Ruin probability for generalized φ-sub-Gaussian fractional Brownian motion
In this paper we investigate the ruin problem for the generalized φ-sub-Gaussian fractional Brownian motion (FBM). Such random process has the same covariation function as FBM but its trajectories belong to the space of φ-sub-Gaussian random variables (i.e. not necessarily Gaussian). For this risk...
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| Цитувати: | Ruin probability for generalized φ-sub-Gaussian fractional Brownian motion / R. Yamnenko // Theory of Stochastic Processes. — 2006. — Т. 12 (28), № 3-4. — С. 261–275. — Бібліогр.: 9 назв.— англ. |
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irk-123456789-44702009-11-24T18:33:31Z Ruin probability for generalized φ-sub-Gaussian fractional Brownian motion Yamnenko, R. In this paper we investigate the ruin problem for the generalized φ-sub-Gaussian fractional Brownian motion (FBM). Such random process has the same covariation function as FBM but its trajectories belong to the space of φ-sub-Gaussian random variables (i.e. not necessarily Gaussian). For this risk process we obtain estimate of the ruin probability. 2006 Article Ruin probability for generalized φ-sub-Gaussian fractional Brownian motion / R. Yamnenko // Theory of Stochastic Processes. — 2006. — Т. 12 (28), № 3-4. — С. 261–275. — Бібліогр.: 9 назв.— англ. 0321-3900 http://dspace.nbuv.gov.ua/handle/123456789/4470 en Інститут математики НАН України |
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In this paper we investigate the ruin problem for the generalized φ-sub-Gaussian fractional Brownian motion (FBM). Such random process has the same covariation function as FBM but its trajectories belong to the space of φ-sub-Gaussian random variables (i.e. not necessarily Gaussian).
For this risk process we obtain estimate of the ruin probability. |
| format |
Article |
| author |
Yamnenko, R. |
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Yamnenko, R. Ruin probability for generalized φ-sub-Gaussian fractional Brownian motion |
| author_facet |
Yamnenko, R. |
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Yamnenko, R. |
| title |
Ruin probability for generalized φ-sub-Gaussian fractional Brownian motion |
| title_short |
Ruin probability for generalized φ-sub-Gaussian fractional Brownian motion |
| title_full |
Ruin probability for generalized φ-sub-Gaussian fractional Brownian motion |
| title_fullStr |
Ruin probability for generalized φ-sub-Gaussian fractional Brownian motion |
| title_full_unstemmed |
Ruin probability for generalized φ-sub-Gaussian fractional Brownian motion |
| title_sort |
ruin probability for generalized φ-sub-gaussian fractional brownian motion |
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Інститут математики НАН України |
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2006 |
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http://dspace.nbuv.gov.ua/handle/123456789/4470 |
| citation_txt |
Ruin probability for generalized φ-sub-Gaussian fractional Brownian motion / R. Yamnenko // Theory of Stochastic Processes. — 2006. — Т. 12 (28), № 3-4. — С. 261–275. — Бібліогр.: 9 назв.— англ. |
| work_keys_str_mv |
AT yamnenkor ruinprobabilityforgeneralizedphsubgaussianfractionalbrownianmotion |
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2025-07-02T07:42:29Z |
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2025-07-02T07:42:29Z |
| _version_ |
1836520203581652992 |
| fulltext |
Theory of Stochastic Processes
Vol. 12 (28), no. 3–4, 2006, pp. 261–275
ROSTYSLAV YAMNENKO
RUIN PROBABILITY FOR GENERALIZED
ϕ-SUB-GAUSSIAN FRACTIONAL BROWNIAN MOTION1
In this paper we investigate the ruin problem for the generalized ϕ-sub-
Gaussian fractional Brownian motion (FBM). Such random process has
the same covariation function as FBM but its trajectories belong to the
space of ϕ-sub-Gaussian random variables (i.e. not necessarily Gauss-
ian). For this risk process we obtain estimate of the ruin probability.
1. Introduction
Such properties of fractional Brownian motion as long-range dependence
and self-similarity make it natural choice in modeling real processes from
financial mathematics and queueing theory.
Recall, that the fractional Brownian motion with index H ∈ (0, 1) is
Gaussian centered process ZH with stationary increments and continuous
paths and covariance function
RH(t, s) = EZH(s)ZH(t) =
1
2
(
t2H + s2H − |s − t|2H
)
.
One of actual tasks of the theory of random processes is finding the esti-
mates of probability that trajectories of a random process exceed the level
specified by some curve. It finds an application in risk theory as classical
problem of the investigation of the ruin probability
P
{
sup
t>0
(X(t) − f(t)) > x
}
for various types of risk process X = (X(t), t ≥ 0) and functions f(t). The
similar problem of finding the buffer overflow probability appears in the
queuing theory for different communication network models.
The tasks of such type were solved for many types of processes, includ-
ing Gaussian ones and aforementioned FBM (see, for example, Norros [1],
Michna [2], Baldi and Pacchiarotti [3], etc.). But since in many cases real
processes are Gaussian only asymptotically or not Gaussian at all, there
arises a problem of introduction of more general class of random processes
than Gaussian one. From the such viewpoint the classes of ϕ-sub-Gaussian
and strictly ϕ-sub-Gaussian random processes are of significant interest as
1Invited lecture.
