Extremes of threshold-dependent Gaussian processes

0.38 MB·

by  L. Bai

EXTREMES OF THRESHOLD-DEPENDENT GAUSSIAN PROCESSES
LONGBAI,KRZYSZTOFDE¸BICKI,ENKELEJDHASHORVA,ANDLANPENGJI
Abstract: Inthiscontributionweareconcernedwiththeasymptoticbehaviour,asu ,ofP sup X (t)>u ,
→∞ t∈[0,T] u
where X (t),t [0,T],u > 0 is a family of centered Gaussian processes with continuous trajecntories. A key applicao-
u
∈
tion of our findings concerns P sup (X(t)+g(t))>u , as u , for X a centered Gaussian process and g
t∈[0,T] → ∞
some measurable trend functionn. Further applications incluode the approximation of both the ruin time and the ruin
7
probability of the Brownian motion risk model with constant force of interest.
1
0 Key Words: Extremes; Gaussian processes; fractional Brownian motion; ruin probability; ruin time.
2
AMS Classification: Primary 60G15; secondary 60G70
n
a
J 1. Introduction

9
1 Let X(t),t 0 be a centered Gaussian process with continuous trajectories. An important problem in applied and
≥

theoretical probability is the determination of the asymptotic behavior of
]
R
P (1) p(u)=P sup (X(t)+g(t))>u , u
h. (t∈[0,T] ) →∞
t
a for some T >0 and g(t),t [0,T] a bounded measurable function. For instance, if g(t)= ct, then in the context of
∈ −
m
risk theory p(u) has interpretationas the ruin probability over the finite-time horizon[0,T]. Dually, in the context of
[
queueing theory, p(u) is related to the buffer overloadproblem; see e.g., [1–5].

1
For the special case that g(t) = 0,t [0,T] the exact asymptotics of (1) is well-known for both locally stationary
v ∈
7 and generalnon-stationaryGaussianprocesses,see e.g.,[6–18]. Commonly, for X a centerednon-stationaryGaussian
8
process it is assumed that the standard deviation function σ is such that t = argmax σ(t) is unique and
3 0 t [0,T]
∈
5 σ(t )=1. Additionally,if the correlationfunctionr andthe standarddeviationfunctionσ satisfy (hereafter means
0
0 ∼
. asymptotic equivalence)
1
0
(2) 1 r(s,t) a t sα, 1 σ(t +t) b tβ, s,t t
7 − ∼ | − | − 0 ∼ | | → 0
1
: for some a,b,β positive and α (0,2], then we have (see [10][Theorem D.3])
v ∈
Xi (3) p(u) C0u(α2−β2)+P X(t0)>u , u ,
∼ { } →∞
r
a where (x) =max(0,x) and
+
a1/αb 1/βΓ(1/β+1) , if α<β,
− α
H
C0 = αb/a, if α=β,
 1P, if α>β.
Here Γ() is the gamma function, and 
·
1
Hα =Tl→im∞TE(t∈s[u0p,T]eW(t)), Pαb/a =E(t∈s[u0,p∞)eW(t)−b/a|t|α), with W(t)=√2Bα(t)−|t|α,
are the Pickands and Piterbarg constants, respectively, where B is a standard fractional Brownian motion (fBm)
α
with self-similarity index α/2 (0,1], see [19–25] for properties of both constants.
∈
The more general case with non-zero g has also been considered in the literature; see, e.g., [1, 26–30]. However, most
ofthe aforementionedcontributionstreatonlyrestrictivetrendfunctionsg. Forinstance,in[26][Theorem3]aHo¨lder-
type conditionfor g is assumed, which excludes important cases of g that appear in applications. The restrictionsare
Date:January20,2017.
1
2 LONGBAI,KRZYSZTOFDE¸BICKI,ENKELEJDHASHORVA,ANDLANPENGJI
often so severe that simple cases such as the Brownian bridge with drift considered in Example 3.11 below cannot be
covered.
A key difficulty when dealing with p(u) is that X+g is not a centered Gaussianprocess. It is howeverpossible to get
rid of the trend function g since for any bounded function g and all u large (1) can be re-written as
X(t)
p (u)=P sup X (t)>u , X (t)= , t [0,T].
T u u
(t∈[0,T] ) 1−g(t)/u ∈
Here X is centered, however it depends on the threshold u, which complicates the analysis.
u
Extremesofthreshold-dependentGaussianprocessesX (t),t Rhavebeenalreadydealtwithinseveralcontributions,
u
∈
see e.g., [2, 3, 30–32]. Our principal result in Theorem 2.4 derives the asymptotics of p (u) for quite general families
T
of centered Gaussian processes X under tractable assumptions on the variance and correlation functions of X . To
u u
this end, in Theorem 2.2 we first derive the asymptotics of
p (u)=P sup X (t)>u , u
∆ u
(t ∆(u) ) →∞
∈
for some short compact intervals ∆(u).
