤ε→0 sup0≤t≤T |q(ε)(t)| <∞ for every T ≥ 0;
(c) limT→∞ lim0≤ε→0h
∑
r≥T/h suprh≤t≤(r+1)h |q(ε)(t)| = 0 for some
h > 0.
It is easy to show that, under conditions A and M1, the following rela-
tion holds,
f (ε) → f (0) = 0 as ε → 0. (7)
Let also assume the following condition that balances the rate at which
time t(ε) approaches infinity, and the convergence rate of the defect f (ε) to
zero as ε→ 0:
B: 0 ≤ t(ε) → ∞ and f (ε) → 0 as ε → 0 in such a way that f (ε)t(ε) → λ,
where 0 ≤ λ ≤ ∞.
The starting point for the research studies presented in book [25] is the
following theorem (Silvestrov, 1976, 1978, 1979).
Theorem 1. Let conditions D1, M1, F1, and B hold. Then,
xε(tε) → e−λ/m
(0)
1
∫∞
0
q(0)(s)ds
m
(0)
1
as ε → 0. (8)
Remark 1. It is worth to note that this theorem reduces to the classical
renewal theorem in the case of non-perturbed renewal equation, i.e., where
the forcing functions q(ε)(t) ≡ q(0)(t) and distribution functions F (ε)(s) ≡
NONLINEARLY PERTURBED STOCHASTIC PROCESSES 135
F (0)(s) do not depend on ε. In particular, condition D1 reduces to the as-
sumption that F (0)(s) is a proper non-arithmetic distribution function; M1
to the assumption that the expectation m
(0)
1 is finite; and F1 to the assump-
tion that the function q(0)(t) is directly Riemann integrable on [0,∞).
In this case, the defect f (ε) ≡ 0 and the balancing condition B holds for
any t(ε) → ∞ as ε → 0 with the parameter λ = 0.
Note that condition D1 does not require and does not provide that the
pre-limit (ε > 0) distribution functions F (ε)(s) are non-arithmetic.
Also, condition F1 does not provide that the pre-limit (ε > 0) forc-
ing functions q(ε)(t) are directly Riemann integrable on [0,∞). However,
this condition does imply that the limit forcing functions q(0)(t) has this
property.
In general case, the balancing condition B restrict the rate of growth
for time t(ε). This restriction becomes unnecessary if an additional Cramér
type condition is imposed on the distributions Fε(s).
In this case, one can also weaken condition D1 and accept also the
possibility for the limit distribution be improper:
D2: (a) F (ε)(·) ⇒ F (0)(·) as ε → 0, where F (0)(t) is a non-arithmetic
distribution function (possibly improper);
(b) f (ε) → f (0) ∈ [0, 1) as ε→ 0.
The Cramér type condition mentioned above takes the following form:
C1: There exists δ > 0 such that:
(a) lim0≤ε→0
∫∞
0
eδsF (ε)(ds) <∞;
(b)
∫∞
0
eδsF (0)(ds) > 1.
Let us introduce the moment generation function,
φ(ε)(ρ) =
∫ ∞
0
eρsF (ε)(ds), ρ ≥ 0.
Consider the following characteristic equation,
φ(ε)(ρ) = 1. (9)
Under condition D2 and C1, there exists ε1 > 0 such that φ(ε)(δ) ∈
(1,∞), and, therefore, equation (9) has a unique non-negative root ρ(ε) and
ρ(ε) ≤ δ, for every ε ≤ ε1. Also,
ρ(ε) → ρ(0) as ε→ 0. (10)
Note also that (a) ρ(0) = 0 if and only if f (0) = 0 and (b) ρ(0) > 0 if and
only if f (0) > 0.
In this case, condition F1 takes the following modified form:
136 DMITRII SILVESTROV
F2: (a) limu→0 lim0≤ε→0 sup|v|≤u |q(ε)(t + v) − q(0)(t)| = 0 almost every-
where with respect to the Lebesgue measure on [0,∞);
(b) lim0≤ε→0 sup0≤t≤T |q(ε)(t)| <∞ for every T ≥ 0;
(c) limT→∞ lim0≤ε→0h
∑
r≥T/h suprh≤t≤(r+1)h e
γt|q(ε)(t)| = 0 for some
h > 0 and γ > ρ(0).
Let us denote,
x̃(ε)(∞) =
∫∞
0
eρ
(ε)sq(ε)(s)m(ds)∫∞
0
seρ(ε)sF (ε)(ds)
,
where m(ds) is the Lebesgue measure on a real line.
Conditions D2, C1, and F2 imply, due to relation ρ(ε) → ρ(0) as ε → 0,
that there exists 0 < ε2 ≤ ε1 such that ρ(ε) < γ and
∫∞
0
eρ
(ε)sq(ε)(s)m(ds) <
∞ for ε ≤ ε2. Thus, the functional x̃(ε)(∞) is well defined for ε ≤ ε2.
The following theorem was also proved in Silvestrov (1976, 1978, 1979).
Theorem 2. Let conditions D2, C1, and F2 hold. Then,
x(ε)(t(ε))
e−ρ(ε)t(ε)
→ x̃(0)(∞) as ε → 0. (11)
The asymptotic relation (11) given in Theorem 2 should be compared
with the asymptotic relation (8) given in Theorem 1, in the case where
ρ(0) = 0.
Indeed, relation (8) can be re-written in the form given in (11), with co-
efficients ρ(ε) = f (ε)/m
(ε)
1 . The Cramér type condition C1 makes it possible
to use in (11) an alternative coefficients ρ(ε) defined as the solution of the
characteristic equation (9). The latter coefficients provide better fitting of
the corresponding exponential approximation for solution of renewal equa-
tion. That is why the asymptotic relation (11) does not restrict the rate
of growth for time t(ε) while the asymptotic relation (8) does impose such
restriction.
Remark 2. It is worth to note that this theorem reduces to the variant of
renewal theorem for improper renewal equation in the case of non-perturbed
renewal equation, also given by Feller (1966). Condition D2 reduces to
the assumption that F (0)(s) is a non-arithmetic distribution function with
defect f (0) ∈ [0, 1); C1 to the assumption that the exponential moment
φ(0)(δ) ∈ (1,∞); and F2 to the assumption that the function eγtq(0)(t) is
directly Riemann integrable on [0,∞) for some γ > ρ(0).
The results formulated in Theorems 1 and 2 created the base for fur-
ther research studies in the area. For example, Shurenkov (1980a, 1980b)
generalised the results of these theorems to the case of matrix renewal equa-
tion using possibility of imbedding the matrix model to the scalar model
considered in Theorems 1 and 2.
NONLINEARLY PERTURBED STOCHASTIC PROCESSES 137
2.2. Exponential expansions in renewal theorem for nonlinearly
perturbed renewal equation. A new improvement was achieved in the
paper Silvestrov (1995) and then in the papers Gyllenberg and Silvestrov
(1998, 1999a, 2000a). Under natural additional perturbation conditions,
which assume that the defect f (ε) and the corresponding moments of the
distribution F (ε)(s) can be expanded in power series with respect to ε up to
and including an order k, explicit expansions for the corresponding charac-
teristic roots were given, and the corresponding exponential expansions were
obtained for solutions of nonlinearly perturbed renewal equations. In Silve-
strov (1995), the case with asymptotically proper distributions F (ε)(s) was
considered, while, in Gyllenberg and Silvestrov (1998, 1999a, 2000a), the
case with asymptotically improper distributions F (ε)(s) was investigated.
These results are presented in Chapter 2 of book [25].
Let us introduce the mixed power-exponential moment generating func-
tions,
φ(ε)(ρ, r) =
∫ ∞
0
sreρsF (ε)(ds), ρ ≥ 0, r = 0, 1, . . . .
Note that, by the definition, φ(ε)(ρ, 0) = φ(ε)(ρ). Under condition D2 and
C1, for any 0 < δ′ < δ, there exists 0 < ε3 < ε2 such that φ(ε)(δ′, r) <∞ for
r = 0, 1, . . . and ε ≤ ε3. Also, φ(ε)(ρ, r) → φ(0)(ρ, r) as ε→ 0 for r = 0, 1, . . .
and ρ ≤ δ′. Let δ′ is chosen such that φ(0)(δ′) ∈ (1,∞). In this case, the
characteristic root ρ(0) < δ′, and, also, there exists 0 < ε4 < ε3 such that
the characteristic roots ρ(ε) < δ′ for ε ≤ ε4.
