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Parisian ruin of the Brownian motion risk model with constant force of interest
Statistics and Probability Letters 120 (2017) 34–44
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Parisian ruin of the Brownian motion risk model with constant force of interest Long Bai a,∗ , Li Luo a,b a
Department of Actuarial Science, University of Lausanne, UNIL-Dorigny, 1015 Lausanne, Switzerland
b
School of Mathematical Sciences and LMPC, Nankai University, Tianjin, 300071, PR China
article
abstract
info
Let B(t ), t ∈ R be a standard Brownian motion. Define a risk process
Article history: Received 23 June 2016 Received in revised form 13 September 2016 Accepted 15 September 2016 Available online 23 September 2016
Rδu (t ) = eδ t
u+c
t
e−δ s ds − σ
t
0
e−δ s dB(s) , t ≥ 0,
(0.1)
0
where u ≥ 0 is the initial reserve, δ ≥ 0 is the force of interest, c > 0 is the rate of premium and σ > 0 is a volatility factor. In this contribution we obtain an approximation of the Parisian ruin probability
MSC: primary 60G15 secondary 60G70
δ
KS (u, Tu ) := P
Keywords: Parisian ruin Ruin probability Ruin time Brownian motion
inf
sup
t ∈[0,S ] s∈[t ,t +Tu ]
δ
Ru (s) < 0 , S ≥ 0,
as u → ∞ where Tu is a bounded function. Further, we show that the Parisian ruin time of this risk process can be approximated by an exponential random variable. Our results are new even for the classical ruin probability and ruin time which correspond to Tu ≡ 0 in the Parisian setting. © 2016 Elsevier B.V. All rights reserved.
1. Introduction In a theoretical insurance model the surplus process Ru (t ) can be defined by Ru (t ) = u + ct − X (t ),
t ≥ 0,
see Embrechts et al. (1997), where u ≥ 0 is the initial reserve, c > 0 is the rate of premium and X (t ), t ≥ 0 denotes the aggregate claims process. More specifically, we assume that the aggregate claims process is a Brownian motion, i.e., X (t ) = σ B(t ), σ > 0. Due to the nature of the financial market, we shall consider a more general surplus process including interest rate, see Rolski et al. (2009), called a risk reserve process with constant force of interest, i.e., Rδu (t ), t ≥ 0, in (0.1). See Rolski et al. (2009), Dębicki et al. (2015a) and He and Hu (2007) for more studies on risk models with force of interest. During the time horizon [0, S ], S ∈ (0, ∞], the classical ruin probability is defined as below δ
ψS (u) := P
∗
δ
inf Ru (t ) < 0 ,
t ∈[0,S ]
Corresponding author. E-mail addresses: [email protected] (L. Bai), [email protected] (L. Luo).
http://dx.doi.org/10.1016/j.spl.2016.09.011 0167-7152/© 2016 Elsevier B.V. All rights reserved.
(1.1)
L. Bai, L. Luo / Statistics and Probability Letters 120 (2017) 34–44
35
see Embrechts et al. (1997), Hüsler and Piterbarg (1999), Hüsler and Piterbarg (2008) and Dieker (2005). In Emanuel et al. δ (1975) and Harrison (1977) the exact formula of ψ∞ (u) for δ > 0 is shown to be
δ ψ∞ (u) =
Ψ
2δ
2c 2
u + σ 2δ ,
σ2
u > 0,
2c 2
Ψ
σ 2δ
where Ψ (x) = 1 − Φ (x) with Φ (·) the distribution function of an N (0, 1) random variable. 0 For δ = 0, the exact value of ψ∞ (u) is well-known (cf. Deelstra, 1994) with 0 ψ∞ (u) = e
− 2cu 2 σ
,
u > 0.
In the literature, there are no results for the classical ruin probability in the case of finite time horizon, i.e., S ∈ (0, ∞). For S ∈ (0, ∞), with motivation from the recent contributions (Dębicki et al., 2015b, 2016) we shall investigate in this paper the Parisian ruin probability over the time period [0, S ] defined as
KSδ (u, Tu ) := P
inf
sup
t ∈[0,S ] s∈[t ,t +Tu ]
Rδu (s) < 0 ,
(1.2)
where Tu ≥ 0 models the pre-specified time. Our assumption on Tu is that lim Tu u2 = T ∈ [0, ∞)
u→∞
and thus ψSδ (u) is a special case of KSδ (u, Tu ) with Tu ≡ 0. Another quantity of interest is the conditional distribution of the ruin time for the surplus process Rδu (t ). The classical ruin time, e.g., Dębicki et al. (2015a), Hashorva and Ji (2015a) and Hüsler and Piterbarg (2008), is defined as
τ (u) = inf{t > 0 : Rδu (t ) < 0}.