ApplicationsofourmainresultsincludederivationofProposition3.1foraclassoflocallystationaryGaussianprocesses
with trend and that of Proposition3.6 for a class of non-stationaryGaussian processes with trend, as well as those of
theircorollaries. Forinstance,adirectapplicationofProposition3.6yieldstheasymptoticsof (1)foranon-stationary
X withstandarddeviationfunction σ andcorrelationfunction r satisfying (2)with t =argmax σ(t). Iffurther
0 t [0,T]
∈
the trend function g is continuous in a neighborhood of t , g(t )=max g(t) and
0 0 t [0,T]
∈
(4) g(t) g(t ) ct t γ, t t
0 0 0
∼ − | − | →
for some positive constants c,γ, then (3) holds with C specified in Proposition 3.9 and β,u being substituted by
0
min(β,2γ) and u g(t ) respectivelly.
0
−
Complementary, we investigate asymptotic properties of the first passage time (ruin time) of X(t)+g(t) to u on the
finite-time interval [0,T], given the process has ever exceeded u during [0,T]. In particular, for
(5) τ =inf t 0:X(t)>u g(t) ,
u
{ ≥ − }
with inf = , we are interested in the approximate distribution of τ τ T,as u . Normal and exponential
u u
{∅} ∞ | ≤ →∞
approximationsofvariousGaussianmodelshavebeendiscussedin[30, 32–35]. Inthispaper,wederivegeneralresults
for the approximationsofthe conditionalpassagetime inPropositions3.3, 3.10. The asymptoticsofp (u)for ashort
∆
compact intervals ∆(u) displayed in Theorem 2.2 plays a key role in the derivation of these results.
Organisation of the rest of the paper: In Section 2, the tail asymptotics of the supremum of a family of centered
Gaussianprocessesindexed by u aregiven. Severalapplications andexamples aredisplayedin Section3. Finally, we
present all the proofs in Section 4 and Section 5.
2. Main Results
Let X (t),t R,u> 0 be a family of threshold-dependent centered Gaussian processes with continuous trajectories,
u
∈
variance functions σ2 and correlation functions r . Our main results concern the asymptotics of slight generalization
u u
of p (u) and p (u) for families of centered Gaussian processes X satisfying some regularity conditions for variance
∆ T u
and coavariance respectivelly.
Let C (E) be the set of continuous real-valued functions defined on the interval E such that f(0) = 0 and for some
0∗
ǫ >ǫ >0
2 1
(6) lim f(t)/tǫ1 = , lim f(t)/tǫ2 =0,
t ,t E | | ∞ t ,t E | |
| |→∞ ∈ | |→∞ ∈
if sup x:x E = or inf x:x E = .
{ ∈ } ∞ { ∈ } −∞
In the following denotes the set of regularly varying functions at 0 with index α R, see [36–38] for details.
α
R ∈
EXTREMES OF THRESHOLD-DEPENDENT GAUSSIAN PROCESSES 3
We shall impose the following assumptions where ∆(u) is a compact interval:
A1: For any large u, there exists a point t R such that σ (t )=1.
u u u
∈
A2: There exists some λ>0 such that
1 1 u2 f(uλt)
(7) lim sup σu(tu+t) − − =0
u→∞t∈∆(u)(cid:12)(cid:12)(cid:12)(cid:16) f(uλt(cid:17))+1 (cid:12)(cid:12)(cid:12)
(cid:12) (cid:12)
holds for some non-negative continuous function(cid:12)f with f(0)=0. (cid:12)
(cid:12) (cid:12)
A3: There exists ρ ,α (0,2] such that
α/2
∈R ∈
1 r (t +s,t +t)
u u u
lim sup − 1 =0
u→∞s,t∈t=∆s(u)(cid:12)(cid:12) ρ2(|t−s|) − (cid:12)(cid:12)
6 (cid:12) (cid:12)
(cid:12) (cid:12)
and η :=lim ρ2(s) [0, ], with λ given in A2.
s→0 s2/λ ∈ ∞
Remark 2.1. If f satisfies f(0)=0 and f(t)>0,t=0, then
6
1 1
lim sup σu(tu+t) − 1 =0
u→∞t∈∆(u),t6=0(cid:12)(cid:12) u−2f(uλt) − (cid:12)(cid:12)
(cid:12) (cid:12)
for some λ>0 implies that (7) is valid. (cid:12) (cid:12)
(cid:12) (cid:12)
Next we introduce some further notation, starting with the Pickands-type constant defined by
α[0,T]=E sup e√2Bα(t)−|t|α , T >0,
H (t [0,T] )
∈
where B is an fBm. Further, define for f C ([S,T]) with S,T R,S <T and a positive constant a
α ∈ 0∗ ∈
Pαf,a[S,T]=E(ts[uSp,T]e√2aBα(t)−a|t|α−f(t)),
∈
and set
f [0, )= lim f [0,T], f ( , )= lim f [S,T].
Pα,a ∞ T Pα,a Pα,a −∞ ∞ S ,T Pα,a
→∞ →−∞ →∞
The finiteness of f [0, ) and f ( , ) is guaranteed under weak assumptions on f, which will be shown in
Pα,a ∞ Pα,a −∞ ∞
the proof of Theorem 2.2, see [2, 3, 5, 7, 15, 25, 39–43] for various properties of and f [0, ).