2000 Mathematics Subject Classification. Primary 91B30.
Key words and phrases. Ruin probability, risk process, phi-sub-Gaussian process, gen-
eralized fractional Brownian motion, buffer overflow.
261
262 ROSTYSLAV YAMNENKO
a natural extension of the class of Gaussian random processes. Detailed
overview of their properties one can found in [4] and [5].
In this paper we investigate the properties of generalized ϕ-sub-Gaussian
fractional Brownian motion process which has the same covariation function
as fractional Brownian motion but its trajectories are not necessarily Gauss-
ian. This process was introduced firstly in [7] under the name of weakly
self-similar stationary increment processes from the space SSubϕ(Ω).
The plan of the paper is as follows. In §2 the general definitions and
some properties of random variables and processes from spaces Subϕ(Ω)
and SSubϕ(Ω) are considered. In §3 we give the definition of generalized
ϕ-sub-Gaussian fractional Brownian motion process (ϕ-GFBM). In §4 the
results from in [8, 9] are used to study the sampling distributions for the ruin
problem for the generalized fractional Brownian motion and for f(t) of the
form f(t) = ctα, c > 0, α ∈ [0, 1]. We obtain the following estimates of the
ruin probability (and of the buffer overflow probability for corresponding
queueing model) for ϕ-GFBM risk process ZH from the class Ψq
x0
which
also includes class of sub-Gaussian random processes (q = 2) and therefore
(Gaussian) FBM.
(i) P
{
sup
a≤t≤b
(ZH(t) − ctα) > x
}
≤
≤ 2
(
e
p
) 1
H
Kb(p, x) exp
{
−(q − 1)x
q
q−1
0 (Cbα + x)
q
q−1
q
q
q−1 b
qH
q−1
}
,
(ii) P
{
sup
t>0
(ZH(t) − ct) > x
}
≤ L(γ, x)x
q(1−H)
(q−1)H exp
{
−κ(γ)x
q(1−H)
q−1
}
,
where Kb(p, x), L(γ, x) are known bounded on x expressions.
2. Space of Subϕ(Ω) random variables: necessary definitions
and some useful properties
2.1. Orlicz N-functions
Let (Ω,F ,P) be a standard probability space.
Definition 1. A continuous even convex function ϕ is an Orlicz N-function
if it is strictly increasing for x > 0, ϕ(0) = 0 and
ϕ(x)
x
→ 0 as x → 0 and
ϕ(x)
x
→ ∞ as x → ∞.
Condition Q. An N -function ϕ satisfies condition Q if
(1) lim inf
x→0
ϕ(x)
x2
= c > 0.
Remark. It may happen that c = ∞.
2.2. ϕ-sub-Gaussian random variables and processes
RUIN PROBABILITY FOR ϕ-GFBM RISK PROCESS 263
Definition 2. [5] Let ϕ be an Orlicz N -function satisfying condition Q.
The random variable ξ belongs to the space Subϕ(Ω) if Eξ = 0, E exp{λξ}
exists for all λ ∈ R and there exists a constant a > 0 such that the following
inequality holds for all λ ∈ R
(2) E exp (λξ) ≤ exp (ϕ(aλ)) .
Theorem 1. [4] The space Subϕ(Ω) is a Banach space with respect to the
norm
(3) τϕ(ξ) = inf
{
a ≥ 0 : E exp(λξ) ≤ exp
(
ϕ(aλ)
)
, λ ∈ R
}
.
When ϕ(x) = x2
2
the space Subϕ(Ω) is called the space of sub-Gaussian
random variables and is denoted by Sub(Ω).
Examples. 1). Centered Gaussian random variable ξ = N(0, σ2) belongs
to space Sub(Ω) and τ(ξ) = (Eξ2)
1
2 . 2). Let ξ be a centered bounded
random variable, i.e. Eξ = 0 and there exists number c > 0 that |ξ| ≤ c
almost surely. Then ξ ∈ Sub(Ω) and τ(ξ) ≤ c.
Let T be a parameter set.
Definition 3. Random process X = (X(t), t ∈ T ) is a ϕ-sub-Gaussian
process if for all t ∈ T X(t) ∈ Subϕ(Ω).
A random ϕ-sub-Gaussian process belongs to Ψq
x0
if
(4) ϕ(x) =
⎧⎨
⎩
xq
xq
0
, |x| > x0,
x2
x2
0
, |x| ≤ x0,
where x0 > 0 and q ≥ 2 are some constants.
2.3. Strictly ϕ-sub-Gaussian random variables and processes
Theorem 2. [4] Let ϕ be an Orlicz N-function satisfying condition Q and
suppose that function ϕ(
√ · ) is convex. Let ξ1, ξ2, . . . , ξn be independent
random variables from the space Subϕ(Ω). Then
(5) τ 2
ϕ
(
n∑
i=1
ξi
)
≤
n∑
i=1
τ 2
ϕ(ξi).
Definition 4. [6] A family of random variables Δ from the space Subϕ(Ω)
is called strictly Subϕ(Ω), if there exists a constant CΔ > 0 such that for
arbitrary finite set I : ξi ∈ Δ, i ∈ I, and for any λi ∈ R the following
inequality takes place
(6) τϕ
(∑
i∈I
λiξi
)
≤ CΔ
(
E
(∑
i∈I
λiξi
)2) 1
2
.