(1.3) δ
Here as in Dębicki et al. (2015b) we define the Parisian ruin time of the risk process Ru (t ) by
η(u) = inf{t ≥ Tu : t − κt ,u ≥ Tu , Rδu (t ) < 0},
with κt ,u = sup{s ∈ [0, t ] : Rδu (s) ≥ 0},
(1.4)
and τ (u) is a special case of η(u) with Tu ≡ 0. Brief organization of the rest of the paper: in Section 2 we first present our main results on the asymptotics of KSδ (u, Tu ) as u → ∞ and then we display the approximation of the Parisian ruin time. All the proofs are relegated to Section 3. 2. Main results Before giving the main results, we shall introduce a generalized Piterbarg constant as
(T ) = lim P (λ, T ), P λ→∞
T ≥ 0,
(2.1)
where for λ, T ≥ 0
(λ, T ) = E P
√
sup
inf e
2B(t −s)−|t −s|−(t −s)
t ∈[0,λ] s∈[0,T ]
.
(0) and P11 [0, ∞) = 2, see Dębicki and Mandjes (2003), Note further that the classical Piterbarg constant P11 [0, ∞) equals P Bai et al. (2016) and Hashorva and Ji (2015b). Throughout this paper ∼ means asymptotic equivalence when the argument tends to 0 or ∞. Recall that Ψ (·) denotes the tail distribution function of an N (0, 1) random variable and Ψ (u) ∼ √ 1 e− 2π u
u2 2
, u → ∞.
Theorem 2.1. For δ > 0, S > 0 and limu→∞ Tu u2 = T ∈ [0, ∞), we have
√ δ
(aT )Ψ K S ( u, T u ) ∼ P 2 −2δ S
where a := σ 22(1δ−ee−2δS )2 .
2δ u + δc (1 − e−δ S )
√ σ 1 − e−2δS
,
u → ∞,
(2.2)
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L. Bai, L. Luo / Statistics and Probability Letters 120 (2017) 34–44
Remark 2.2. (a) When Tu ≡ 0, KSδ (u, Tu ) reduces to the classical ruin probability ψSδ (u), and by Theorem 2.1 with T = 0
√ δ
δ
KS (u, 0) = ψS (u) ∼ 2Ψ
2δ u + δc (1 − e−δ S )
√ σ 1 − e−2δS
,
u → ∞.
(b) If δ = 0
KS0 (u, Tu ) = P
inf
sup
t ∈[0,S ] s∈[t ,t +Tu ]
(bT )Ψ ∼P
(u + cs − σ B(s)) < 0
u + cS
√ σ S
,
u → ∞,
(2.3)
where b := 2σ 12 S 2 and we used the result of corollary 3.4(ii) in Dębicki et al. (2016). Further, if δ = 0 and Tu ≡ 0, by (2.3) with T = 0, we get the asymptotic result of the classical ruin probability
ψS0 (u) ∼ 2Ψ
u + cS
√
σ S
,
u → ∞.
(2.4)
In fact, Deelstra (1994) gave the exact result of ψS0 (u), u > 0, i.e.,
cS − u − 2cu + e σ2 Φ √ σ S σ S u + cS ∼ 2Ψ , u → ∞, √ σ S
ψS0 (u) = Ψ
u + cS
√
which follows from e lim
u→∞
− 2cu 2 σ
Ψ
Φ
cS√ −u σ S
u+ √cS σ S
− 2cu 2
= lim
− σ2c2 e
σ
Φ
u→∞
−σ
cS√ −u σ S
√1
2π S
−
√1
e
−
σ 2π S 2 − u+√cS /2
e
u+ √cS σ S
2
/2
= 1.
σ S
Our next result discusses the approximation of the conditional ruin time. Theorem 2.3. Let η(u) satisfy (1.4), under the assumptions of Theorem 2.1, we have for any x > 0 and δ ≥ 0,
exp (−ax) , P u (S + Tu − η(u)) > x η(u) ≤ S + Tu ∼ exp (−bx) ,
2
if δ > 0, if δ = 0,
u → ∞,
2 −2δ S
where a := σ 22(1δ−ee−2δS )2 and b := 2σ 12 S 2 . Remark 2.4. If Tu ≡ 0, then η(u) = τ (u) and by Theorem 2.3, we obtain as u → ∞
P u2 (S − τ (u)) > x | τ (u) ≤ S ∼
exp (−ax) , exp (−bx) ,
if δ > 0, if δ = 0.