Hα Pα,a ∞
Denote by I the indicator function. For the regularly varying function ρ(), we denote by ←−ρ() its asymptotic
{·} · ·
inverse (which is asymptotically unique). Throughout this paper, we set 0 = 0 and u = 0 if u > 0. Let
−∞
·∞
Ψ(u):=P >u , with a standard normal random variable.
{N } N
In the next theorem we shall consider two functions x (u),x (u),u R such that x (1) , x (1) with
1 2 ∈ 1 t ∈ Rµ1 2 t ∈ Rµ2
µ ,µ λ, and
1 2
≥
(8) lim uλx (u)=x [ , ],i=1,2, with x <x .
i i 1 2
u ∈ −∞ ∞
→∞
Theorem 2.2. Let X (t),t R be a family of centered Gaussian processes with variance functions σ2 and correlation
u ∈ u
functions r . If A1-A3 are satisfied with ∆(u) = [x (u),x (u)], and f C ([x ,x ]), then for M satisfying M
u 1 2 ∈ 0∗ 1 2 u u ∼
u,u , we have
→∞
I
(9) P sup Xu(tu+t)>Mu C uλ←−ρ(u−1) −{η=∞}Ψ(Mu), u ,
(t ∆(u) )∼ →∞
∈ (cid:0) (cid:1)
where
x2e f(t)dt, if η = ,
Hα x1 − ∞
(10) C = Psuαfp,ηR[x1,x2]e, f(t), iiff ηη ∈=(00,,∞),
t∈[x1,x2] −
and Pαf,η(−∞,∞)∈(0,∞). 
4 LONGBAI,KRZYSZTOFDE¸BICKI,ENKELEJDHASHORVA,ANDLANPENGJI
Remark 2.3. Let α (0,2],a> 0 be given. If f C ([x ,x ]) for x ,x ,y R,x <x , as shown in Appendix, we
∈ ∈ 0∗ 1 2 1 2 ∈ 1 2
have, with f (t):=f(y+t),t R
y
∈
(11) f [x ,x ]= fy [x y,x y], f [x , )= fy [x y, ).
Pα,a 1 2 Pα,a 1− 2− Pα,a 1 ∞ Pα,a 1− ∞
In particular, if f(t)=ct,c>0, then for any x R
∈
ct [x, )= cx+ct[0, )=e cx ct [0, ).
Pα,a ∞ Pα,a ∞ − Pα,a ∞
Next, for any fixed T (0, ), in order to analyse p (u) we shall suppose that:
T
∈ ∞
A1’: For all large u, σ (t) attains its maximum over [0,T] at a unique point t such that
u u
σ (t )=1 and lim t =t [0,T].
u u u 0
u ∈
→∞
A4: For all u large enough
1 p(lnu)q
inf 1+
t∈[0,T]\(tu+∆(u))σu(t) ≥ u2
holds for some constants p>0,q>1.
A5: For some positive constants G,ς >0
E (X (t) X (s))2 Gt sς
u u
− ≤ | − |
holds for all s,t x [0,T]:σ(x)=0 an(cid:8)d X (t)= Xu(t). (cid:9)
∈{ ∈ 6 } u σu(t)
Below we define for λ given in A2 and ν,d positve
[0,δ ] if t 0,
u u
≡
 [ tu,δu], if tu du−ν and ν λ,
(12) ∆(u)= [[−−δδu,,Tδu], t ], iiff Ttu ∼∼tdu−νduorνTa−n≥dtuν∼dλu,−ν when ν <λ, or t0 ∈(0,T),
u u u −
− − − ∼ ≥
where δ = (lnu)q λ withq[g−ivδeun,0i]n A4. if tu =T,
u u
(cid:16) (cid:17)
Theorem 2.4. Let X (t),t [0,T] be a family of centered Gaussian processes with variance functions σ2 and
u ∈ u
correlation functions r . Assume that A1’,A2-A5 are satisfied with ∆(u)=[c (u),c (u)] given in (12) and
u 1 2
lim c (u)uλ =x [ , ],i=1,2, x <x .
i i 1 2
u ∈ −∞ ∞
→∞
If f C ([x ,x ]), then for M suc that lim M /u=1 we have
∈ 0∗ 1 2 u u→∞ u
I
(13) P sup Xu(t)>Mu C uλ←−ρ(u−1) −{η=∞}Ψ(Mu), u ,
(t [0,T] )∼ →∞
∈ (cid:0) (cid:1)
where C is the same as in (10) if η (0, ] and C =1 if η =0.
∈ ∞
Remark 2.5. Theorem 2.4 generalises both [26][Theorem 1] and [32][Theorem 4.1].
3. Applications
3.1. Locally stationary Gaussian processes with trend. In this section we consider the asymptotics of (1) for
X(t),t [0,T] a centered locally stationary Gaussian process with unit variance and correlationfunction r satisfying
∈
1 r(t,t+h)
(14) lim sup − =1
h→0t∈[0,T](cid:12)(cid:12) a(t)|h|α (cid:12)(cid:12)
with α (0,2], a() a positive continuous function o(cid:12)n [0,T] and fur(cid:12)ther
∈ · (cid:12) (cid:12)
(15) r(s,t)<1, s,t [0,T] and s=t.