The basic role plays the following nonlinear perturbation condition:
P
(k)
1 : φ(ε)(ρ(0), r) = φ(0)(ρ(0), r) + b1,rε + · · · + bk−r,rεk−r + o(εk−r) for r =
0, . . . , k, where |bn,r| <∞, n = 1, . . . , k − r, r = 0, . . . , k.
It is convenient to define b0,r = φ(0)(ρ(0), r), r = 0, 1, . . .. From the
definition of ρ(0) it is clear that b0,0 = φ(0)(ρ(0), 0) = 1.
It should be noted that, in the case f (0) = 0, where characteristic root
ρ(0) = 0, the perturbation condition P
(k)
1 involves usual power moments
of distributions F (ε)(s). While, in the case f (0) > 0, where characteristic
root ρ(0) > 0, the perturbation condition involve mixed power-exponential
moments of distributions F (ε)(s).
Let also formulate the following condition that balances the rate at which
time t(ε) approaches infinity, and the convergence rate of perturbation in
different asymptotic zones, for 1 ≤ r ≤ k:
B(r): 0 ≤ t(ε) → ∞ in such a way that εrt(ε) → λr, where 0 ≤ λr <∞.
The following theorem given in Silvestrov (1995) and Gyllenberg and
Silvestrov (1998, 1999a, 2000a).
138 DMITRII SILVESTROV
Theorem 3. Let conditions D2, C1, and P
(k)
1 hold. Then,
(i) The root ρ(ε) of equation (9) has the asymptotic expansion
ρ(ε) = ρ(0) + a1ε+ · · ·+ akε
k + o(εk), (12)
where the coefficients an are given by the recurrence formulas a1 =
−b1,0/b0,1 and, in general, for n = 1, . . . , k,
an = − b−1
0,1(bn,0 +
n−1∑
q=1
bn−q,1aq
+
∑
2≤m≤n
n∑
q=m
bn−q,m ·
∑
n1,...,nq−1∈Dm,q
q−1∏
p=1
anp
p /np!), (13)
where Dm,q, for every 2 ≤ m ≤ q < ∞, is the set of all nonnegative,
integer solutions of the system
n1 + · · ·+ nq−1 = m, n1 + · · · + (q − 1)nq−1 = q. (14)
(ii) If bl,0 = 0, l = 1, . . . , r, for some 1 ≤ r ≤ k, then a1, . . . , ar = 0.
If bl,0 = 0, l = 1, . . . , r − 1 but br,0 < 0, for some 1 ≤ r ≤ k, then
a1, . . . , ar−1 = 0 but ar > 0.
(iii) If, additionally, conditions B(r), for some 1 ≤ r ≤ k, and F2 hold,
then the following asymptotic relation holds:
x(ε)(t(ε))
exp{−(ρ(0) + a1ε+ · · ·+ ar−1εr−1)t(ε)}
→ e−λrar x̃(0)(∞) as ε→ 0. (15)
The asymptotic relation (15) given in Theorem 3 should be compared
with the asymptotic relation (11) given in Theorem 2.
The asymptotic relation (11) looks nicely but has actually a serious
drawback. Indeed, the exponential normalisation with the coefficient ρ(ε) is
not so effective because of this coefficient is given us only as the root of the
nonlinear equation (9), for every ε ≥ 0.
Relation (15) essentially improve the asymptotic relation (11) replacing
this simple convergence relation by the corresponding asymptotic expansion.
The exponential normalisation with the coefficient ρ(0) +a1ε+ · · ·+ar−1ε
r−1
involves the root ρ(0). To find it one should solve only one nonlinear equation
(9), for the case ε = 0. As far as the coefficients a1, . . . , ar−1 are concerned,
they are given in the explicit algebraic recurrence form.
NONLINEARLY PERTURBED STOCHASTIC PROCESSES 139
Moreover, the root ρ(0) = 0 in the most interesting case, where f (0) = 0,
i.e., the limit renewal equation is proper. Here, the non-linear step con-
nected with finding of the root of equation (9) can be omitted.
If there exist 0 < ε′ ≤ e5 such that the conditions listed in Remark 2
holds for the distribution function F (ε)(s) and the forcing function q(ε)(t) for
every ε ≤ ε′, then according Theorem 2, the following asymptotic relation
holds for every ε ≤ ε′,
x(ε)(t)
e−ρ(ε)t
→ x̃(ε)(∞) as t→ ∞. (16)
Let us now define for the mixed power-exponential moment functionals
for the forcing functions,
ω(ε)(ρ, r) =
∫ ∞
0
sreρsq(ε)(s)m(ds), ρ ≥ 0, r = 0, 1, . . . .
Under conditions C1 and F2, for any 0 < γ′ < γ, there exists 0 < ε6 < ε5
such that ω̄(ε)(γ′, r) =
∫∞
0
sreγ
′s|q(ε)(s)|m(ds) < ∞ for r = 0, 1, . . . and
ε ≤ ε6. Also, ω(ε)(ρ, r) → ω(0)(ρ, r) as ε → 0 for r = 0, 1, . . . and ρ ≤ γ′.
Let γ′ is chosen such that ρ(0) < γ′. In this case, there exists 0 < ε7 < ε6
such that the characteristic roots ρ(ε) < γ′ for ε ≤ ε7.
Note that the renewal limit x̃(ε)(∞) is well defined for ε ≤ ε7 even
without the assumptions made above in order to provide asymptotic relation
(16) and, moreover,
x̃(ε)(∞) =
ω(ε)(ρ(ε), 0)
φ(ε)(ρ(ε), 1)
. (17)
Let us now formulate a perturbation condition for mixed power-exponen-
tial moment functionals for the forcing functions:
P
(k)
2 : ω(ε)(ρ(0), r) = ω(0)(ρ(0), r) + c1,rε + · · · + ck−r,rεk−r + o(εk−r) for r =
0, . . . , k, where |cn,r| <∞, n = 1, . . . , k − r, r = 0, . . . , k.
It is convenient to set c0,r = ω(0)(ρ(0), r), r = 0, 1, . . ..
The following theorem supplements Theorem 3.
Theorem 4. Let conditions D2, C1, F2, P
(k+1)
1 , and P
(k)
2 hold. Then the
functional x̃(ε)(∞) has the following asymptotic expansions:
x̃(ε)(∞) =
ω(0)(ρ(0), 0) + f ′
1ε+ · · · + f ′
kε
k + o(εk)
φ(0)(ρ(0), 1) + f ′′
1 ε+ · · ·+ f ′′
k ε
k + o(εk)
= x̃(0)(∞) + f1ε+ · · · + fkε
k + o(εk), (18)
140 DMITRII SILVESTROV
where the coefficients f ′
n, f
′′
n are given by the formulas f ′
0 = ω(0)(ρ(0), 0) =
c0,0, f
′
1 = c1,0 + c0,1a1, f
′′
0 = φ(0)(ρ(0), 1) = b0,1, f
′′
1 = b1,1 + b0,2a1, and in
general for n = 0, . . . , k,
f ′
n = cn,0 +
n∑
q=1
cn−q,1aq
+
∑
2≤m≤n
n∑
q=m
cn−q,m ·
∑
n1,...,nq−1∈Dm,q
q−1∏
p=1
anp
p /np!, (19)
and
f ′′
n = bn,1 +
n∑
q=1
bn−q,2aq
+
∑
2≤m≤n
n∑
q=m
bn−q,m+1 ·
∑
n1,...,nq−1∈Dm,q
q−1∏
p=1
anp
p /np!, (20)
and the coefficients fn are given by the recurrence formulas f0 = x̃(0)(∞) =
f ′
0/f
′′
0 and in general for n = 0 . . . , k,
fn = (f ′
n −
n−1∑
q=0
f ′′
n−qfq)/f
′′
0 . (21)
It should be noted that one should require the perturbation condition
P
(k+1)
1 stronger than P
(k)
1 in Theorem 4. This is because of the former
condition is needed to get the corresponding expansion for φ(ε)(ρ(ε), 1) in an
asymptotic power series with respect to ε up to and including the order k.