3. Proofs Hereafter we assume that Ci , i ∈ N are some positive constants. Proof of Theorem 2.1. For S > 0 and u large enough δ
KS (u, Tu ) = P
sup
inf
t ∈[0,S ] s∈[t ,t +Tu ]
=P
sup
sup
0
fu (S )
0
> fu (S )
inf
X (s)
inf
Xu (s) > fu (S ) ,
t ∈[0,S ] s∈[t ,t +Tu ]
=P
s s −δ z −δ z σ e dB(z ) − c e dz > u
t ∈[0,S ] s∈[t ,t +Tu ]
fu (s)
(2.5)
L. Bai, L. Luo / Statistics and Probability Letters 120 (2017) 34–44
37
with X ( s) = σ
X (s)
s
e−δ z dB(z ),
X (s) =
σX (s)
0
fu (s) =
,
u + δc (1 − e−δ s )
σX (s)
and
Xu (s) = X (s)
fu (S ) fu (s)
,
where σX2 (s) is the variance of X (s) with σX2 (s) = σ2δ (1 − e−2δ s ). 2 Set ρ(u) = lnuu and for any λ > 0, Bonferroni inequality yields 2
Π0 (u) := P
sup
inf
s∈[t ,t +Tu ] t ∈[S −λu−2 ,S ]
Xu (s) > fu (S )
≤ KSδ (u, Tu ) ≤ Π0 (u) + Π1 (u) + Π2 (u),
(3.1)
where
Π 1 ( u) = P
sup
inf
t ∈[0,S −ρ(u)] s∈[t ,t +Tu ]
Xu (s) > fu (S ) ,
Π2 (u) = P
sup
t ∈[S −ρ(u),S −λu−2 ]
inf
s∈[t ,t +Tu ]
Xu (s) > fu (S ) .
First we give some upper bounds of Π1 (u) and Π2 (u) which finally show that
Π1 (u) + Π2 (u) = o (Π0 (u)) ,
u → ∞.
(3.2)
For all u large
2 f ( S ) f ( S ) u u ) X (t1 ) E (Xu (t1 ) − Xu (t2 ))2 = E − X ( t 2 u + δc 1 − e−δ t1 u + δc 1 − e−δ t2 2 2 c c −δ S −δ S t2 u + u + ( 1 − e ) ( 1 − e ) δ δ ≤ C1 E e−δ z dB(z ) + C2 − u + δc (1 − e−δ t1 ) u + δc (1 − e−δ t2 ) t1 t1 < t2 , t1 , t2 ∈ (0, S ].
≤ C3 |t1 − t2 |, Moreover, sup
t ∈[0,S −ρ(u)]
Var (Xu (t )) =
sup
fu ( S )
=
fu (t )
t ∈[0,S −ρ(u)]
fu2 (S )
2 fu2
(S − ρ(u))
,
where we use the fact that fu (t ) is a decreasing function for t ∈ [0, S ] when u large enough. Therefore, by Theorem 8.1 in Piterbarg (1996), we obtain
Π 1 ( u) ≤ P
sup
t ∈[0,S −ρ(u)]
Xu (t ) > fu (S )
≤ C4 u2 Ψ (fu (S − ρ(u))) ,
(3.3)
and direct calculation yields that u Ψ (fu (S − ρ(u))) ≤ √
u2
2
−
2π fu (S )
2 −a(ln u)2
∼u e
e
fu2 (S ) 2
fu2 (S −ρ(u)) −1 fu2 (S )
e−
fu2 (S ) 2
Ψ (fu (S )) = o (Ψ (fu (S ))) ,
u → ∞,
2 −2δ S
where a = σ 22(1δ−ee−2δS )2 and we use the fact that
δ e−2δS t, 1 − e−2δ S
t → 0.
△k = kλu−2 , (k + 1)λu−2 ,
k ∈ N,
1−
fu (S ) fu (S − t )
∼
(3.4)
Set and N (u) = λ−1 ρ(u)u2 ,
where ⌊·⌋ stands for the ceiling function, then
Π 2 ( u) ≤ P
sup
t ∈[S −ρ(u),S −λu−2 ]
Xu (t ) > fu (S )
=P
sup
t ∈[λu−2 ,ρ(u)]
Xu (S − t ) > fu (S )
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L. Bai, L. Luo / Statistics and Probability Letters 120 (2017) 34–44
≤
N (u)
P sup Xu (S − t ) > fu (S ) t ∈∆k
k=1 N (u)
≤
P sup X (S − t ) > fu (S − kλu−2 )
t ∈∆0
k=1 N (u)
≤
P
k=1
sup X (S − u−2 t ) > fu (S − kλu−2 ) .
(3.5)
t ∈[0,λ]
Clearly, inf
1≤k≤N (u)
fu (S − kλu−2 ) → ∞,
u → ∞,
(3.6)
and for t1 < t2 , t1 , t2 ∈ [0, S ],
rX (t1 , t2 ) := E X (t1 )X (t2 ) =
1 − e−2δ t1 1 − e−2δ t2
.