∀ ∈ 6
We refer to e.g., [9, 10, 44–46] for results on locally stationary Gaussian processes. Extensions of this class to α(t)-
locally stationary processes are discussed in [13, 47, 48].
EXTREMES OF THRESHOLD-DEPENDENT GAUSSIAN PROCESSES 5
Regarding the continuous trend function g, we define g =max g(t) and set
m t [0,T]
∈
H := s [0,T]:g(s)=g .
m
{ ∈ }
Set below, for any t [0,T]
0
∈
, if t (0,T),
(16) Q =1+I , w = −∞ 0 ∈
t0 {t0∈(0,T)} t0 ( 0, if t0 =0 or t0 =T.
Proposition 3.1. Suppose that (14) and (15) hold for a centered locally stationary Gaussian process X(t),t [0,T]
∈
and let g :[0,T] R be a continuous function.
→
i) If H = t and (4) holds, then as u
0
{ } →∞
(17) P(ts[u0p,T](X(t)+g(t))>u)∼Ct0u(α2−γ1)+Ψ(u−gm),
∈
where (set with a=a(t ))
0
Q a1/αc 1/γΓ(1/γ+1) , if α<2γ,
t0 − Hα
Ct0 = P1,αc|,ta|γ[wt0,∞), iiff αα=>22γγ,.
ii) If H =[A,B]⊂[0,T] with 0≤A<B ≤T, then as u→∞
B
P sup (X(t)+g(t))>u α (a(t))1/αdtuα2Ψ(u gm).
(t∈[0,T] )∼H ZA −
Remarks 3.2. i) If H = t ,...,t , then as mentioned in [10], the tail distribution of the corresponding supremum
1 n
{ }
is easily obtained assuming that for each t the assumptions of Proposition 3.1 statement i) hold, implying that
i
n
P(ts[u0p,T](X(t)+g(t))>u)∼ Ctj u(α2−γ1)+Ψ(u−gm), u→∞.
∈ (cid:16)Xj=1 (cid:17)
ii) The novelty of Proposition 3.1 statement i) is that for the trend function g only a polynomial local behavior around
t is assumed. In the literature so far only the case that (4) holds with γ =2 has been considered (see [28]).
0
iii) By the proof of Proposition 3.1 statement i), if g(t) is a measurable function which is continuous in a neighborhood
of t and smaller than g ε for some ε>0 in the rest part over [0,T], then the results still hold.
0 m
−
We present below the approximation of the conditional passage time τ τ T with τ defined in (5).
u u u
| ≤
Proposition 3.3. Suppose that (14) and (15) hold for a centered locally stationary Gaussian process X(t),t [0,T].
∈
Let g :[0,T] R be a continuous function, H = t and (4) holds.
0
→ { }
i) If t [0,T), then for any x (w , )
0 ∈ ∈ t0 ∞
γc1/γRx e−c|t|γdt
wt0 , if α<2γ,
 Qt0Γ(1/γ)
Pnu1/γ(τu−t0)≤x(cid:12)τu ≤To∼ PPαcαc|,|t,at|a|γγ[[wwtt00,,∞x]), if α=2γ,
(cid:12) sup e ctγ, if α>2γ,
ii) If t =T, then for any x ( ,0)  t∈[wt0,x] − ||
0
∈ −∞
γc1/γR∞e−c|t|γdt
−x , if α<2γ,
Γ(1/γ)

Pnu1/γ(τu−t0)≤x(cid:12)τu ≤To∼ PPαcα|c,ta|,|taγ|γ[−[0x,∞,∞)), if α=2γ,
(cid:12) e cxγ, if α>2γ.
 − | |
6 LONGBAI,KRZYSZTOFDE¸BICKI,ENKELEJDHASHORVA,ANDLANPENGJI
Example 3.4. Let X(t),t [0,T] be a centered stationary Gaussian process with unit variance and correlation
∈
function r that satisfies r(t) = 1 atα(1+o(1)), t 0 for some a > 0, α (0,2], and r(t) < 1, for all t (0,T].
− | | → ∈ ∈
Let τ be defined as in (5) with g(t)= ct,c>0. Then we have
u
−
c 1a1/α , α (0,2),
P max (X(t) ct)>u u(α2−1)+Ψ(u) − Hα ∈
t [0,T] − ∼ ( ct [0, ), α=2,
(cid:26)∈ (cid:27) Pα,a ∞
and for any x positive
1 e cx, α (0,2),
P uτ x τ T − − ∈
u ≤ u ≤ ∼ Pαct,a[0,x] , α=2.
n (cid:12)(cid:12) o  Pαct,a[0,∞)
Example 3.5. Let X(t),t > 0 be a stand(cid:12)ardized fBm,i.e., X(t) = Bα(t)/tα/2 with Bα an fBm. Let c,T be positive
constants. Then for any n N, we have
∈
n
P t [Tm,(na+x1)T] X(t)+csin 2Tπt >u ∼ ajα1Hα√T2cπuα2−12Ψ(u−c),
(cid:26)∈ (cid:18) (cid:18) (cid:19)(cid:19) (cid:27) j=1
X
 
α
where a = 1 (4j+1)T − ,j =1,...,n.