Chapter 2 of book [25] also contains asymptotic results based on more
general perturbation conditions.
It worth to mention that discrete time analogues of some of the re-
sults presented above are given in papers by Englund and Silvestrov (1997),
Englund (2000, 2001), and Silvestrov (2000b). Also, exponential asymp-
totic expansions for renewal equation with non-polynomial perturbations
are studied in papers by Englund and Silvestrov (1997), Englund (2001),
and Ni, Silvestrov, and Malyarenko (2008).
3. Nonlinearly perturbed stochastic processes
3.1. Nonlinearly perturbed regenerative processes. Method of
asymptotic analysis of nonlinearly perturbed renewal equation can be di-
rectly used in studies of quasi- and pseudo-stationary asymptotics for non-
linearly perturbed regenerative processes. The corresponding mixed ergodic
NONLINEARLY PERTURBED STOCHASTIC PROCESSES 141
and limit theorems and mixed ergodic and large deviation theorems for non-
linearly perturbed regenerative processes are given in a form of exponen-
tial asymptotics for joint distributions of positions of regenerative processes
and regenerative stopping times. These distributions satisfy some renewal
equations, and the corresponding theorems may be obtained by applying
the above methods of asymptotic analysis for perturbed renewal equations.
The corresponding results are presented in Chapter 3 of book [25].
Let ξ(ε)(t), t ≥ 0 be a regenerative process with a measurable phase
space X and regeneration times τ
(ε)
n , n = 1, 2, . . ., and μ(ε) be a regenerative
stopping time that regenerates jointly with the process ξ(ε)(t), at times τ
(ε)
n .
Both the regenerative process ξ(ε)(t) and the regenerative stopping time
μ(ε) are assumed to depend on a small perturbation parameter ε ≥ 0. The
processes ξ(ε)(t), for ε > 0 are considered as a perturbation of the process
ξ(0)(t), and therefore we assume some weak continuity conditions for certain
characteristic quantities of these processes regarded as functions of ε at point
ε = 0.
As far as the regenerative stopping times are concerned, we consider two
cases. The first one is a pseudo-stationary case, where the random variables
μ(ε) are stochastically unbounded, i.e., μ(ε) tend to ∞ in probability as
ε → 0. The second one is the quasi-stationary case, where the random
variables μ(ε) are stochastically bounded as ε→ 0.
The object of studies is the probabilities P (ε)(t, A) = P{ξ(ε)(t) ∈ A,
μ(ε) > t}. These probabilities satisfy the following renewal equation,
P (ε)(t, A) = q(ε)(t, A) +
∫ ∞
0
P (ε)(t− s, A)F (ε)(ds), t ≥ 0, (22)
where the forcing function q(ε)(t, A) = P{ξ(ε)(t) ∈ A, τ
(ε)
1 ∧ μ(ε) > t} and
distribution function F (ε)(s) = P{τ (ε)
1 ≤ s, μ(ε) ≥ τ
(ε)
1 }.
Note that the distribution F (ε)(s) has the defect f (ε) = P{μ(ε) < τ
(ε)
1 }.
In this case, the mixed power-exponential moment generating function
φ(ε)(ρ, r) = E(τ
(ε)
1 )reρτ
(ε)
1 χ(μ(ε) ≥ τ
(ε)
1 ) and the characteristic equation (9)
takes the form φ(ε)(ρ, 1) = 1. The corresponding perturbation condition
assumes that function φ(ε)(ρ, r) (taken in point ρ(0) which is the root of
the limit characteristic equation) can be expanded in a power series with
respect to ε up to and including the order k − r for every r = 1, . . . , k.
The direct application of Theorems 1–4 to the renewal equation (22)
yields the exponential asymptotic expansions of type (3) for the probabil-
ities P (ε)(t, A) as well as the corresponding asymptotic expansions for the
renewal limits P̃ (ε)(∞, A) = limt→∞ eρ
(ε)tP (ε)(t, A) and then the asymp-
totic expansions of type (5) for the quasi-stationary distributions π(ε)(A) =
P̃ (ε)(∞, A)/P̃ (ε)(∞, X). Finally, these asymptotic results is expanded to
the model of nonlinearly perturbed regenerative processes with transition
period.
142 DMITRII SILVESTROV
3.2. Perturbed semi-Markov processes. The asymptotic results
obtained in Chapters 1–3 play the key role in further studies. They let one
make a very detailed analysis of pseudo- and quasi-stationary phenomena
for perturbed semi-Markov processes with a finite set of states and absorp-
tion. It can be done by using the fact that a semi-Markov process can be
considered as a regenerative process with regeneration times which are sub-
sequent return moments to any fixed state j
= 0. The first hitting time
to the absorption state 0 is, in this case, the regenerative stopping time.
The asymptotic results mentioned above are obtained by applying the cor-
responding results for regenerative processes given in Chapter 3. These
results are presented in Chapters 4 and 5 of book [25].
A semi-Markov process η(ε)(t), t ≥ 0, with a phase space X = {0, . . . , N}
and transition probabilities Q
(ε)
ij (u) is the object of studies. The semi-
Markov process η(ε)(t) is assumed to depend on a perturbation parameter
ε ≥ 0. The processes η(ε)(t), for ε > 0, are considered as perturbations of
the process η(0)(t), and, therefore, some weak continuity conditions are im-
posed on transition characteristics. Namely, the assumptions are made that
moment functionals of transition probabilities are continuous, if regarded
as functions of ε, at the point ε = 0.
Let us denote by μ
(ε)
j is the first hitting time (as result of a jump) for the
process η(ε)(t) to a state j ∈ X. It is also assumed that 0 is an absorption
state. The first hitting time μ
(ε)
0 to the state 0 is an absorption time. In
applications, the absorption times μ
(ε)
0 are often interpreted as transition
times for different stochastic systems described by semi-Markov processes;
these are occupation times or waiting times in queueing systems, lifetimes
in reliability models, extinction times in population dynamic models, etc.
The object of studies is probabilities P
(ε)
ij (t) = Pi{η(ε)(t) = j, μ
(ε)
0 > t}.
Not only the generic case, where the limiting semi-Markov process has
one communication class of recurrent-without absorption states, is consid-
ered in details, but also the case, where the limiting semi-Markov process
has one communication class of recurrent-without absorption states and, ad-
ditionally, the class of non-recurrent-without absorption states. The latter
model covers a significant part of applications.
Semi-Markov processes possess a regeneration property at moments of
hitting to a fixed state. This let one write down the following renewal type
equations for the above probabilities,
P
(ε)
ij (t) = q
(ε)
ij (t) +
∫ ∞
0
P
(ε)
jj (t− s) 0G
(ε)
ij (ds), t ≥ 0, i, j
= 0, (23)
where the forcing function q
(ε)
ij (t) = P{η(ε)(t) = j, μ
(ε)
j ∧ μ
(ε)
0 > t} and
distribution function 0G
(ε)
ij (s) = Pi{μ(ε)
j ≤ s, μ
(ε)
j ≤ μ
(ε)
0 }.
In fact, the relations above supply a renewal equation for every j
= 0
NONLINEARLY PERTURBED STOCHASTIC PROCESSES 143
if to choose i = j and a transition renewal type relation in the case where
i
= 0, j.
In Chapter 4, asymptotic results for probabilities P
(ε)
ij (t), which follows
from the basic Theorems 1–2 and are based on the weak perturbation as-
sumption about weak convergence of transition probabilities Q
(ε)
ij (u) and
minimal moment conditions involving only first moments of these transition
probabilities, are given as well as asymptotic relations involving Cramér type
conditions imposed on mixed power-exponential moments of these transi-
tion probabilities.
There exist analogs of these results in the literature, for example, those
obtained in papers by Alimov and Shurenkov (1990a, 1990b), Shurenkov
and Degtyar (1994), and Elĕıko and Shurenkov (1995).
However, Chapter 4 also contains a number of new asymptotic solidar-
ity propositions for perturbed semi-Markov processes that clarify the role of
initial state in the corresponding asymptotic relations as well as solidarity
properties of cyclic absorption probabilities, and moment generating func-
tions for hitting times for perturbed semi-Markov processes. They show
that the corresponding conditions for convergence formulated in terms of
cyclic characteristics of return times, and the form of the corresponding
asymptotic relations are invariant with respect to the choice of the recurrent-
without-absorption state used to construct the regeneration cycles. Also, a
number of auxiliary propositions dealing with convergence of hitting prob-
abilities, distributions, expectations, and exponential moments of hitting
times is formulated and proved.