Further,
−2 −2 2 −2 Var X (S − u t1 ) − X (S − u t2 ) − 1 lim sup sup fu (S − kλu ) u→∞ 1≤k≤N (u) t ̸=t , 2a|t1 − t2 | 1 2 t1 ,t2 ∈[0,λ] 2 − 2rX (S − u−2 t1 , S − u−2 t2 ) = lim sup sup fu2 (S − kλu−2 ) − 1 u→∞ 1≤k≤N (u) t ̸=t , 2a|t1 − t2 | 1 2 t1 ,t2 ∈[0,λ]
= 0,
(3.7)
and sup
sup
1≤k≤N (u) |t1 −t2 |<ε t1 ,t2 ∈[0,λ]
≤ C5 u2
fu2 (S − kλu−2 )E
X (S − u−2 t1 ) − X (S − u−2 t2 ) X (S )
rX (S − u−2 t1 , S ) − rX (S − u−2 t2 , S )
sup |t1 −t2 |<ε t1 ,t2 ∈[0,λ]
≤ C6 u2
sup |t1 −t2 |<ε t1 ,t2 ∈[0,λ]
≤ C7
−2 −2 1 − e−2δ(S −u t1 ) − 1 − e−2δ(S −u t2 )
|t1 − t2 | → 0,
sup
u → ∞, ε → 0.
(3.8)
|t1 −t2 |<ε t1 ,t2 ∈[0,λ]
According to (3.6), (3.7), (3.8) and Lemma 5.3 of De¸bicki et al. (2015), (3.5) is followed by
Π2 (u) ≤ C8 λ
N (u)
∞ Ψ fu (S − kλu−2 ) ≤ C9 Ψ (fu (S ))λ e−C10 kλ = o (Ψ (fu (S ))) ,
k =1
u → ∞, λ → ∞,
(3.9)
k=1
where the last inequality follows from (3.4). Next we give the asymptotic behavior of Π0 (u) as u → ∞ based on an appropriate application of the Appendix in Dębicki et al. (2016). For any ε1 > 0 and u large enough
Π0 (u) = P
sup
t ∈[S −λu−2 ,S ]
inf
s∈[t ,t +Tu ]
Xu (s) > fu (S )
≤P
sup
inf
t ∈[S −λu−2 ,S ] s∈[t ,t +(1−ε1 )Tu
−2 ]
Xu (s) > fu (S )
= P sup inf Xu (S + u s − u t ) > fu (S ) s∈[0,(1−ε1 )T ] t ∈[0,λ] = P sup inf Yu (t , s) > fu (S ) −2
t ∈[0,λ] s∈[0,(1−ε1 )T ]
=: Π0+ (u)
−2
L. Bai, L. Luo / Statistics and Probability Letters 120 (2017) 34–44
39
and
Π 0 ( u) ≥ P
sup
inf
t ∈[0,λ] s∈[0,(1+ε1 )T ]
Y u ( t , s) > f u ( S )
=: Π0− (u),
where Yu (t , s) := Xu (S + u−2 s − u−2 t ), for (t , s) ∈ [0, λ] × [0, (1 + ε1 )T ]. Since
σYu (t , s) :=
Var (Yu (t , s)) =
Var(Xu (S + u−2 s − u−2 t )) =
fu (S ) fu (S + u−2 s − u−2 t )
−2δ S
and (3.4), there exists d(t , s) = 1δ−ee−2δS (t − s) such that lim
sup
u→∞ (t ,s)∈[0,λ]×[0,(1+ε )T ] 1
2 u (1 − σY (t , s)) − d(t , s) = 0. u
(3.10)
Moreover, for (t1 , s1 ), (t2 , s2 ) ∈ [0, λ] × [0, (1 + ε1 )T ] and s1 − t1 > s2 − t2 , Var(Yu (t1 , s1 ) − Yu (t2 , s2 ))
X (S + u−2 s1 − u−2 t1 )
=
fu2
(S )E
=
fu2
(S )(J1 (u) + J2 (u) + J3 (u)),
u + δc (1 − e−δ(S +u
−2 s −u−2 t ) 1 1
)
−
2
X (S + u−2 s2 − u−2 t2 ) u + δc (1 − e−δ(S +u
−2 s −u−2 t ) 2 2
)
where
J1 (u) = E
X (S + u−2 s1 − u−2 t1 ) − X (S + u−2 s2 − u−2 t2 ) −2 s −u−2 t ) 1 1
u + δc (1 − e−δ(S +u
2
)
− e−δ(S +u s2 −u t2 ) ) J2 (u) = 2 − 2 − 2 −2 −2 u + δc (1 − e−δ(S +u s1 −u t1 ) ) u + δc (1 − e−δ(S +u s2 −u t2 ) ) X (S + u−2 s1 − u−2 t1 ) − X (S + u−2 s2 − u−2 t2 ) −2 −2 X (S + u s2 − u t2 ) = 0, ×E −2 −2 u + δc (1 − e−δ(S +u s1 −u t1 ) ) 2 −2 −2 −2 −2 c 2 (e−δ(S +u s1 −u t1 ) − e−δ(S +u s2 −u t2 ) ) δ E X (S + u−2 s2 − u−2 t2 ) . J3 (u) = − 2 − 2 − 2 − 2 c c u + δ (1 − e−δ(S +u s1 −u t1 ) ) u + δ (1 − e−δ(S +u s2 −u t2 ) ) c