j 2 4
(cid:16) (cid:17)
3.2. Non-stationary Gaussian processes with trend. In this section we consider the asymptotics of (1) for
X(t),t [0,T] a centered Gaussian process with non-constant variance function σ2. Define below whenever σ(t)=0
∈ 6
X(t)
X(t):= , t [0,T],
σ(t) ∈
and set for a continuous function g
σ(t)
(18) m (t):= , t [0,T], u>0.
u
1 g(t)/u ∈
−
Proposition 3.6. Let X and g be as above. Assume that t = argmax m (t) is unique with lim t = t
u t [0,T] u u u 0
∈ →∞
and σ(t ) = 1. Further, we suppose that A2-A5 are satisfied with σ (t) = mu(t) , r (s,t) = r(s,t), X (t) = X(t)
0 u mu(tu) u u
and ∆(u)=[c (u),c (u)] given in (12). If in A2 f C ([x ,x ]) and
1 2 ∈ 0∗ 1 2
lim c (u)uλ =x [ , ],i=1,2, x <x ,
i i 1 2
u ∈ −∞ ∞
→∞
then we have
(19) P sup (X(t)+g(t))>u C uλ←−ρ(u−1) −I{η=∞}Ψ u−g(tu) , u ,
(t∈[0,T] )∼ (cid:18) σ(tu) (cid:19) →∞
(cid:0) (cid:1)
where C is the same as in (10) when η (0, ] and C =1 when η =0.
∈ ∞
Remarks 3.7. i) Proposition 3.6 extends [26][Theorem 3] and the results of [1] where (1) was analyzed for special X
with stationary increments and special trend function g.
ii) The assumption that σ(t )=1 is not essential in the proof. In fact, for the general case where σ(t )=1 we have
0 0
6
that (19) holds with
σ0−α2Hα xx12e−σ0−2f(t)dt, if η =∞,
C = 1P,ασ,0−σ20−f2η[Rx1,x2], iiff ηη =∈(00,,∞), σ0 =σ(t0).

Proposition 3.8. Under the notation and assumptions of Proposition 3.6 without assuming A3,A5, if X is differ-
entiable in the mean square sense such that
r(s,t)<1,s=t, E X′2(t0) >σ′2(t0),
6
and E X 2(t) σ2(t) is continuous in a neighborhood of(cid:8)t , then(cid:9)(19) holds with
′ ′ 0
−
(cid:8) (cid:9) α=2, ρ2(t)= 1 E X′2(t0) σ′2(t0) t2.
2 −
(cid:16) n o (cid:17)
EXTREMES OF THRESHOLD-DEPENDENT GAUSSIAN PROCESSES 7
The next result is an extension of a classical theorem concerning the extremes of non-stationary Gaussian processes
discussed in the Introduction, see [10][Theorem D.3].
Proposition3.9. LetX(t),t [0,T]bea centeredGaussian process with correlation function r and variance function
∈
σ2 such that t = argmax σ(t) is unique with σ(t ) = σ > 0. Suppose that g is a bounded measurable function
0 t [0,T] 0
∈
being continuous in a neighborhood of t such that (4) holds. If further (2) is satisfied, then
0
u g(t )
(20) P sup (X(t)+g(t))>u C0u(α2−β2∗)+Ψ − 0 ,
(t∈[0,T] )∼ (cid:18) σ (cid:19)
where β =min(β,2γ),
∗
σ−2/αa1/αHα w∞t0 e−f(t)dt, if α<β∗,
C0 = P1,αf,σ−2a[wt0,∞R), iiff αα>=ββ∗,,
∗
with f(t)= σb3|t|βI{β=β∗}+ σc2|t|γI{2γ=β∗} and wt0 defined in (16).
Proposition 3.10. i) Under the conditions and notation of Proposition 3.6, for any x [x ,x ] we have
1 2
∈
Rx e−f(t)dt
x1 , if η = ,
 Rxx12e−f(t)dt ∞
(21) P(cid:8)uλ(τu−tu)≤x(cid:12)τu ≤T(cid:9)∼ PPαfαf,,ηη[[xx11,,xx2]], if η ∈(0,∞),
(cid:12)
sup e f(t), if η =0.
ii) Under the conditions and notation of Proposition3.9, iftt∈[x1,x[0],T−), then for x (w , )
0 ∈ ∈ t0 ∞
Rx e−f(t)dt
Rw∞t0e−f(t)dt, if α<β∗,
 wt0
Pnu2/β∗(τu−t0)≤x(cid:12)τu ≤To∼ PPαfαf,,aa[[wwtt00,,∞x]), if α=β∗,
(cid:12)
and if t =T, then for x ( ,0)  supt∈[wt0,x]e−f(t), if α>β∗,
0
∈ −∞
R∞e−f(t)dt
−x , if α<β ,
R∞e−f(t)dt ∗
 0
Pnu2/β∗(τu−t0)≤x(cid:12)τu ≤To∼ PPαfαf,a,a[−[0x,∞,∞)), if α=β∗,
(cid:12)
e f(x), if α>β .