If Δ is a family of strictly Subϕ(Ω) random variables, then linear closure Δ
of the family Δ in the space L2(Ω) also is strictly Subϕ(Ω) family of random
264 ROSTYSLAV YAMNENKO
variables. Linearly closed families of strictly Subϕ(Ω) random variables form
a space of strictly ϕ-sub-Gaussian random variables. This space is denoted
by SSubϕ(Ω).
When ϕ(x) = x2
2
the space SSubϕ(Ω) is called the space of strictly sub-
Gaussian random variables and is denoted by SSub(Ω).
The space of jointly Gaussian random variables belongs to space SSub(Ω).
Definition 5. A random process X = (X(t), t ∈ T ) is a strictly ϕ-sub-
Gaussian process if the corresponding family of random variables belongs
to the space SSubϕ(Ω).
Example. [6] Let ϕ be such an Orlicz N -function that the function ϕ(
√ · )
is convex and
X(t) =
∞∑
k=1
ξkφk(t),
where series
∞∑
k=1
ξkφk(t) converges in mean square sense for all t ∈ T and
family {ξk, k ≥ 1} belongs to the space SSubϕ(Ω), for instance {ξk, k ≥ 1}
are independent random variables from SSubϕ(Ω). Then X(t) is a strictly
ϕ-sub-Gaussian random process.
2.4. Probability of overrunning for ϕ-sub-Gaussian random
process
Let (T, ρ) be a pseudometrical (metrical) compact space with pseudomet-
ric (metric) ρ.
Suppose there exists such continuous monotonically increasing function
σ = {σ(h), h > 0}, that σ(h) → 0, as h → 0, and the following inequality is
true
(7) sup
ρ(t,s)≤h
τϕ(Y (t) − Y (s)) ≤ σ(h).
Let β > 0 be some number such that β ≤ σ
(
inf
s∈T
sup
t∈T
ρ(t, s)
)
, γ(u) =
τϕ(Y (u)), NT (u) denotes the least number of closed ρ-balls with radius u
needed to cover T .
Theorem 3. [9] Let Y = {Y (t), t ∈ T} be a separable random process from
the space Subϕ(Ω) and f = {f(t), t ∈ T} be such a continuous function that
|f(u) − f(v)| ≤ δ(ρ(u, v)), where δ = {δ(s), s > 0} is some monotonically
increasing nonnegative function, and X(t) = Y (t) − f(t). Let r = {r(u) :
u ≥ 1} be such a continuous function that r(u) > 0 as u > 1 and the
function s(t) = r(exp{t}), t ≥ 0, is convex. If
β∫
0
r(NT (σ(−1)(u)))du < ∞,
RUIN PROBABILITY FOR ϕ-GFBM RISK PROCESS 265
then for all p ∈ (0; 1) and x > 0 the following inequalities hold true
P
{
sup
t∈T
X(t) > x
}
≤ inf
λ>0
Zr(λ, p, β),(8)
P
{
inf
t∈T
X(t) < −x
}
≤ inf
λ>0
Zr(λ, p, β),(9)
P
{
sup
t∈T
|X(t)| > x
}
≤ 2 inf
λ>0
Zr(λ, p, β),(10)
where
Zr(λ, p, β) =
= exp
{
θϕ(λ, p) + pϕ
(
λβ
1 − p
)
+ λ
( ∞∑
k=2
δ(σ(−1)(βpk−1)) − x
)}
×
×r(−1)
⎛
⎝ 1
βp
βp∫
0
r(NT (σ(−1)(u)))du
⎞
⎠ ,
θϕ(λ, p) = sup
u∈T
(
(1 − p)ϕ
(
λγ(u)
1 − p
)
− λf(u)
)
.
3. Process of ϕ-sub-Gaussian generalized fractional
Brownian motion
Definition 6. [7] We call the process ZH = (ZH(t), t ∈ T ) ϕ-sub-Gaussian
generalized fractional Brownian motion (ϕ-GFBM) with Hurst index H ∈
(0, 1) if ZH is ϕ-sub-Gaussian process with stationary increments and co-
variance function
RH(t, s) = EZH(s)ZH(t) =
1
2
(
t2H + s2H − |s − t|2H
)
.
Example. Let {ηn, n = 1, 2, . . .} be a sequence of independent random
variables such that Eηn = 0, Eη2
n = 1 and ηn ∈ SSubϕ(Ω), where ϕ is such
an N -function that function ϕ(
√·) is convex and τϕ(ηn) ≤ τ < +∞. Then
the process
ZH(t) =
∞∑
n=1
λnηnψn(t)
is a centered strictly ϕ-sub-Gaussian random process with covariance func-
tion RH , where λn are eigenvalues and ψn are corresponding eigen-functions
of the following integral equation
ψ(s) =
1
λ2
∫ T
0
RH(t, s)ψ(t)dt.
266 ROSTYSLAV YAMNENKO
4. Main results
It is easy to obtain the following corollary for the process of ϕ-GFBM
from theorem 3 (see also [8,9]).