δ
(e−δ(S +u
−2 s −u−2 t ) 1 1
,
−2
−2
Since
2
lim u2 fu2 (S )J1 (u) = lim fu2 (S )E X (S + u−2 s1 − u−2 t1 ) − X (S + u−2 s2 − u−2 t2 )
u→∞
u→∞
= lim = =
2δ e
1 − e−2δ S 2δ e−2δ S 1 − e−2δ S
σ 2 −2δ(S +u−2 s2 −u−2 t2 ) −2 −2 (e − e−2δ(S +u s1 −u t1 ) ) − e−2δS ) 2δ u2
u→∞ σ 2 (1 2δ −2δ S
((s1 − s2 ) − (t1 − t2 )) Var (B(s1 − t1 ) − B(s2 − t2 )) , −2 s −u−2 t ) 1 1
lim u2 fu2 (S )J3 (u) ≤ lim C11 (e−δ(S +u
u→∞
u→∞
−2 s −u−2 t ) 2 2
− e−δ(S +u
2 )E X (S + u−2 s2 − u−2 t2 ) = 0,
thus lim u2 Var(Yu (t1 , s1 ) − Yu (t2 , s2 )) =
u→∞
2δ e−2δ S 1 − e−2δ S
Var (B(s1 − t1 ) − B(s2 − t2 )) .
(3.11)
Further, there exist some constants G, u0 > 0, such that for any u > u0 u2 Var(Yu (t1 , s1 ) − Yu (t2 , s2 )) ≤ G(|t1 − t2 | + |s1 − s2 |)
(3.12)
holds uniformly with respect to (t1 , s1 ), (t2 , s2 ) ∈ [0, λ] × [0, (1 + ε1 )T ]. By (3.10), (3.11), (3.12), Lemma 5.1 in Dębicki et al. (2016) and limu→∞ fu (S )/u = 1/σX (S ), we obtain
(aλ, a(1 + ε1 )T )Ψ (fu (S )), Π0− (u) ∼ P
u → ∞.
(3.13)
40
L. Bai, L. Luo / Statistics and Probability Letters 120 (2017) 34–44
Similarly
(aλ, a(1 − ε1 )T )Ψ (fu (S )), Π0+ (u) ∼ P
u → ∞.
Letting ε1 → 0 and λ → ∞, we have
(aT )Ψ (fu (S )), Π0 (u) ∼ P
u → ∞.
The above combined with (3.3) and (3.9) derives (3.2), therefore by (3.1) the proof is complete.
Proof of Theorem 2.3. Case 1 δ > 0: According to the definition of conditional probability, for any x, u > 0
P u2 (S + Tu − η(u)) > x | η(u) ≤ S + Tu
sup
P
inf
t ∈[0,S −xu−2 ]
=
s∈[t ,t +Tu ]
P
sup
inf
t ∈[0,S ] s∈[t ,t +Tu ]
s s −δz σ 0 e dB(z ) − c 0 e−δz dz > u
s s σ 0 e−δz dB(z ) − c 0 e−δz dz > u
.
(3.14)
Using the same notation of X (s), X (s), fu (s), Xu (s), σX (s) as in the proof of Theorem 2.1, we have for u large enough
sup
P
inf
s∈[t ,t +Tu ] t ∈[0,S −xu−2 ]
s s −δ z −δ z σ e dB(z ) − c e dz > u 0
0
=P
sup
inf
s∈[t ,t +Tu ] t ∈[0,S −xu−2 ]
X (s)
fu (S ) fu (s)
> fu (S )
=P Set ρ(u) =
sup
inf
t ∈[0,S −xu−2 ]
ln u 2 u
s∈[t ,t +Tu ]
Xu (s) > fu (S ) .
. For any λ > 0, Bonferroni inequality yields
Π0 (u) ≤ P ∗
sup
inf
s∈[t ,t +Tu ] t ∈[0,S −xu−2 ]
Xu (s) > fu (S )
≤ Π0∗ (u) + Π1∗ (u) + Π2∗ (u),
(3.15)
where
Π0 (u) = P
∗
sup
inf
t ∈[S −xu−2 −λu−2 ,S −xu−2 ]
s∈[t ,t +Tu ]
Π1 (u) = P ∗
Xu (s) > fu (S ) ,
sup
inf
t ∈[0,S −ρ(u)] s∈[t ,t +Tu ]
Xu (s) > fu (S ) ,
Π2 (u) = P ∗
sup
inf
t ∈[S −ρ(u),S −xu−2 −λu−2 ]
s∈[t ,t +Tu ]
Xu (s) > fu (S ) .