Example 3.11. Let X(t) = B(t) tB(1),t [0,1], where B(t) i−s a standard Brownia∗n motion and suppose that τ
u
− ∈
is defined by (5) with g(t)= ct. Then
−
(22) P sup (X(t) ct)>u e−2(u2+cu),
(t [0,1] − )∼
∈
u
P u τ x τ 1 Φ(4x), x ( , ).
u u
− c+2u ≤ ≤ ∼ ∈ −∞ ∞
(cid:26) (cid:18) (cid:19) (cid:12) (cid:27)
(cid:12)
We note that according to [49][Lemma 2.7], the resu(cid:12)lt in (22) is actually exact, i.e. for any u>0,
P sup (X(t) ct)>u =e 2(u2+cu).
t∈[0,1] − −
Nonw, let T =1/2. It appearos that the asymptotics in this case is different, i.e.,
(23) P sup (X(t) ct)>u Φ(c)e 2(u2+cu),
−
(t [0,1/2] − )∼
∈
8 LONGBAI,KRZYSZTOFDE¸BICKI,ENKELEJDHASHORVA,ANDLANPENGJI
and
u 1 Φ(4x)
P u τ x τ , x ( ,c/4].
u u
− c+2u ≤ ≤ 2 ∼ Φ(c) ∈ −∞
(cid:26) (cid:18) (cid:19) (cid:12) (cid:27)
(cid:12)
Similarly, we have (cid:12)
c 1
(24) P sup X(t)+ c t >u 2Ψ(c)e−2(u2−cu)
(t∈[0,1](cid:18) 2 − (cid:12)(cid:12) − 2(cid:12)(cid:12)(cid:19) )∼
(cid:12) (cid:12)
and (cid:12) (cid:12)
4x (|t|+c)2
P u τu 1 x τu 1 −∞e− 2 dt, x ( , ).
(cid:26) (cid:18) − 2(cid:19)≤ (cid:12) ≤ (cid:27)∼ R 2√2πΨ(c) ∈ −∞ ∞
(cid:12)
Weconcludethissectionwithanapplication(cid:12)ofProposition3.6tothecalculationoftheruinprobabilityofaBrownian
motion risk model with constant force of interest over infinite-time horizon.
3.3. Ruin probability in Gaussian risk model. Consider risk reserve process U(t), with interest rate δ, modeled
by
t t
U(t)=ueδt+c eδ(t v)dv σ eδ(t v)dB(v), t 0,
− −
− ≥
Z0 Z0
where c,δ,σ are some positive constants and B is a standard Brownian motion. The corresponding ruin probability
over infinite-time horizon is defined as
p(u)=P inf U(t)<0 .
t [0, )
(cid:26)∈ ∞ (cid:27)
For this model we also define the ruin time τ =inf t 0:U(t)<0 . Set below
u
{ ≥ }
δ 2 c
h(t)= t+r2 r , t [0, ), r = .
σ2 − ∈ ∞ δ
(cid:16)p (cid:17)
We present next approximations of the ruin probability and the conditional ruin time τ τ < as u .
u u
| ∞ →∞
Proposition 3.12. As u
→∞
1
(25) p(u) h r2, Ψ 2δu2+4cu
∼P1,δ/σ2 − ∞ σ
(cid:18) (cid:19)
(cid:2) (cid:1) p
and for x ( r2, )
∈ − ∞
P(u2 e−2δτu −(cid:18)δuc+c(cid:19)2!≤x τu <∞)∼ PP1h1h,,δδ//σσ22[(cid:2)−−rr22,,∞x(cid:3)).
(cid:12)
(cid:12)
Remark 3.13. According to [50] (see also [51]) we have
√2δ √2c
(26) P inf U(t)<0 =Ψ (u+r) Ψ .
(cid:26)t∈[0,∞) (cid:27) σ !. σ√δ!
By (25) and (11)
2δ δ 2 δ 1
P inf U(t)<0 E sup exp B(t) t+r2 r t Ψ 2δu2+4cu
(cid:26)t∈[0,∞] (cid:27) ∼ (t∈[−r2,∞) rσ2 − σ2 (cid:16)p − (cid:17) − σ2| |!) (cid:18)σp (cid:19)
c2 2c c2 √2δ
E sup exp √2B(t) t+ + t+ t Ψ (u+r)
∼ t∈[−σc22δ,∞) −(cid:18) σ2δ(cid:19) σ√δr σ2δ −| |! σ !
= E sup exp √2B(t) 2t+ 2c √t Ψ √2δ (u+r) , 
(t∈[0,∞) (cid:18) − σ√δ (cid:19)) σ !
which combined with (26) implies that
1
2c √2c −
(27) E sup exp √2B(t) 2t+ √t = Ψ .
(t∈[0,∞) (cid:18) − σ√δ (cid:19)) σ√δ!!
EXTREMES OF THRESHOLD-DEPENDENT GAUSSIAN PROCESSES 9
4. Proofs
In the proofs presented in this section C ,i N are some positive constants which may be different from line to line.
i
∈
We first give two preliminary lemmas, which play an important role in the proof of Theorem 2.2.