In this case, the distribution function 0G
(ε)
jj (t) of the return-without-
absorption time in a state j
= 0 generates the renewal equation. The
corresponding characteristic equation takes the form
∫∞
0
eρs 0G
(ε)
jj (ds) = 1.
In particular, it is shown that the characteristic root ρ(ε) of this equation
does not depend on the choice of a recurrent-without-absorption state j
= 0.
3.3. Nonlinearly perturbed semi-Markov processes. The asymp-
totic results presented in Chapter 4 play a preparatory role. They can be
improved to the form of exponential expansions for joint distributions of
positions of semi-Markov processes and absorption times. Such expansions
were obtained in papers Gyllenberg and Silvestrov (1998, 1999a, 2000a),
Silvestrov (2000b, 2007a). They are presented in more advanced form in
Chapter 5 of book [25]. Also, asymptotic expansions for quasi-stationary
distributions of nonlinearly perturbed semi-Markov processes are given in
this chapter.
We consider the same model of perturbed semi-Markov processes as in
Chapter 4. Cramér type conditions of type C1 and nonlinear perturbation
conditions are imposed on the transition probabilities of the semi-Markov
processes Q
(ε)
ij (u). These conditions assume that mixed power-exponential
moment generation functions p
(ε)
ij [ρ, r] =
∫∞
0
sreρsQ
(ε)
ij (ds), i
= 0, j ∈ X
144 DMITRII SILVESTROV
(taken in point ρ(0) which is the root of the corresponding characteristic
equation for the limit case ε = 0) can be expanded in a power series with
respect to ε up to and including the order k − r for every r = 1, . . . , k.
As above, the relationship between the rate with which ε tends to zero
and the time t tends to infinity has a delicate influence upon the results.
The balance between the rate of perturbation and the rate of growth of time
is characterised by the following asymptotic relation εrt(ε) → λr < ∞ as
ε → 0 that is assumed to hold for some 1 ≤ r ≤ k.
With the above assumptions, some natural additional non-periodicity
conditions for the transition probabilities Q
(ε)
ij (u), the following asymptotic
relations are obtained,
Pi{η(ε)(t(ε)) = j, μ
(ε)
0 > t(ε)}
e−(ρ(0)+a1ε+···+ar−1εr−1)t(ε)
→ ˜̃π
(0)
ij (ρ(0))e−λrar as ε → 0, i, j
= 0. (24)
There are to alternatives in this relation, where the characteristic root
(a) ρ(0) = 0 and (b) ρ(0) > 0.
The first alternative (a) holds if an absorption in 0 is impossible for
the limiting process η(0)(t). In such a case, the absorption times μ
(ε)
0 tend in
probability to ∞ as ε→ 0. The asymptotic relation (24) describes pseudo-
stationary phenomena for perturbed semi-Markov processes.
The second alternative (b) holds if an absorption in 0 is possible for the
limiting process η(0)(t). In this case, the absorption times μ
(ε)
0 are stochas-
tically bounded as ε → 0. The asymptotic relation (24) describes, in this
case, quasi-stationary phenomena for perturbed semi-Markov processes.
Relations (24) let us get a new type of so-called mixed ergodic (for the
process η(ε)(t)) and large deviation (for the lifetime μ
(ε)
0 ) theorems.
Let us, for simplicity, restrict the presentation by the basic case, where
the set of non-absorbing states X1 = {1, . . . , N} is one communicative class
of recurrent-without absorption states for the limit semi-Markov process.
Let us consider the pseudo-stationary case, where (a) ρ(0) = 0. In this
case, the limiting coefficients ˜̃π
(0)
ij (0) = π̃
(0)
j , where π̃
(0)
j are the stationary
probabilities of the limiting process η(0)(t).
If k = 1, then the only case r = 1 is possible for the balancing condition.
In this case, (24) is equivalent to the following asymptotic relation
Pi{η(ε)(t(ε)) = j, εμ
(ε)
0 > εt(ε)} → π̃
(0)
j (0)e−λ1a1 as ε → 0, i, j
= 0. It shows
that the position of the semi-Markov process η(t(ε)) and the normalised ab-
sorption time εμ
(ε)
0 are asymptotically independent and have, in the limit, a
stationary distribution and an exponential distribution, respectively. This
can be interpreted as a mixed ergodic theorem (for the regenerative pro-
cesses) and a limit theorem (for regenerative stopping times).
If k = 2, then two cases, r = 1 and r = 2, are possible for the balancing
condition.
NONLINEARLY PERTURBED STOCHASTIC PROCESSES 145
The case r = 1 was already commented and interpreted above. In
this case, relation (24) can be given in the equivalent alternative form,
Pij(t
(ε))/π̃
(0)
j e−a1εt
(ε) → 1 as ε → 0. It shows that probability Pij(t
(ε)) can
be approximated by the exponential type mixed tail probability π̃
(0)
j e−a1εt
(ε)
,
with the zero asymptotic relative error, in every asymptotic time zone which
is determined by the relation εt(ε) → λ1 as ε→ 0, where 0 ≤ λ1 <∞.
In the case r = 2, the asymptotic relation (24) reduces to the asymptotic
relation Pij(t
(ε))/π̃
(0)
j e−a1εt
(ε) → e−a2λ2 as ε → 0. It shows that probability
Pij(t
(ε)) can be approximated by the the exponential type mixed tail proba-
bility π̃
(0)
j e−a1εt
(ε)
as ε → 0, with the asymptotic relative error 1− e−a2λ2 , in
every asymptotic time zone which is determined by the relation ε2t(ε) → λ2
as ε→ 0, where 0 ≤ λ2 <∞.
If λ2 = 0, then εt(ε) = o(ε−1), and the asymptotic relative error is 0.
Note that this case also covers the situation where εt(ε) is bounded, which
corresponds to the asymptotic relation (24) with r = 1. This is already an
extension of this asymptotic relation since it is possible that εt(ε) → ∞.
If λ2 > 0, then εt(ε) = O(ε−1), and the asymptotic relative error is
1 − e−λ2a2 . It differs from 0. Therefore, o(ε−1) is an asymptotic bound for
the large deviation zone with zero asymptotic relative error in the above
approximation.
To get the approximation with zero asymptotic relative error in the
asymptotic time zone which are determined by the relation ε2t(ε) → λ2 one
should approximate the mixed tail probabilities Pij(t
(ε)) by the exponential
type mixed tail probabilities π̃
(0)
j e−(a1ε+a2ε2)t(ε) ∼ π̃
(0)
j e−a1εt
(ε)−a2λ2 , i.e., to
introduce the corresponding corrections for the parameters in the exponents.
The comments above let one interpret relation (24) in the case r = 2 as
a new type of mixed ergodic and large deviation theorem for the nonlinearly
perturbed process η(ε)(t) and the lifetime μ
(ε)
0 .
A similar interpretation can be made for the asymptotic relation (24) if
k > 2. As above, relation (24) can also be interpreted as a mixed ergodic
and limit theorem if r = 1. On the other hand, this relation is naturally
to consider as a mixed ergodic and large deviation theorem for the corre-
sponding asymptotic time large deviation zones if 1 < r ≤ k.
A similar interpretation remains also in the quasi-stationary case, where
(b) ρ(0) > 0.
As was mentioned above, in the pseudo-stationary case (a), where ρ(0) =
0, the limiting coefficients ˜̃π
(0)
ij (0) do not depend of initial states i and co-
incides with the corresponding stationary probabilities π̃
(0)
j for the limiting
process η(0)(t). The situation is not so simple in the quasi-stationary case
(b). In this case, under some natural conditions, there exists a so-called
quasi-stationary distribution for the semi-Markov process η(ε)(t) given by
146 DMITRII SILVESTROV
the formulas π
(ε)
j (ρ(ε)) = ˜̃π
(ε)
ij (ρ(ε))/
∑
r �=0
˜̃π
(ε)
ir (ρ(ε)), j
= 0. In this case, the
coefficients ˜̃π
(ε)
ij (ρ(ε)), which are defined as in (24) but for a fixed ε ≥ 0,
may depend on the initial state i
= 0. But, the quasi-stationary probabil-
ities π
(ε)
j (ρ(ε)), j
= 0, do not depend on the choice of the state i
= 0 in the
formulas above.