By (3.3) and (3.9) in the proof of Theorem 2.1, we know
Π1∗ (u) = o (Ψ (fu (S ))) ,
u → ∞,
(3.16)
and
Π2 (u) ≤ P ∗
sup
t ∈[S −ρ(u),S −λu−2 ]
inf
s∈[t ,t +Tu ]
Xu (s) > fu (S )
= o (Ψ (fu (S ))) ,
u → ∞, λ → ∞.
Next we give the asymptotic behavior of Π0∗ (u) as u → ∞. For any ε1 > 0 and u large enough
Π0 (u) = P ∗
sup t ∈[S −xu−2 −λu−2 ,S −xu−2 ]
inf
s∈[t ,t +Tu ]
X (s)
=P
sup
inf
s∈[t ,t +Tu ] t ∈[S −xu−2 −λu−2 ,S −xu−2 ]
X (s)
fu (S ) f u ( s)
> fu (S )
fu (S − xu−2 ) fu (s)
> fu (S − xu ) −2
(3.17)
L. Bai, L. Luo / Statistics and Probability Letters 120 (2017) 34–44
≤P
sup
inf
t ∈[S −xu−2 −λu−2 ,S −xu−2 ] s∈[t ,t +(1−ε1 )Tu
=P
sup
fu (s)
X ( S + u− 2 s − u− 2 t − u− 2 x )
inf
Yu∗ (t , s) > fu (S − xu−2 )
inf
Yu∗ (t , s) > fu (S − xu−2 )
t ∈[0,λ] s∈[0,(1−ε1 )T ]
sup
fu (S − xu−2 )
inf
=P
−2 ]
X (s)
t ∈[0,λ] s∈[0,(1−ε1 )T ]
41
> fu (S − xu−2 )
fu (S − xu−2 ) fu (S + u−2 s − u−2 t − u−2 x)
> fu (S − xu−2 )
=: Π0∗+ (u), and
Π0∗ (u) ≥ P
sup
t ∈[0,λ] s∈[0,(1+ε1 )T ]
=: Π0∗− (u),
f (S −xu−2 ) where Yu∗ (t , s) := X (S + u−2 s − u−2 t − u−2 x) f (S +u−u 2 s−u−2 t −u−2 x) , (t , s) ∈ [0, λ] × [0, (1 + ε1 )T ] and σY2∗ (t , s) := u u
Var(Yu∗ (t , s)) =
fu (S −xu−2 ) fu (S +u−2 s−u−2 t −u−2 x)
2
.
Using the similar argumentation as (3.10) in the proof of Theorem 2.1, we have lim
sup
u→∞ (t ,s)∈[0,λ]×[0,(1+ε )T ] 1
2 u (1 − σY ∗ (t , s)) − d(t , s) = 0,
(3.18)
u
−2δ S
with d(t , s) = 1δ−ee−2δS (t − s). Moreover, (3.11), (3.12) still hold for Yu∗ (t , s) and (t1 , s1 ), (t2 , s2 ) ∈ [0, λ] × [0, (1 + ε1 )T ]. By Lemma 5.1 in Dębicki et al. (2016) and limu→∞ fu (S )/u = 1/σX (S ), we obtain
(aλ, a(1 + ε1 )T )Ψ (fu (S − xu−2 )) ∼ e−ax P (aλ, a(1 + ε1 )T )Ψ (fu (S )), Π0∗− (u) ∼ P
u → ∞.
Similarly,
(aλ, a(1 − ε1 )T )Ψ (fu (S )), Π0∗+ (u) ∼ e−ax P
u → ∞.
Letting ε1 → 0 and λ → ∞, we have
(aT )Ψ (fu (S )), Π0∗ (u) ∼ e−ax P
u → ∞.
The above combined with (3.15)–(3.17) derives that
sup
P
Xu (s) > fu (S )
inf
t ∈[0,S −xu−2 ]
s∈[t ,t +Tu ]
(aT )Ψ (fu (S )), ∼ e−ax P
u → ∞.
Thus, the claim follows by using the results of Theorem 2.1 and (3.14). Case 2 δ = 0:
sup
P
inf
s∈[t ,t +Tu ] t ∈[0,S −xu−2 ]
P u2 (S + Tu − η(u)) > xη(u) ≤ S + Tu =
P
sup
inf
(σ B(s) − cs) > u
(σ B(s) − cs) > u
.
t ∈[0,S ] s∈[t ,t +Tu ]
For u large enough
P
sup
inf
t ∈[0,S −xu−2 ]
(σ B(s) − cs) > u = P
s∈[t ,t +Tu ]
sup
inf
s∈[t ,t +Tu ] t ∈[0,S −xu−2 ]
Xu (s) > fu (S ) ,
with X (s) = σ B(s), Set ρ(u) =
ln u 2 u
B(s) X ( s) = √ , s
u + cs
√
σ s
and Xu (s) = X (s)
fu (S ) fu (s)
.