Theorem 4. Let ZH = (ZH(t), t ∈ [a, b]) be a process of strictly ϕ-GFBM
with Hurst parameter H ∈ (0, 1), C > 0 be some constant. Then for all
x > 0, numbers a, b such that 0 ≤ a < b < ∞, p ∈ (0, 1), β ∈
(
0,
(
b−a
2
)H]
and λ > 0 the following inequality holds true
P
{
sup
a≤t≤b
(ZH(t) − ctα) > x
}
≤ (b − a)
(
e
βp
) 1
H
×
× exp
{
λc(βp)
α
H
CΔ(1 − p
α
H )
+ pϕ
(
λβ
1 − p
)
+ (1 − p)θϕ(λ, p) − λx
CΔ
}
,(11)
where θϕ(λ, p) = sup
a≤u≤b
(
ϕ
(
λuH
1−p
)
− λcuα
CΔ(1−p)
)
, and CΔ is the constant from
definition 4 of the space SSubϕ(Ω).
Theorem 5. Let ZH = (ZH(t), t ∈ [a, b], 0 ≤ a < b < ∞), be a process of
strictly ϕ-GFBM from the class Ψq
x0
with Hurst parameter H ∈ (1
2
, 1). Let
C > 0, p ∈ (0, 1) and α ∈ [0, 1] be some constants and suppose that if q > 2
then the following condition holds
(12) max
{
v−H ;
2H
(b − a)H
}
≤
(
x0C(b − a)
(bqH − aqH)
) 1
q−1
,
where v = a if a > 0, or v = b if a = 0.
Then for all ε > 0 the following estimate is true
(13) P
{
sup
a≤t≤b
(
1
CΔ
ZH(t) − Ctα
)
> ε
}
≤ 2
(
e
p
) 1
H
Wa, b(ε)Ka, b(p, ε),
where
Wa,b(ε) = exp
{
−
(
Cxq
0(b
α − aα)
(bqH − aqH)
) 1
q−1
(
ε + Caαbα bqH−α − aqH−α
bqH − aqH
)}
,
Ka,b(p, ε) = exp
{
p
(
Cxq
0(b
α − aα)
(bqH − aqH)
) 1
q−1
(
ε + Caαbα bqH−α − aqH−α
bqH − aqH
+
C(b − a)αp
α−H
H (1 − p)
2(1 − p
α
H )
+
C(b − a)qH(bα − aα)
2qH(bqH − aqH)
)}
.
Also if
(14) ε ≥ εb =
qC(bα − aα)
bqH − aqH
(
bqH +
(b − a)qHp
2qH(1 − p)
)
+
C(b − a)αp
α
H
2(1 − p
α
H )
− Cbα,
RUIN PROBABILITY FOR ϕ-GFBM RISK PROCESS 267
then the following estimate, which is better than (13), holds true
(15) P
{
sup
a≤t≤b
(
1
CΔ
ZH(t) − Ctα
)
> ε
}
≤ 2
(
e
p
) 1
H
Γb(ε)Kb(p, ε),
where
Γb(ε) = exp
{
−(q − 1)x
q
q−1
0 (Cbα + ε)
q
q−1
q
q
q−1 b
qH
q−1
}
,
Kb(p, ε) = exp
{
p x
q
q−1
0 (Cbα + ε)
1
q−1
q
q
q−1 b
qH
q−1
(
(Cbα + ε)(b − a)qH
2qHbqH(1 − p)
+
+
p
α
H
−1qC(b − a)α
2(1 − p
α
H )
+ (q − 1)(Cbα + ε)
)}
.
Remarks. 1) If q = 2 then condition (12) is unnecessary. 2) Since p can
be chosen small enough the expression Kb(p, ε) can be bounded for any ε.
Proof. For simplicity put β = (b−a)H
2H . In order to unambiguously determine
function ϕ(·), consider such λ ≥ λ0 > 0 that λ0(b−a)H
2H(1−p)
≥ x0 and λ0aH
1−p
≥ x0 if
a > 0, or λ0bH
1−p
≥ x0 if a = 0. Then we can put
λ0 = (1 − p)x0 max
{
v−H;
2H
(b − a)H
}
,
where v = a for a > 0 or else v = b if a = 0.
Since ϕ(x) is strictly convex then
(16) θϕ(λ, C, p) =
{
λqaqH
xq
0(1−p)q − λCaα
1−p
, λ0 ≤ λ ≤ λ∗;
λqbqH
xq
0(1−p)q − λCbα
1−p
, λ > λ∗,
where λ∗ =
(
C(bα − aα)
(bqH − aqH)
) 1
q−1
(1 − p)x
q
q−1
0 .
For λ ≥ λ0 consider exponential part from estimate (11) in the theorem 4.
Γ(λ, p, ε) = exp
{
λq
(
dqH
xq
0(1 − p)q−1
+
(b − a)qHp
xq
02
qH(1 − p)q
)
−
− λ
(
Cdα + ε − C(b − a)αp
α
H
2(1 − p
α
H )
)}
:= exp {λqAd − λBd} ,(17)
where d =
{
a, if λ ≤ λ∗,
b, if λ > λ∗.