In Chapter 5, the asymptotic expansions for the quasi-stationary distri-
butions for nonlinearly perturbed semi-Markov processes with absorption
also are given,
π
(ε)
j (ρ(ε)) = π
(0)
j (ρ(0)) + gj[1]ε+ · · ·+ gj[k]ε
k + o(εk), j
= 0. (25)
Both asymptotic expansions (24) and (25) are provided by the explicit
algorithms for calculating the coefficients in these expansions.
Mixed large deviation and ergodic theorems for nonlinearly perturbed
semi-Markov processes presented above are obtained by application expo-
nential expansions in mixed ergodic and large deviation theorems developed
in Chapter 3 for nonlinearly perturbed regenerative processes. The fact that
any semi-Markov processes is a regenerative process with the return times
to some fixed state being regeneration times can be employed. The time
of absorption (the first time of hitting the absorption state 0) is a regener-
ative stopping time. A semi-Markov process is characterized by prescrib-
ing its transition probabilities. It is natural to formulate the perturbation
conditions in terms of these transition probabilities. However, the condi-
tions and expansions formulated for regenerative processes are specified in
terms of expansions for the moments of regeneration times. Therefore, the
corresponding asymptotic expansions for absorption probabilities and the
moments of return and hitting times for perturbed semi-Markov processes
must be developed.
Thus, as the first step, asymptotic expansions for hitting probabilities,
power and mixed power-exponential moments of hitting times are con-
structed using a procedure that is based on recursive systems of linear
equations for hitting probabilities and moments of hitting times. These
moments satisfy recurrence systems of linear equations with the same per-
turbed coefficient matrix and the free terms connected by special recurrence
systems of relations. In these relations, the free terms for the moments of a
given order are given as polynomial functions of moments of lower orders.
This permits to build an effective recurrence algorithm for constructing the
corresponding asymptotic expansions. Each sub-step in this recurrence al-
gorithm is of a matrix but linear type, where the solution of the system
of linear equations with nonlinearly perturbed coefficients and free terms
should be expanded in asymptotic series. These expansions are also pro-
vided with a detailed analysis of their pivotal properties. These results have
their own values and possible applications beyond the problems studied in
the book.
NONLINEARLY PERTURBED STOCHASTIC PROCESSES 147
As soon as asymptotic expansions for moments of return-without-absorp-
tion times are constructed, the second nonlinear but scalar step of construc-
tion asymptotic expansions expansions for the characteristic root ρ(ε) can
be realised.
The separation of two steps described above, the first one matrix and
recurrence but linear and the second one nonlinear but scalar, significantly
simplify the whole algorithm.
Asymptotic expansion for the characteristic root ρ(ε) let one combine
it with asymptotic relations for probabilities Pi{η(ε)(t(ε)) = j, μ
(ε)
0 > t(ε)}
given in Chapter 4 and obtain the exponential expansion (24).
The asymptotic expansion (25) for the quasi-stationary distributions
π
(ε)
j (ρ(ε)) requires two more steps, which are needed for constructing asymp-
totic expansions at the point ρ(ε) for the corresponding moment generation
functions, giving expressions for quasi-stationary probabilities in the quo-
tient form, and for transforming the corresponding asymptotic quotient ex-
pressions to the form of power asymptotic expansions.
As a result, one get an effective algorithm for a construction of the
asymptotic expansions given in relations (24) and (25). It seems, that the
method used for obtaining the expansions mentioned above has its own
value and great potential for future studies.
As a particular but important example, the model of nonlinearly per-
turbed continuous time Markov chains with absorption is also considered.
In this case, it is more natural to formulate the perturbation conditions in
terms of generators of the perturbed Markov chains. Here, an additional
step in the algorithms is needed, since the initial perturbation conditions for
generators must be expressed in terms of the moments for the corresponding
semi-Markov transition probabilities. Then, the basic algorithms obtained
for nonlinearly perturbed semi-Markov processes can be applied.
Chapters 1–5 present a theory that can be applied in studies of pseudo-
and quasi-stationary phenomena in nonlinearly perturbed stochastic sys-
tems.
4. Applications
4.1. Nonlinearly perturbed stochastic systems. Chapter 6 of
book [25] deals with applications of the results obtained in Chapters 1–5
to an analysis of pseudo- and quasi-stationary phenomena in nonlinearly
perturbed stochastic systems. This chapter is partly based on the results of
the papers Gyllenberg and Silvestrov (1994, 1999a, 2000a).
Examples of stochastic systems under consideration are queueing sys-
tems, epidemic, and population dynamics models with finite lifetimes. In
queueing systems, the lifetime is usually the time at which some kind of a
fatal failure occurs in the system. In epidemic models, the time of extinc-
tion of the epidemic in the population plays the role of the lifetime, while
148 DMITRII SILVESTROV
in population dynamics models, the lifetime is usually the extinction time
for the corresponding population.
Several classical models being the subject of long term research studies
were selected. These models serve nowadays mainly as platforms for demon-
stration of new methods and innovation results. Our goal also is to show
what kind of new types results related to quasi-stationary asymptotics can
be obtained for such models with nonlinearly perturbed parameters.
All types of asymptotic results studied in Chapters 1–5 are given. They
include the following: (i) mixed ergodic theorems (for the state of the sys-
tem) and limit theorems (for the lifetimes) that describe transition phenom-
ena; (ii) mixed ergodic and large deviation theorems that describe pseudo-
and quasi-stationary phenomena; (iii) exponential expansions in mixed er-
godic and large deviation theorems; (iv) theorems on convergence of quasi-
stationary distributions; and (v) asymptotic expansions for quasi-stationary
distributions. In all the examples, we try to specify and to describe, in a
more explicit form, conditions and algorithms for calculating the limit ex-
pressions, the characteristic roots, and the coefficients in the corresponding
asymptotic expansions.
As the first example, a M/M queueing system with highly reliable main
servers is considered. The simplest variant of such system with nonlinearly
perturbed parameters was the subject of discussion in the first three subsec-
tions above. This queueing system is our first choice because of its function
can be described by some nonlinearly perturbed continuous time Markov
chains with absorption. Here, all conditions take a very explicit and clear
form.
Also a M/G queueing system with quick service and a bounded queue
buffer is considered. In this case, the perturbed stochastic processes, which
describe the dynamics of the queue in the system, belong to the class of
so-called stochastic processes with semi-Markov modulation. These pro-
cesses admit a construction of imbedded semi-Markov processes and are
more general than semi-Markov processes. This example was chosen be-
cause it shows in which way the main results obtained in the book can be
applied to stochastic processes more general than semi-Markov processes,
in particular, to stochastic processes with semi-Markov modulation.
The next example is based on classical semi-Markov and Markov birth-
and-death type processes. Some classical models of queueing systems, epi-
demic or population dynamic models can be described with the use of such
processes. We show in which way nonlinear perturbation conditions should
be used and what form will take advanced quasi- and pseudo-stationary
asymptotics developed in Chapters 1–5.
Finally, an example of nonlinearly perturbed metapopulation model is
considered. This example is interesting since it brings, for the first time,
the discussion on advanced quasi- and pseudo-stationary asymptotics in this
NONLINEARLY PERTURBED STOCHASTIC PROCESSES 149
actual area of research in mathematical biology.
4.2. Nonlinearly perturbed risk processes. The classical risk pro-
cesses are still the object of intensive research studies as show, for example,
references given in the bibliography. Of course, the purpose of these studies
is not any more to derive formulas relevant for field applications. These
studies intend to illustrate new methods and types of results that can later
be expanded to more complex models. The same approach was used by us
when choosing this model. The aim was to show that the innovative meth-
ods of analysis for nonlinearly perturbed processes developed in the book
can yield new results for this classical models.