. For any λ > 0, Bonferroni inequality yields
0 (u) ≤ P Π
fu (s) =
sup
inf
s∈[t ,t +Tu ] t ∈[0,S −xu−2 ]
0 (u) + Π 1 (u) + Π 2 (u), Xu (s) > fu (S ) ≤ Π
(3.19)
42
L. Bai, L. Luo / Statistics and Probability Letters 120 (2017) 34–44
where
0 (u) = P Π
sup
inf
t ∈[S −xu−2 −λu−2 ,S −xu−2 ]
1 (u) = P Π
sup
inf
t ∈[0,S −ρ(u)] s∈[t ,t +Tu ]
s∈[t ,t +Tu ]
Xu (s) > fu (S ) ,
Xu (s) > fu (S ) ,
2 (u) = P Π
sup
inf
t ∈[S −ρ(u),S −xu−2 −λu−2 ]
s∈[t ,t +Tu ]
Xu (s) > fu (S ) .
Notice that for u large enough
1 E ( Xu (t1 ) − Xu (t2 ))2 = E
u + cS u + ct1
S
B(t1 ) −
≤ C12 E (B(t1 ) − B(t2 ))
≤ C14 |t1 − t2 |,
2
u + cS
B(t2 )
u + ct2
+ C13
2
u + cS u + ct1
−
u + cS
2
u + ct2
t1 < t2 , t1 , t2 ∈ (0, S ],
and Var Xu (t ) =
sup
t ∈[0,S −ρ(u)]
sup
fu (S ) fu (t )
t ∈[0,S −ρ(u)]
fu2 (S )
2 =
fu2
(S − ρ(u))
,
where we use the fact that fu (t ) is a decreasing function for t ∈ [0, S ] when u large enough. Moreover, 1−
fu ( S )
∼
fu (S − t )
inf
1≤k≤N (u)
1 2S
t,
t → 0,
fu (S − kλu−2 ) → ∞,
u → ∞,
and for t1 < t2 , t1 , t2 ∈ [0, S ], r X (t1 , t2 ) := E X (t1 )X (t2 ) =
t1 t2
.
Then lim
sup
u→∞ 1≤k≤N (u)
= lim
sup t1 ̸=t2 , t1 ,t2 ∈[0,λ]
sup
u→∞ 1≤k≤N (u)
−2 −2 2 −2 Var X (S − u t1 ) − X (S − u t2 ) − 1 fu (S − kλu ) 2b|t1 − t2 |
sup t1 ̸=t2 , t1 ,t2 ∈[0,λ]
−2 −2 2 f (S − kλu−2 ) 2 − 2rX (S − u t1 , S − u t2 ) − 1 = 0, u 2b|t − t | 1
(3.20)
2
where b = 2σ 12 S 2 , and sup
sup
1≤k≤N (u) |t1 −t2 |<ε t1 ,t2 ∈[0,λ]
≤ C15 u2
fu2 (S − kλu−2 )E
sup |t1 −t2 |<ε t1 ,t2 ∈[0,λ]
≤ C16 u2
sup |t1 −t2 |<ε t1 ,t2 ∈[0,λ]
≤ C17
sup
X (S − u−2 t1 ) − X (S − u−2 t2 ) X (S )
r(S − u−2 t1 , S ) − r(S − u−2 t2 , S ) X
X
S − u−2 t1 − S − u−2 t2
|t1 − t2 | → 0,
u → ∞, ε → 0.
(3.21)
|t1 −t2 |<ε t1 ,t2 ∈[0,λ]
By Theorem 8.1 in Piterbarg (1996) and Lemma 5.3 in De¸bicki et al. (2015), using the similar argumentation as in the proof of Theorem 2.1, we derive
1 (u) + Π 2 (u) = o (Ψ (fu (S ))) , Π
u → ∞, λ → ∞.