It is obvious that if λ∗ ≤
(
Bb
qAb
) 1
q−1
, that is if
(18) ε ≥ εb =
qC(bα − aα)
bqH − aqH
(
bqH +
(b − a)qHp
2qH(1 − p)
)
+
C(b − a)αp
α
H
2(1 − p
α
H )
− Cbα,
268 ROSTYSLAV YAMNENKO
then the function Γ(λ, p, ε) reaches its minimum value at the point λ =(
Bb
qAb
) 1
q−1
. Also for the unambiguity of the function ϕ we need λ0 ≤ λ∗, i.e.
max
{
v−H ;
2H
(b − a)H
}
≤
(
x0C(bα − aα)
(bqH − aqH)
) 1
q−1
.
Consider the inequalities
(1 + x′)α′
< 1 + α′x′, 0 < α′ < 1, x′ ≥ 0;(19)
(1 − x′′)α′′ ≥ 1 − α′′x′′, α′′ ≥ 1, 0 ≤ x′′ ≤ 1.(20)
Let x′ =
(b − a)qHp
2qHbqH(1 − p)
, α′ = 1
q−1
and x′′ =
C(b − a)αp
α
H
2(Cbα + ε)(1 − p
α
H )
, α′′ =
q
q−1
. From (18) follows that x′′ ≤ 1. Applying (19) and (20) to (17), using
inequality 1
1+z
≤ 1 for z ≥ 0, and the identity − z1
z2(1+z3)
= −z1
z2
+ z2
1+z2
for
some positive z1, z3 and z2, we have
min
λ>0
Γ(λ, p, ε) ≤ min
λ≥λ0
Γ(λ, p, ε) = exp
⎧⎨
⎩−(q − 1)B
q
q−1
b
q
q
q−1 A
1
q−1
b
⎫⎬
⎭ =
= exp
⎧⎪⎪⎪⎨
⎪⎪⎪⎩
−
(q − 1)x
q
q−1
0 (Cbα + ε)
q
q−1
(
1 − C(b−a)αp
α
H
2(Cbα+ε)
(
1−p
α
H
)) q
q−1
q
q
q−1 b
qH
q−1
(
1 + (b−a)qHp
(2b)qH (1−p)
) 1
q−1
⎫⎪⎪⎪⎬
⎪⎪⎪⎭
×
× exp
⎧⎪⎪⎪⎨
⎪⎪⎪⎩
p(q − 1)x
q
q−1
0
(
Cbα + ε − C(b−a)αp
α
H
2 1−p
α
H
) q
q−1
q
q
q−1
(
bqH + (b−a)qH p
2qH(1−p)
) 1
q−1
⎫⎪⎪⎪⎬
⎪⎪⎪⎭
≤
≤ exp
⎧⎪⎪⎨
⎪⎪⎩−
(q − 1)x
q
q−1
0 (Cbα + ε)
q
q−1
(
1 − qC(b−a)αp
α
H
2(q−1)(Cbα+ε) 1−p
α
H
)
q
q
q−1 b
qH
q−1
(
1 + (b−a)qH p
(q−1)(2b)qH (1−p)
)
⎫⎪⎪⎬
⎪⎪⎭×
× exp
{
p(q − 1)x
q
q−1
0 (Cbα + ε)
q
q−1
q
q
q−1 b
qH
q−1
}
≤
≤ exp
{
−(q − 1)x
q
q−1
0 (Cbα + ε)
q
q−1
q
q
q−1 b
qH
q−1
}
exp
{
px
q
q−1
0 (Cbα + ε)
1
q−1
q
q
q−1 b
qH
q−1
×
(
(Cbα + ε)(b − a)qH
2qHbqH(1 − p)
+
p
α
H
−1qC(b − a)α
2(1 − p
α
H )
+ (q − 1)(Cbα + ε)
)}
.(21)
RUIN PROBABILITY FOR ϕ-GFBM RISK PROCESS 269
Alternatively, if ε < εb
min
λ>0
Γ(λ, p, ε) ≤ Γ(λ∗, p, ε) = exp
{(
C(bα − aα)
(bqH − aqH)
) q
q−1
×
×(1 − p)qx
q2
q−1
0
(
bqH
xq
0(1 − p)q−1
+
(b − a)qHp
xq
02
qH(1 − p)q
)
−
−
(
C(bα − aα)
(bqH − aqH)
) 1
q−1
(1 − p)x
q
q−1
0
(
Cbα + ε − C(b − a)αp
α
H
2(1 − p
α
H )
)}
=
= exp
{
−
(
C(bα − aα)
(bqH − aqH)
) 1
q−1
x
q
q−1
0
(
ε + Caαbα bqH−α − aqH−α
bqH − aqH
)}
×
× exp
{
p
(
C(bα − aα)
(bqH − aqH)
) 1
q−1
x
q
q−1
0
(
ε + Caαbα bqH−α − aqH−α
bqH − aqH
+
+
C(b − a)αp
α−H
H (1 − p)
2(1 − p
α
H )
+
C(b − a)qH(bα − aα)
2qH(bqH − aqH)
)}
.(22)
It is obvious that the latter estimate holds true also for ε ≥ εb. So from
(21) and (22) we have the assertion of the theorem. �
Theorem 6. Let ZH = (ZH(t), t ≥ 0) be random processes from class Ψq
x0
with Hurst parameter H ∈ [0.5, 1), qH > 1. Let C > 0 and γ > 1 be some