Lemma 4.1. Let ξ(t),t R be a centered stationary Gaussian process with unit variance and correlation function r
∈
satisfying
(28) 1 r(t) aρ2(t), t 0,
− ∼ | | →
with a>0, and ρ , α (0,2]. Let f be a continuous function, K be a family of index sets and
α/2 u
∈R ∈
ξ(←−ρ(u−1)t)
Z (t):= , t [S ,S ],
u 1+u−2f(←−ρ(u−1)uλt) ∈ 1 2
where λ>0 and <S <S < . If M (u),k K is such that
1 2 k u
−∞ ∞ ∈
M (u)
k
(29) lim sup 1 =0,
u→∞k∈Ku(cid:12)(cid:12) u − (cid:12)(cid:12)
then we have (cid:12) (cid:12)
(cid:12) (cid:12)
1
(30) lim sup P sup Z (t)>M (u) f[S ,S ] =0,
u→∞k∈Ku(cid:12)(cid:12)Ψ(Mk(u)) (t∈[S1,S2] u k )−Rη 1 2 (cid:12)(cid:12)
(cid:12) (cid:12)
where (cid:12) (cid:12)
(cid:12) (cid:12)
[a1/αS ,a1/αS ] f() 0,
Rfη[S1,S2]:=E(t∈[sSu1p,S2]e√2aBα(t)−a|t|α−f(η−1/αt))=( PHαhα,a[S1,S12] 2 oth·erw≡ise,
with η :=lim ρ2(t) (0, ] and h(t)=f(η 1/αt) for η (0, ), h(t)=f(0) for η = .
t↓0 t2/λ ∈ ∞ − ∈ ∞ ∞
Proof of Lemma4.1: We set η 1/α =0 if η = . The prooffollows by checking the conditions of[52][Theorem2.1]
−
∞
where the results still holds if we omit the requirements f(0)=0 and [S ,S ] 0. By (29)
1 2
∋
lim inf M (u)= .
k
u→∞k∈Ku ∞
By continuity of f we have
(31) ul→im∞k∈Kus,tu∈p[S1,S2](cid:12)Mk2(u)u−2f(←−ρ(u−1)uλt)−f(η−1/αt)(cid:12)=0.
(cid:12) (cid:12)
Moreover,(28) implies (cid:12) (cid:12)
Var(ξ(←−ρ(u−1)t) ξ(←−ρ(u−1)t′))=2 2r ←−ρ(u−1)(t t′) 2aρ2 ←−ρ(u−1)(t t′) , u ,
− − − ∼ − →∞
(cid:0)(cid:12) (cid:12)(cid:1) (cid:0)(cid:12) (cid:12)(cid:1)
holds for t,t′ [S1,S2]. Thus (cid:12) (cid:12) (cid:12) (cid:12)
∈
(32) lim sup sup M2(u)Var(ξ(←−ρ(u−1)t)−ξ(←−ρ(u−1)t′)) 1 =0.
u→∞k∈Kut=6 t′∈[S1,S2](cid:12)(cid:12) k 2au2ρ2(|←−ρ(u−1)(t−t′)|) − (cid:12)(cid:12)
Since ρ2 α which satisfies the uniform(cid:12)(cid:12)convergence theorem (UCT) for regularly v(cid:12)(cid:12)arying function, see, e.g., [53],
∈ R
i.e.,
(33) lim sup u2ρ2 ←−ρ(u−1)(t t′) t t′ α =0,
u→∞t,t′∈[S1,S2](cid:12) (cid:0)(cid:12) − (cid:12)(cid:1)−| − | (cid:12)
and further by the Potter’s bound for ρ2, se(cid:12)e [53] w(cid:12)e have (cid:12) (cid:12)
(34) lium→s∞upt,t′∈st=[uStp1′,S2]u2ρ2(cid:0)(cid:12)(cid:12)|←t−ρ−(ut−′|α1)−(εt1−t′)(cid:12)(cid:12)(cid:1) ≤C1max(cid:16)|S1−S2|α−ε1,|S1−S2|α+ε1(cid:17)<∞,
6
where ε (0,min(1,α)). We know that for α (0,2]
1
∈ ∈
(35) tα t′ α C2 t t′ α∧1, t,t′ [S1,S2].