Chapter 7 of book [25] contains results that extend the classical Cramér-
Lundberg and diffusion approximations for the ruin probabilities to a model
of nonlinearly perturbed risk processes. Both approximations are presented
in a unified way using the techniques of perturbed renewal equations devel-
oped in Chapters 1 and 2. This chapter is partly based on the results of
the papers Gyllenberg and Silvestrov (1999b, 2000b) and Silvestrov (2000a,
2007b).
The main new element in the results presented in Chapter 7 is a high
order exponential asymptotic expansion in these approximations for non-
linearly perturbed risk processes. Correction terms are obtained for the
Cramér-Lundberg and diffusion type approximations, which provide the
right asymptotic behaviour of relative errors in the perturbed model. We
study the dependence of these correction terms on the relations between the
rate of perturbation and the rate of growth of the initial capital.
Also various variants of the diffusion type approximation, including the
asymptotics for increments and derivatives of the ruin probabilities are
given.
Finally, we give asymptotic expansions in the Cramér-Lundberg and
diffusion type approximations for distribution of the capital surplus prior
and at ruin for nonlinearly perturbed risk processes.
It seems to us that results presented in Chapters 6 and 7 illustrate well
a potential of asymptotic methods developed in the book.
The works by Englund (1999a, 1999b, 2001) and Ni, Silvestrov, and
Malyarenko (2008) may also be mentioned. They also deal with applications
of methods based on perturbed renewal to asymptotic analysis of nonlinearly
perturbed queuing systems and nonlinearly perturbed risk processes.
5. New directions for the research
The last Chapter 8 of book [25] contains three supplements. The first
one gives some basic arithmetic operation formulas for scalar and matrix
asymptotic expansions.
The book contains also extended and carefully gathered bibliography
that has more than 1000 references to works in the areas related to the
150 DMITRII SILVESTROV
subject of the book. The third supplement in Chapter 8 contains the cor-
responding brief bibliographical remarks.
In the second supplement, some new directions in the research concerned
pseudo- and quasi-stationary phenomena for perturbed stochastic systems
that relate to the theory developed in this book are discussed and com-
mented on. There is a hope that this discussion will be especially useful for
young researchers and stimulate their interest to research studies in these
areas.
5.1. Nonlinearly perturbed stochastic systems with discrete
time. There is no doubt that all results on continuous time stochastic pro-
cesses and systems presented in the book should have their analogues for
discrete time stochastic processes and systems. It should be noted that
discrete time models are interesting by themselves and have important ap-
plications.
Some results concerned asymptotic expansions in mixed ergodic and
large deviation theorems for nonlinearly perturbed regenerative processes
with discrete time can be found in Englund and Silvestrov (1997), Englund
(2000, 2001), and in Silvestrov (2000b) for discrete time Markov chains with
absorption. However, these results give discrete time analogues just for a
small part of the results from the theory developed in the present book for
continuous time processes.
A development of a similar complete theory for nonlinearly perturbed
stochastic processes and system with discrete time requires an additional
comprehensive research.
5.2. Asymptotic expansions based on non-polynomial systems
of infinitesimals. Asymptotic expansions in mixed ergodic and limit/large
deviation theorems and asymptotic expansions for quasi-stationary distri-
butions can also be obtained for nonlinearly perturbed stochastic processes
and systems where the expansions in the initial perturbation conditions are
based on different systems of infinitesimals.
In this book, all expansions are based on the integer powers ϕn(ε) =
εn, n = 0, 1, . . ., for the simplest infinitesimal ϕ1(ε) = ε. This is a very nat-
ural system of infinitesimals that uses Taylor type expansions for nonlinearly
perturbed characteristics of the corresponding processes and systems.
There is a conjecture that analogous asymptotic expansions can also be
constructed for a model where the corresponding expansions include prod-
ucts of integer powers ϕn̄(ε) =
∏m
i=1(ϕi(ε))
ni, n̄ = (n1, . . . , nm), n1, . . . , nm =
0, 1, . . ., m = 1, 2, . . ., of infinitesimals taken from a finite or countable base
set {ϕi(ε), i = 1, 2, . . .}. The only condition should be imposed on these
infinitesimals that any two infinitesimals of the product form given above,
ϕ′
n̄′(ε) and ϕ′′
n̄′′(ε), should be asymptotically comparable, i.e., there should
be a constant 0 ≤ c ≤ ∞ (determined by these infinitesimals) such that
ϕ′
n̄′(ε)/ϕ′′
n̄′(ε) → c as ε → 0. The difference in the corresponding expan-
NONLINEARLY PERTURBED STOCHASTIC PROCESSES 151
sions should mainly be caused by new forms in arithmetics for asymptotic
expansions, i.e., in formulas for coefficients in sums, products, quotients, as
well as in the power, exponential, and other functions in the asymptotic
expansions. Such expansions permit to obtain a more dense net of infinites-
imals in the corresponding expansions and also to get the expansions in
models with corresponding types of perturbations.
Examples of models with non-polynomial perturbations can be found in
Englund and Silvestrov (1997) and Englund (1999a, 1999b, 2000, 2001) and
Ni, Silvestrov, and Malyarenko (2008). In the first five works, the model of
nonlinear perturbations with the base set of the infinitesimals {ε, e−a/ε} is
considered. The last paper deals with the model of nonlinear perturbations
with the base set of the infinitesimals {ε, εω}, where ω > 1 is some irrational
number, is considered.
5.3. Asymptotic expansions in mixed ergodic and large de-
viation theorems for semi-Markov type processes with countable
and general phase spaces. The asymptotic results obtained in Chapters
4 and 5 of this book relate to nonlinearly perturbed Markov chains and
semi-Markov processes with finite phase spaces. These processes possess
a regeneration property at return moments in a fixed state. This makes it
possible to use the asymptotic results for nonlinearly perturbed regenerative
processes presented in Chapter 3.
Markov chains and semi-Markov processes with countable phase spaces
possess similar regeneration properties. Moreover, the method of artificial
regeneration developed in the works of Kovalenko (1977), Nummelin (1978),
Athreya and Ney (1978), permits to construct regeneration moments for
Markov and semi-Markov processes with a general phase space. In this
way, the results concerning the asymptotic analysis of pseudo- and quasi-
stationary phenomena for nonlinearly perturbed regenerative processes can
be applied to nonlinearly perturbed Markov chains and semi-Markov pro-
cesses with countable and general phase spaces.
There exists a large number of works devoted to ergodic theorems, limit
theorems for random functionals of hitting time types, as well as large de-
viation theorems for such functionals in non-mixed or mixed forms (with
ergodic theorems). These works relate to Markov and semi-Markov type
processes with finite, countable, and general phase spaces. The results
presented in Chapter 4 are related to this direction. The corresponding
references are listed and commented on in the bibliographical remarks of
book [25].
It should be noted that the results given in Chapter 4 play only a
preparatory role with respect to the results related to exponential asymp-
totic expansions in mixed ergodic and large deviation theorem, and asymp-
totic expansions for quasi-stationary distributions given in Chapter 5. These
theorems also require many auxiliary results that are important by them-
152 DMITRII SILVESTROV
selves. Such results, for example, include asymptotic cyclic solidarity prop-
erties for the corresponding processes and asymptotic expansions for absorp-
tion and hitting probabilities, power and mixed power-exponential moments
for the first hitting times and related functionals, and others. Most of these
results still have no analogues for Markov and semi-Markov type processes
with countable and general phase spaces, and they may constitute a new
direction for the future research. As usual, operator analogues for matrix
techniques used in the models with finite phase spaces should be employed
for models with countable and general phase spaces, which significantly
complicates the problem under consideration.
The corresponding references are given in the bibliographical remarks
of book [25]. Here, we would like only to mention some originating papers
related research studies of quasi-stationary phenomena Vere-Jones (1962),
Kingman (1963), Darroch and Seneta (1965), and some books related to
these problems, Seneta (1981), Stewart and Sun Ji Guang (1990), Meyn
and Tweedie (1993), Kijima (1997), and Stewart (1998, 2001).
5.4. Mixed ergodic and large deviation theorems for semi-Mar-
kov type processes with asymptotically uncoupled phase space.
Results presented in Chapters 4 and 5 are related to a basic model of non-
linearly perturbed semi-Markov processes with one limit class of recurrent-
without-absorption states. It would be very interesting to extend the theory
to models in which the set of non-absorption states for perturbed processes
asymptotically splits into several classes of recurrent-without-absorption
states which do not communicate with each others.