constants. Then for all
(23) ε ≥ 2
H(q−1)
1−H γMq(1 − H)(γqH−1 − 1)
x
1
1−H
0 C
H
1−H (qH − 1)
max
{
1
γqH − 1
;
(γqH − 1)
H
1−H
(γ − 1)
H(q−1)
1−H
}
and
ζ ∈
(
0, ε
q(1−H)
(q−1)H
)
,
where M is such an integer that
(24) M ≥ 1 +
q − 1
q(1 − H)
log
(
q − 1
q(1 − H)
(γ − 1)
1
q−1
(γqH − 1)
1
q−1
)
,
the following inequality holds true
(25)
P
{
sup
t>0
(
1
CΔ
ZH(t) − Ct
)
> ε
}
≤ L(γ, ε)ε
q(1−H)
(q−1)H exp
{
−κ(γ)ε
q(1−H)
q−1
}
,
270 ROSTYSLAV YAMNENKO
where
κ(γ) =
x
q
q−1
0 C
qH
q−1 (q − 1)(γ − 1)
1
q−1 (γqH − γ)
qH−1
q−1
(q − qH)
q−qH
q−1 (qH − 1)
qH−1
q−1 (γqH − 1)
qH
q−1
,(26)
L(γ, ε) = 2ζ−1e
1
H (K0(γ) + K1(γ)S1(γ, ε)
+ γ
q(1−H)M
(q−1)H KM(γ) + KM+1(γ)S2(γ, ε)) < ∞,
(27)
S1(γ, ε) =
M−1∑
k=1
γ
q(1−H)k
(q−1)H exp
{
−κ(γ)ε
q(1−H)
q−1
}Sk,M (γ)
,(28)
S2(γ, ε) =
∞∑
k=M+1
γ
q(1−H)k
(q−1)H exp
{
−κ(γ)ε
q(1−H)
q−1
}Sk,M (γ)
,(29)
Sk,M(γ) =
q(1 − H)
q − 1
γ
qH−1
q−1
(M−k) +
qH − 1
q − 1
γ
q(1−H)
q−1
(k−M) − 1,(30)
K0(γ) = exp
{
x
q
q−1
0 ζHC
qH
q−1
(
γMq(1 − H)(γqH−1 − 1)
(qH − 1)(γqH − 1)
) qH−1
q−1
×
(
1 +
γ−M(qH − 1)(γqH − 1)
q(1 − H)(γqH−1 − 1)
(
1
2qH
+
H
2
))}
,
(31)
Kk(γ) = exp
{
x
q
q−1
0 ζHC
qH
q−1
(
γ − 1
γqH − 1
) 1
q−1
×
×
(
γMq(1 − H)(γqH − γ)
(qH − 1)(γqH − 1)
) qH−1
q−1
×
×
(
γ−k +
γ−M(qH − 1)
q(1 − H)
+
γ−M(γ − 1)qH+1(qH − 1)
2qH(γqH − γ)q(1 − H)
+
γ−M(γ − 1)(γqH − 1)(qH − 1)
2(γqH − γ)q(1 − H)γ
q(1−H)2k
(q−1)H
(
1 − γ
− q(1−H)k
(q−1)H
)
⎞
⎟⎠
⎫⎪⎬
⎪⎭ , k ≥ 1.
(32)
Remark. The ruin probability can be minimized by appropriate selection
of parameters γ and ζ .
Proof. Let us consider the following partition: [0,∞) =
∞⋃
k=0
[ak, bk], where
a0 = 0, b0 = a, bk = ak+1 = γka, k ≥ 1, a > 0, γ > 1 and apply for each
interval theorem 5. Then for any k ≥ 1
Wak , bk
(ε) = Wk(γ, ε) =
= exp
{
− C
1
q−1 x
q
q−1
0
a
qH−1
q−1 γ
qH−1
q−1
(k−1)
(
γ − 1
γqH − 1
) 1
q−1
(
ε + Cγka
γqH−1 − 1
γqH − 1
)}
.
RUIN PROBABILITY FOR ϕ-GFBM RISK PROCESS 271
Consider the function
j(n) = γ
1−qH
q−1
(n−1)
(
ε + Cγna
γqH−1 − 1
γqH − 1
)
as continuous relative to it’s argument n. Then
dj(n)
dn
=
(
1 − qH
q − 1
(
ε +
Cγna(γqH−1 − 1)
γqH − 1
)
+
Caγn(γqH−1 − 1)
γqH − 1
)
× γ
1−qH
q−1
(n−1) log γ
=
(
ε(1 − qH)
q − 1
+
Cγna(γqH−1 − 1)q(1 − H)
(γqH − 1)(q − 1)
)
γ
1−qH
q−1
(n−1) log γ,
dj(n)
dn
= 0 ⇔ a =
εγ−n
C
qH − 1
q(1 − H)
γqH − 1
γqH−1 − 1
.
If the previous equality holds true then Wn(γ, ε) takes maximal value.
Choose such an a that Wk(γ, ε) takes maximal values for k = M for some
M ≥ 1. Then
(33) a =
εγ−M
C
qH − 1
q(1 − H)
γqH − 1
γqH−1 − 1
.