| | −| | ≤ | − | ∈
(cid:12) (cid:12)
(cid:12) (cid:12)
10 LONGBAI,KRZYSZTOFDE¸BICKI,ENKELEJDHASHORVA,ANDLANPENGJI
By (28) for any small ǫ>0, when u large enough
(36) r(←−ρ(u−1)t) 1 ρ2(←−ρ(u−1) t)(1 ǫ), r(←−ρ(u−1)t) 1 ρ2(←−ρ(u−1) t)(1+ǫ)
≤ − | | − ≥ − | |
hold for t [S ,S ], then by (29) for u large enough
1 2
∈
sup sup Mk2(u)E [ξ(←−ρ(u−1)t)−ξ(←−ρ(u−1)t′)]ξ(0)
k∈Ku|t−t′|<ε,t,t′∈[S1,S2] (cid:8) (cid:9)
C3u2 sup r(←−ρ(u−1)t) r(←−ρ(u−1)t′)
≤ −
|t−t′|<ε,t,t′∈[S1,S2](cid:12) (cid:12)
C3 sup u(cid:12)2ρ2(←−ρ(u−1) t) u2ρ2(←−ρ((cid:12)u−1) t′ ) +ǫ u2ρ2(←−ρ(u−1) t) +ǫ u2ρ2(←−ρ(u−1) t′ )
≤ | | − | | | | | |
C3|t−t′|<εs,tu,tp′∈[S1,S2](cid:0)(cid:12)(cid:12)u2ρ2 ←−ρ(u−1)(t) tα + u2ρ2 ←(cid:12)(cid:12)−ρ(u−(cid:12)(cid:12)1)(t′) t′ α +(cid:12)(cid:12) tα(cid:12)(cid:12) t′ α (cid:12)(cid:12)(cid:1)
≤ −| | −| | | | −| |
|t−t′|<ε,t,t′∈[S1,S2](cid:0)(cid:12) (cid:0)(cid:12) (cid:12)(cid:1) (cid:12) (cid:12) (cid:0)(cid:12) (cid:12)(cid:1) (cid:12) (cid:12) (cid:12)
(37) +C4ǫ tα−ε1 + t′ α−(cid:12)ε1 (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)
| | | |
(38) C εα (cid:16)1+C ǫ, u (cid:17)(cid:17)
5 ∧ 6
≤ →∞
0,ε 0,ǫ 0,
→ → →
where in (37) we use (34) and (38) follows from (33) and (35).
Hence the proof follows from [52][Theorem 2.1]. (cid:3)
Lemma4.2. LetZ (s,t),(s,t) R2 beacenteredstationaryGaussianfieldwithunitvarianceandcorrelationfunction
u
∈
r (, ) satisfying
Zu · ·
s α/2 t α/2
(39) 1−rZu(s,t)=au−2(cid:18)(cid:12)←−ρ(u−1)(cid:12) +(cid:12)←−ρ(u−1)(cid:12) (cid:19), (s,t)∈R2,
with a>0, ρ2 α and α (0,2]. Let Ku b(cid:12)(cid:12)e some in(cid:12)(cid:12)dex se(cid:12)(cid:12)ts. Then(cid:12)(cid:12), for Mk(u),k Ku satisfying (29) and for any
∈R ∈ ∈
S ,S ,T ,T 0 such that max(S ,S )>0,max(T ,T )>0, we have
1 2 1 2 1 2 1 2
≥
1
lim sup P sup Z (s,t)>M (u) (S ,S ,T ,T ) =0,
u k 1 2 1 2
u→∞k∈Ku(cid:12)(cid:12)Ψ(Mk(u)) ((s,t)∈D(u) )−F (cid:12)(cid:12)
(cid:12) (cid:12)
where D(u)=[−←−ρ(u−1)S1,(cid:12)(cid:12)←−ρ(u−1)S2]×[−←−ρ(u−1)T1,←−ρ(u−1)T2] and (cid:12)(cid:12)
(S ,S ,T ,T )= [ a2/αS ,a2/αS ] [ a2/αT ,a2/αT ].
1 2 1 2 α/2 1 2 α/2 1 2
F H − H −
Proof of Lemma 4.2: The proof follows by checking the conditions of [35][Lemma 5.3].
For D =[ S ,S ] [ T ,T ] we have
1 2 1 2
− × −
P sup Zu(s,t)>Mk(u) =P sup Zu(←−ρ(u−1)s,←−ρ(u−1)t)>Mk(u) .
((s,t)∈Du ) ((s,t)∈D )
Since by (39)
Var(Zu(←−ρ(u−1)s,←−ρ(u−1)t)−Zu(←−ρ(u−1)s′,←−ρ(u−1)t′)) = 2−2rZu ←−ρ(u−1)(s−s′),←−ρ(u−1)(t−t′)
= au 2 s(cid:0) s α/2+ t t α/2 (cid:1)
− ′ ′
| − | | − |
(cid:16) (cid:17)
we obtain
(40) lim sup sup M2(u)Var(Zu(←−ρ(u−1)s,←−ρ(u−1)t)−Zu(←−ρ(u−1)s′,←−ρ(u−1)t′)) 1 =0.
u→∞k∈Ku(s,t)6=(s′,t′)∈D(cid:12)(cid:12) k 2a(|s−s′|α/2+|t−t′|α/2) − (cid:12)(cid:12)
Further, since for α/2 (0,1] (cid:12) (cid:12)
(cid:12) (cid:12)
∈
tα/2 t α/2 C t t α/2, sα/2 s α/2 C s s α/2
′ 1 ′ ′ 2 ′
| | −| | ≤ | − | | | −| | ≤ | − |
(cid:12) (cid:12) (cid:12) (cid:12)
holds for t,t [ T ,T(cid:12)],s,s [ S ,(cid:12)S ], we have by (39)(cid:12) (cid:12)
′ ∈ − 1 2(cid:12) ′ ∈ − 1 (cid:12) 2 (cid:12) (cid:12)
sup sup Mk2(u)E [Zu(←−ρ(u−1)s,←−ρ(u−1)t)−Zu(←−ρ(u−1)s′,←−ρ(u−1)t′)]Zu(0,0)
k∈Ku|((ss,,tt))−,((ss′,′t,t′)′)|<Dε (cid:8) (cid:9)
∈