It should be noted that most of the results for models with an asymptot-
ically uncoupled phase space are related to weak convergence limit theorems
for distributions of hitting times for pseudo-stationary models mixed with
the corresponding ergodic theorems and related theorems on asymptotic
aggregation of Markov and semi-Markov type processes. Also a few re-
sults concerned asymptotic expansions for linearly perturbed Markov and
semi-Markov type processes are known.
At the same time, a study of large deviation theorems for hitting times,
as well as exponential asymptotic expansions in mixed ergodic and large
deviation theorems for hitting times and asymptotic expansions for quasi-
stationary distributions in models with asymptotically uncoupled sets of
recurrent-without-absorption states for linearly and all the more nonlinearly
perturbed and Markov and semi-Markov type processes, have not been con-
ducted before. Such theorems certainly constitute an additional prospective
direction for the future research.
In principle, the method based on applying asymptotic results to per-
turbed regenerative processes developed in Chapter 3 can be used in com-
bination with the method of recurrence asymptotic analysis for power and
mixed power-exponential moments of hitting times developed in Chapter
NONLINEARLY PERTURBED STOCHASTIC PROCESSES 153
5. However, one should expect that pseudo-absorption effects caused by
asymptotic vanishing of communication between different recurrent-without-
absorption classes of states may complicate the asymptotic analysis for
power and mixed power-exponential moments of hitting times for models
with asymptotically uncoupled phase space.
The corresponding references are in the bibliographical remarks of book
[25]. Here, we would like only to mention some originating papers in this
area that are Dobrushin (1953), Meshalkin (1958), Simon and Ando (1961),
Hanen (1963), and Korolyuk (1969) as well as the latest books in the area,
Stewart and Sun Ji Guang (1990), Stewart (1998, 2001), Yin and Zhang
(1998), Koroliuk and Limnios (2005), and Anisimov (2008).
5.5. Mixed ergodic and large deviation theorems for stochas-
tic processes with semi-Markov modulation (switchings). Methods
of the asymptotic analysis of pseudo- and quasi-stationary phenomena in
nonlinearly perturbed Markov chains and semi-Markov processes also can
be applied to more general processes of Markov and semi-Markov types.
In order to be able to apply the methods of asymptotic analysis developed
for perturbed renewal equations, such processes should possess appropriate
regeneration properties. In particular, these are so-called stochastic pro-
cesses with semi-Markov modulation (switchings), which possess imbedded
semi-Markov processes. In the Markov case, these processes are also known
as Markov processes with discrete components. The books of Gikhman
and Skorokhod (1975), Silvestrov (1980), Koroliuk and Limnios (2005), and
Anisimov (2008) contain descriptions of these classes of stochastic processes.
In Chapter 6 of book [25], a detailed description of a typical example
of M/G queueing systems with quick service and a bounded queue buffer,
which can be described with the use of processes with semi-Markov modu-
lation, is given. This example shows in which way the methods developed
in the book can be applied to the asymptotic analysis of pseudo- and quasi-
stationary phenomena in nonlinearly perturbed processes with semi-Markov
modulation. The main point in the corresponding asymptotic analysis is to
express the perturbation and other conditions in terms of local characteris-
tics related to the individual semi-Markov regeneration cycles.
There are no doubts that the main results of the theory presented in
this book can be extended to this class of processes that are more general
than regenerative and semi-Markov processes.
5.6. Double asymptotic expansions in limit and large devi-
ation theorems for lifetimes in nonlinearly perturbed stochastic
processes and systems. The expansions obtained in Chapter 5, ap-
plied to marginal distributions of absorption times, have the following form
P{εμ(ε) > t/εr−1}/πe−(ρ0+a1ε+···+arεr)t/εr
= 1 + o(1) for r = 1, . . . , k. These
expansions describe the behaviour of relative errors in large deviation zones
of different orders, when approximating the distribution of absorption times
154 DMITRII SILVESTROV
with exponential type distributions that have specially fitted parameters.
Our conjecture is that it should also be possible to expand the residual
term o(1) in the asymptotic relation given above. This hope is based on
the results given in Korolyuk, Penev and Turbin (1972, 1981), Polǐsčuk
and Turbin (1973), and Abadov (1984), Silvestrov and Abadov (1984, 1991,
1993), where such expansions have been obtained in pseudo-stationary mo-
del (ρ0 = 0), respectively, for the case r = 1 in the first three papers and for
the cases r = 1 and r = 2 in the last four papers. Note that the case r = 1
corresponds to asymptotic expansions in usual weak limit theorem while
the case r = 2 corresponds asymptotic expansions large deviation theorems
for the zone of large deviation of the order O(ε−1). In the cases r > 2,
the question about such “double” expansions is an open problem. These
remarks can be also related to exponential expansions in mixed ergodic and
large deviation theorems.
5.7. Explicit upper bounds for the remainder terms in asymp-
totic expansions for nonlinearly perturbed stochastic systems. The
problem of finding explicit upper bounds for remainder terms in exponential
asymptotic expansions in mixed ergodic and limit/large deviation theorems
for nonlinearly perturbed stochastic processes and systems is of a special im-
portance. For example, such explicit upper bounds for the remainder terms
are obtained in the asymptotic expansions given in the papers Silvestrov
and Abadov (1991, 1993) mentioned above.
Of course, the investigation of the rates of convergence in limit theorems
for lifetime functionals for regenerative processes and Markov- and semi-
Markov type processes (which correspond to the problem of finding explicit
upper bounds for the remainder terms in zero-order expansions) continue
to be actual.
Some explicit upper bounds for the remainder terms in asymptotic ex-
pansions of stationary and quasi-stationary distributions are known for lin-
early perturbed Markov- and semi-Markov processes. Similar problems
related to explicit upper bounds for the remainder terms in asymptotic
expansions of stationary and quasi-stationary distributions for nonlinearly
perturbed stochastic processes and systems have not been conducted so far.
The corresponding references are given in the bibliographical remarks
of book [25]. Here, we would like only to mention some papers and books,
which contain explicit estimates in stability problems, rates of approxima-
tion for the perturbation of stationary distributions and related characteris-
tics for Markov chains and semi-Markov type processes. These are, Solov’ev
(1983), Kalashnikov and Rachev (1988), Meyn and Tweedie (1993), Kar-
tashov (1996), and Kalashnikov (1997).
5.8. Quasi-stationary phenomena in nonlinearly perturbed que-
ueing and reliability systems. Asymptotic analysis of pseudo- and quasi-
stationary phenomena in nonlinearly perturbed queueing systems and net-
NONLINEARLY PERTURBED STOCHASTIC PROCESSES 155
works of M/M, M/G, and G/M types, which can be described with the
use of Markov chains, semi-Markov processes, and stochastic processes with
semi-Markov modulation, is an unlimited area for applications of the theory
developed in the book. Examples of such applications are given in Chapter
6 of book [25]. These examples show that, in such applications, the main
problems translate to technical and sometimes non-trivial and interesting
calculations connected with a re-formulation of the perturbation conditions
and rewriting formulas for the coefficients of the asymptotic expansions and
quasi-stationary distributions in terms of local characteristics used to define
the corresponding queueing systems.
It would be interesting to develop an asymptotic analysis of pseudo-
and quasi-stationary phenomena for nonlinearly perturbed queueing sys-
tems with a more general structure of input flows of customers and service
processes, e.g., systems with input flows of group income of customers and
group service, different models of vacations, systems with parameters in
input flows and service processes depending on a queue, systems with mod-
ulated input flows and switching service regimes, systems with several kinds
of customers, systems with several types of servers, with and without reser-
vation, etc.
As well known, queueing systems of G/G type can also be described
with the use of regenerative processes. Here the moments when the queue
becomes empty play, as a rule, the role of regeneration times. Thus, we
can use the methods of asymptotic analysis of pseudo- and quasi-stationary
phenomena for perturbed regenerative processes. However, the analysis of
regeneration cycles and a reduction of the conditions and formulas that are
based on the global characteristics connected with the regeneration cycles
to, correspondingly, conditions and formulas that would be based on local
characteristics used to define the corresponding systems is expected to be
much more difficult for such systems.