After substituting a from (33) in Wk(γ, ε) we have
W
(M)
k (γ, ε) = exp
{
− C
qH
q−1 x
q
q−1
0 ε
q(1−H)
q−1 γ
qH−1
q−1
(M+1−k)
(
q(1 − H)
qH − 1
) qH−1
q−1
×
× (γ − 1)
1
q−1 (γqH−1 − 1)
qH−1
q−1
(γqH − 1)
qH
q−1
(
1 + γk−M qH − 1
q(1 − H)
)}
for k ≥ 1 and
W
(M)
0 (γ, ε) = exp
{
− C
qH
q−1 x
q
q−1
0 ε
q(1−H)
q−1 γ
(qH−1)M
q−1 ×
×
(
q(1 − H)
qH − 1
) qH−1
q−1
(
γqH−1 − 1
γqH − 1
) qH−1
q−1
}
.
Let’s define by
W (γ, ε) = W
(M)
M (γ, ε) = exp
{
−κ(γ)ε
q(1−H)
q−1
}
,(34)
κ(γ) =
x
q
q−1
0 C
qH
q−1 (q − 1)(γ − 1)
1
q−1 (γqH − γ)
qH−1
q−1
(q − qH)
q−qH
q−1 (qH − 1)
qH−1
q−1 (γqH − 1)
qH
q−1
.(35)
It is obvious that W (γ, ε) ≥ W
(M)
0 (γ, ε) if
(36) M ≥ 1 +
q − 1
q(1 − H)
log
(
q − 1
q(1 − H)
(γ − 1)
1
q−1
(γqH − 1)
1
q−1
)
.
272 ROSTYSLAV YAMNENKO
Consider the following ratio
W
(M)
k (γ, ε)
W (γ, ε)
= exp
{
− C
qH
q−1 x
q
q−1
0 ε
q(1−H)
q−1 γ
qH−1
q−1
(q − 1)(qH − 1)
1−qH
q−1
(q(1 − H))
q(1−H)
q−1
× (γ − 1)
1
q−1 (γqH−1 − 1)
qH−1
q−1
(γqH − 1)
qH
q−1
×
(
q(1 − H)
q − 1
γ
qH−1
q−1
(M−k) +
qH − 1
q − 1
γ− q(1−H)
q−1
(M−k) − 1
)}
= W (γ, ε)Sk,M (γ),
where Sk,M(γ) are defined in (30). It is easy to see that Sk,M(γ) > 0 for
any γ > 1 and k ≥ 1. In order to estimate the expressions consider the
inequalities
ex ≥ 1 + x + x2
and
e−x ≥ 1 − x
for x ≥ 0. Then
Sk,M(γ) ≥ (qH − 1)q2(1 − H)2 log2(γ)(k − M)2
(q − 1)2
,
k > M,
and the series
S2(γ, ε) =
∞∑
k=M+1
γ
q(1−H)k
(q−1)H W (γ, ε)Sk,M(γ)
≤
∞∑
k=M+1
γ
q(1−H)k
(q−1)H
× exp
{
−κ(γ)ε
q(1−H)
q−1
(qH − 1)q2(1 − H)2 log2 γ
(q − 1)2
}(k−M)2
converges for any ε > 0 and γ > 1.
RUIN PROBABILITY FOR ϕ-GFBM RISK PROCESS 273
Consider Ka, b(p, ε) from theorem 5 in details after substituting a from
(33).
Kak, bk
(p, ε) = exp
{
px
q
q−1
0
(
C(bk − ak)
(bqH
k − aqH
k )
) 1
q−1
(
ε+
+Cakbk
bqH−1
k − aqH−1
k
bqH
k − aqH
k
+
C(bk − ak)p
1−H
H (1 − p)
2(1 − p
1
H )
+
C(bk − ak)
qH+1
2qH(bqH
k − aqH
k )
)}
= exp
{
px
q
q−1
0
(
C(γ − 1)
(γqH − 1)
) 1
q−1
(
εγ−M(qH − 1)(γqH − 1)
Cq(1 − H)(γqH−1 − 1)
) 1−qH
q−1
×
×γ
(1−qH)(k−1)
q−1
(
ε +
εγk−M(qH − 1)
q(1 − H)
+
εγk−M(γ − 1)qH+1(qH − 1)
2qH(γqH − γ)q(1 − H)
+
+
εγk−M(γ − 1)(γqH − 1)(qH − 1)p
1−H
H (1 − p)
2(γqH − γ)q(1 − H)(1 − p
1
H )
)}
.(37)
For k ≥ 0 let’s put p = pk = ζH
ε
q(1−H)
q−1 γ
q(1−H)k
q−1
, where ζ is such a positive
constant that for all k ≥ 0 pk < 1, i.e. 0 < ζ < ε
q(1−H)
q−1 . Then for k ≥ 1
Kak , bk
(pk, ε) ≤ Kk(γ), Kk(γ) specified in (32).
It easy to check that fraction p
1−H
H (1−p)
1−p
1
H
monotone increases for p < 1 and
p
1−H
H (1−p)
1−p
1
H
↗ H if p ↗ 1. So in the same way for k = 0
Ka0, b0(p0, ε) = exp
{
px
q
q−1
0 C
1
q−1 a
1−qH
q−1
(
ε +
Ca
2qH
+
Cap
1−H
H (1 − p)
2(1 − p
1
H )
)}
≤ K0(γ).
Let’s investigate fulfilment of the condition (12) from theorem 5 for each
interval of the partition. For the first interval [0, a] we have
max{a−H ; 2Ha−H} =
2H
aH
≤ (x0Ca1−qH
) 1
q−1 .
And after substituting (33) we have the following inequality.
(38) ε ≥ 2
H(q−1)
1−H γMq(1 − H)(γqH−1 − 1)
x
1
1−H
0 C
H
1−H (qH − 1)(γqH − 1)
.
For the intervals [ak, bk] = [aγk−1, aγk], k ≥ 1 in similar fashion
max
{
a−Hγ−kH;
2H
aHγH(k−1)(γ − 1)H
}
=
2H
aHγH(k−1)(γ − 1)H
≤
(
x0Ca1−qH(γ − 1)
q(γqH − 1)
) 1
q−1
274 ROSTYSLAV YAMNENKO
After simple transformations we have
(39) ε ≥ 2
H(q−1)
1−H γM−k+1q(1 − H)(γqH−1 − 1)(γqH − 1)
H
1−H
x
1
1−H
0 C
H
1−H (qH − 1)(γ − 1)
H(q−1)
1−H
From (38) and (39) follows (23).
Then from all the above
P
{
sup
t>0
(
1
CΔ
ZH(t) − Ct
)
> ε
}
≤
≤
∑
k≥0
P
{
sup
t∈[ak ,bk]
(
1
CΔ
ZH(t) − Ct
)
> ε
}
≤
≤
∑
k≥0
2
(
e
pk
) 1
H
W
(M)
k (γ, ε)Kak,bk
(pk, ε) ≤
≤ 2ζ−1e
1
H ε
q(1−H)
(q−1)H
(
K0(γ)W
(M)
0 (γ, ε) +
M−1∑
k=1
γ
q(1−H)k
(q−1)H W
(M)
k (γ, ε)Kk(γ)
+ γ
q(1−H)M
(q−1)H W (γ, ε)KM(γ) +
∞∑
k=M+1
γ
q(1−H)k
(q−1)H W
(M)
k (γ, ε)Kk(γ)
)
≤
≤ 2ζ−1e
1
H ε
q(1−H)
(q−1)H W (γ, ε)
(
K0(γ) + K1(γ)
M−1∑
k=1
γ
q(1−H)k
(q−1)H
W
(M)
k (γ, ε)
W (γ, ε)
+
+ γ
q(1−H)M
(q−1)H KM(γ) + KM+1(γ)
∞∑
k=M+1
γ
q(1−H)k
(q−1)H
W
(M)
k (γ, ε)
W (γ, ε)
))
≤
≤ ε
q(1−H)
(q−1)H W (γ, ε)2ζ−1e
1
H ×
× (K0(γ) + K1(γ)S1(γ, ε) + γ
q(1−H)k
(q−1)H KM(γ) + KM+1(γ)S2(γ, ε)). �
References
1. I. Norros. On the use of Fractional Brownian motions in the Theory of Connec-
tionless Networks. IEEE Journal on selected areas in communications, Vol.13,
No.6. 953–962, 1995.
2. Z. Michna. Self-similar processes in collective risk theory. Journal of Applied
Mathematics and Stochastic Analysis, 11(4), 429–448, 1998.
3. P. Baldi and B. Pacchiarotti. Importance sampling for the ruin problem for gen-
eral Gaussian processes. Preprint. 2004.
4. V.V. Buldygin and Yu.V. Kozachenko, Metric Characterization of Random Vari-
ables and Random Processes. AMS, Providence, RI, 2000.
5. R. Giuliano-Antonini, Yu. Kozachenko, T. Nikitina, Spaces of φ-Subgaussian
Random Variables, Rendiconti, Academia Nazionale delle Scienze detta dei XL,
Memorie di Matematica e Applicazioni, 121o(2003), Vol. XXVII, fasc.1, 95–124.
RUIN PROBABILITY FOR ϕ-GFBM RISK PROCESS 275
6. Yu.V. Kozachenko, Yu.A. Kovalchuk, Boundary value problems with random ini-
tial conditions, and functional series from Subϕ(Ω). I. Ukrainian Math. Journal.
50 (4), 504–515, 1998.
7. Yu. Kozachenko, T. Sottinen and O. Vasilik. Weakly self-similar stationary incre-
ment processes from the space SSubϕ(Ω). Probab. Theory and Math. Statistics,
65, 2001.
8. Yu. Kozachenko, O. Vasylyk and R. Yamnenko. On the probability of exceeding
some curve by ϕ-subgaussian random process. Theory of Stoch. Processes. 9(25),
No. 3-4, 70–80, 2003.
9. Yu. Kozachenko, O. Vasylyk and R. Yamnenko. Upper estimate of overrunning
by Subϕ(Ω) random process the level specified by continuous function. Random
Oper. and Stoch. Equ., Vol. 13, No. 2, 101–118, 2005.
Department of Probability Theory and Mathematical Statistics, Kyiv
National Taras Shevchenko University, Kyiv, Ukraine
E-mail address: yamnenko@univ.kiev.ua
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