The corresponding references are given in the bibliographical remarks
of book [25]. Here, we would like only to mention some surveys and latest
books related to asymptotic problems for queueing systems; these are As-
mussen (1987, 2003), Kalashnikov and Rachev (1988), Kalashnikov (1994),
Kovalenko (1994), Kovalenko, Kuznetsov, and Pegg (1997), Borovkov (1998),
Limnios and Oprişan (2001), Whitt (2002), and Anisimov (2008).
5.9. Quasi-stationary phenomena in nonlinearly perturbed mo-
dels of biological type systems. Another unlimited area for applica-
tions of the asymptotic results dealing with pseudo- and quasi-stationary
phenomena in stochastic systems are nonlinearly perturbed epidemic and
population dynamics models which can be described with the use of Markov
chains, semi-Markov processes with finite phase spaces. Examples of such
applications are given in Chapter 6 of book [25]. Similarly to the examples
from the queueing theory, the main problems are to reformulate the per-
156 DMITRII SILVESTROV
turbation conditions and to express the formulas for the coefficients in the
asymptotic expansions and quasi-stationary distributions in terms of the
local characteristic used to define the corresponding epidemic or population
dynamics models.
Nonlinearly perturbed epidemic and population dynamics models with
different types, e.g., sex, age, health status, etc., of individuals in the pop-
ulation, forms of interactions between individuals, environmental modula-
tion, complex spatial structure, etc., make interesting objects for asymptotic
analysis of pseudo- and quasi-stationary phenomena.
The related references are given and commented in the bibliographical
remarks of book [25].
5.10. Asymptotics of probabilities and other characteristics
connected with ruins for perturbed risk type processes. Chapter 7
of the book is devoted to studies of quasi-stationary phenomena for non-
linearly perturbed risk processes. Asymptotic results for ruin probabilities,
densities of non-ruin probabilities, and distributions of the capital surplus
prior to and at the time of a ruin are given. However, there are many
other functional characteristics connected with risk processes that satisfy
renewal equations. In such cases, analogous asymptotic results can be ob-
tained with the use of asymptotic methods developed for perturbed renewal
equations in Chapters 1 and 2. More general risk type processes may also
supply models with functional characteristics satisfying renewal equations.
For example, such are risk processes with investment components described
by Brownian motions, more general Lévy type risk processes, risk processes
with modulated claim flows, etc.
We give corresponding references in the bibliographical remarks. Here,
we would like only to mention some surveys and latest books related to
asymptotic problems for risk processes; these are Embrechts, Klüppelberg,
and Mikosch (1997), Rolski, Schmidli, Schmidt, and Teugels (1999), As-
mussen (2000), and Bening and Korolev (2002).
5.11. Markov and semi-Markov processes describing dynam-
ics of credit ratings. Finally, we would like also to mention a possible
prospective area of financial applications for asymptotic methods developed
in the book. The methods of asymptotic analysis of pseudo- and quasi-
stationary characteristics may be also applied to studies of default type
distributions for Markov and semi-Markov processes with absorption which
become common to use for description of dynamics of credit ratings.
5.12. Numerical studies connected with asymptotic expansions
describing quasi-stationary phenomena in nonlinearly perturbed
stochastic systems. Numerical and simulation studies make a natural
supplement to analytical results, especially such as asymptotic expansions
for functional characteristics of perturbed stochastic processes and systems.
For example, asymptotic results presented in Chapter 7 of book [25] yield
NONLINEARLY PERTURBED STOCHASTIC PROCESSES 157
formulas for approximations of ruin probabilities based on higher order mo-
ments of the claim distributions. These approximations are asymptotically
optimal in the sense that the corresponding relative errors tend to zero. It
would be interesting to compare them with other known approximations.
This, however, would require to carry out additional comprehensive an-
alytical, numerical, and simulation studies that are beyond the frame of
the present book. There is a hope that these experimental studies will be
realised in a future for this model as well for other perturbed models of
stochastic processes and systems.
5.13. Asymptotics for perturbed equations of renewal type.
The program for studies of pseudo- and stationary phenomena in nonlin-
early perturbed stochastic processes and systems realised in the book is
based on the method of the asymptotic analysis for perturbed renewal equa-
tions presented in Chapters 1 and 2. There are no doubts that the main
asymptotic results for the perturbed renewal equations presented in these
chapters should hold for more general equations of renewal type. In many
cases, such results can be achieved by a suitable reduction to the case of
classical renewal equation treated in the book. For example, Shurenkov
(1980a, 1980b, 1980c) gave such a generalisation of some results presented
in Chapter 1 to the case of a matrix renewal equation. Chapter 4 contains
a more through and extended presentation of matrix version of asymptotic
results for perturbed renewal equation given in Chapter 1 for the model
case of perturbed semi-Markov processes with absorption. Chapter 5 essen-
tially improves these asymptotic results to the much more advanced form
of the corresponding asymptotic expansions. In fact, this chapter contains
a matrix generalisation of the results of Chapter 2 to the model case of
nonlinearly perturbed semi-Markov processes with absorption.
There exist other versions of renewal type equations, for example, the
so-called renewal equations with waits, operator renewal type equations,
renewal equations on a whole line etc. It would be interesting to extend the
theory presented in the present book to such equations.
There are also other areas, beyond ergodic, pseudo- and quasi-stationary
problems, where the renewal equation and the corresponding asymptotic
results play an essential role. For example, these are asymptotic problems
for moment functionals appearing in the theory of branching processes and
many others. We hope that the book will stimulate future works directed
to new applications of the theory presented in this book.
5.14. Bibliography. The bibliography of book [25] contains more
than 1000 references to works in related areas, dealing with ergodic and
quasi-ergodic theorems, stability theorems, limit and large deviation theo-
rems for lifetime-type functionals and asymptotic aggregation theorems for
regenerative, Markov, and semi-Markov type processes, as well as applica-
tions of such theorems to queueing systems, models of population dynamics,
158 DMITRII SILVESTROV
epidemic models, and other stochastic systems.
This bibliography also contains a representative sample of references
to works, where alternative methods and results on asymptotic expansions
for characteristic of perturbed stochastic processes. Here, we would like
only to point the books already mentioned above and some additional
relevant books related to perturbation problems for stochastic processes.
These are Korolyuk and Turbin (1976, 1978), Wentzell and Freidlin (1979),
Seneta (1981, 2006), Asmussen (1987, 2000, 2003), Kalashnikov and Rachev
(1988), Stewart and Sun Ji Guang (1990), Ho and Cao (1991), Meyn and
Tweedie (1993), Kalashnikov (1994), Kartashov (1996), Embrechts, Klüp-
pelberg, and Mikosch (1997), Borovkov (1998), Stewart (1998, 2001), Yin
and Zhang (1998), Korolyuk V.S. and Korolyuk, V.V. (1999), Latouche and
Ramaswami (1999), Whitt (2002), Silvestrov (2004), Koroliuk and Limnios
(2005), Anisimov (2008), and Gyllenberg and Silvestrov (2008).
6. Conclusion
Quasi-stationary phenomena and related problems are a subject of inten-
sive studies during several decades. However, the development of theory of
quasi-stationary phenomena is still far from its completion. The part of the
theory related to conditions of existence of quasi-stationary distributions is
comparatively well developed while computational aspects of the theory are
underdeveloped. The content of the book [25] is concentrated in this area.
The book presents new effective methods for asymptotic analysis of pseudo-
and quasi-stationary phenomena for nonlinearly perturbed stochastic pro-
cesses and systems. Moreover, the results presented in the book unite, for
the first time, research studies of pseudo- and quasi-stationary phenomena
in the frame of one theory. Methods of asymptotic analysis for nonlinearly
perturbed stochastic processes and systems developed in the book have their
own values and possible applications beyond the problems studied in the
book.
The results presented in the book will be interesting to specialists, who
work in such areas of the theory of stochastic processes as ergodic, limit,
and large deviation theorems, analytical and computational methods for
Markov chains, regenerative, Markov, semi-Markov, risk and other classes
of stochastic processes, renewal theory, and their queueing, reliability, pop-
ulation dynamics, and other applications. There is a hope that the book
will also attract attention of those researchers, who are interested in new an-
alytical methods of analysis for nonlinearly perturbed stochastic processes
and systems, especially those who like serious analytical work.
References
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