(3.22)
L. Bai, L. Luo / Statistics and Probability Letters 120 (2017) 34–44
43
0 (u) as u → ∞. For any ε1 > 0 and u large enough Next we give the asymptotic behavior of Π 0 (u) = P Π
sup
inf
s∈[t ,t +Tu ] t ∈[S −xu−2 −λu−2 ,S −xu−2 ]
X ( s)
=P
sup
inf
s∈[t ,t +Tu ] t ∈[S −xu−2 −λu−2 ,S −xu−2 ]
sup
fu (s)
> fu (S )
fu (S − xu−2 ) fu (s)
> fu (S − xu−2 )
inf
Yu (t , s) > fu (S − xu−2 )
inf
0− (u), Yu (t , s) > fu (S − xu−2 ) =: Π
≤P
X ( s)
fu (S )
t ∈[0,λ] s∈[0,(1−ε1 )T ]
0+ (u) =: Π and
0 (u) ≥ P Π
sup
t ∈[0,λ] s∈[0,(1+ε1 )T ]
f (S −xu−2 )
where Yu (t , s) := X (S + u−2 s − u−2 t − u−2 x) f (S +u−u 2 s−u−2 t −u−2 x) , for (t , s) ∈ [0, λ] × [0, (1 + ε1 )T ]. u Using the similar argumentation as (3.10)–(3.12) in the proof of Theorem 2.1, we obtain that lim
sup
u→∞ (t ,s)∈[0,λ]×[0,(1+ε )T ] 1
with d(t , s) =
1 2S
2 u (1 − σ (t , s)) − d(t , s) = 0, Y
(t − s) and σYu (t , s) :=
lim u2 Var( Yu (t1 , s1 ) − Yu (t2 , s2 )) =
u→∞
(3.23)
u
Var( Yu (t , s)), 1 S
Var (B(s1 − t1 ) − B(s2 − t2 )) ,
and for some constant G and all u large enough u2 Var( Yu (t1 , s1 ) − Yu (t2 , s2 )) ≤ G(|t1 − t2 | + |s1 − s2 |) uniformly for (t1 , s1 ), (t2 , s2 ) ∈ [0, λ] × [0, (1 + ε1 )T ]. By Lemma 5.1 in Dębicki et al. (2016) and limu→∞ fu (S )/u =
1 √ , σ S
we obtain
0− (u) ∼ P (bλ, b(1 + ε1 )T )Ψ (fu (S − xu−2 )) ∼ e−bx P (bλ, b(1 + ε1 )T )Ψ (fu (S )), Π
u → ∞.
Similarly,
0+ (u) ∼ e−bx P (bλ, b(1 − ε1 )T )Ψ (fu (S )), Π
u → ∞.
Letting ε1 → 0 and λ → ∞, we have
0 (u) ∼ e−bx P (bT )Ψ (fu (S )), Π
u → ∞.
The above combined with (3.19) and (3.22) leads to
P
sup t ∈[0,S −xu−2 ]
inf
s∈[t ,t +Tu ]
(bT )Ψ (fu (S )), Xu (s) > fu (Sx (u)) ∼ e−bx P
u → ∞.
Using the above asymptotic equality and (b) of Remark 2.2, we obtain the results.
Acknowledgment Thanks to Swiss National Science Foundation Grant no. 200021-166274. References Bai, L., De¸bicki, K., Hashorva, E., Luo, L., 2016. On generalised Piterbarg constants. Manuscript. Dębicki, K., Hashorva, E., Ji, L., 2015a. Gaussian risk model with financial constraints. Scand. Actuar. J. 2015 (6), 469–481. Dębicki, K., Hashorva, E., Ji, L., 2015b. Parisian ruin of self-similar Gaussian risk processes. J. Appl. Probab. 52 (3), 688–702. Dębicki, K., Hashorva, E., Ji, L., 2016. On Parisian ruin over a finite-time horizon. Sci. China Math. 59 (3), 557–572. De¸bicki, K., Hashorva, E., Liu, P., 2015. Ruin probabilities and passage times of γ -reflected Gaussian processes with stationary increments. http://arXiv.org/ abs/1511.09234. Dębicki, K., Mandjes, M., 2003. Exact overflow asymptotics for queues with many Gaussian inputs. J. Appl. Probab. 40 (3), 704–720. Deelstra, G., 1994. Remarks on boundary crossing result for Brownian motion. Blätter der DGVFM 21 (4), 449–456. Dieker, A.B., 2005. Extremes of Gaussian processes over an infinite horizon. Stochastic Process. Appl. 115 (2), 207–248.
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Emanuel, D.C., Harrison, J.M., Taylor, A.J., 1975. A diffusion approximation for the ruin function of a risk process with compounding assets. Scand. Actuar. J. 1975 (4), 240–247. Embrechts, P., Klüppelberg, C., Mikosch, T., 1997. Modelling Extremal Events. In: Applications of Mathematics (New York), Springer-Verlag, Berlin. Harrison, J.M., 1977. Ruin problems with compounding assets. Stochastic Process. Appl. 5 (1), 67–79. Hashorva, E., Ji, L., 2015a. Approximation of passage times of γ -reflected processes with fBm input. J. Appl. Probab. 51 (3), 713–726. Hashorva, E., Ji, L., 2015b. Piterbarg theorems for chi-processes with trend. Extremes 18 (1), 37–64. He, X., Hu, Y., 2007. Ruin probability for the integrated Gaussian process with force of interest. J. Appl. Probab. 44, 685–694. Hüsler, J., Piterbarg, V.I., 1999. Extremes of a certain class of Gaussian processes. Stochastic Process. Appl. 83 (2), 257–271. Hüsler, J., Piterbarg, V.I., 2008. A limit theorem for the time of ruin in a Gaussian ruin problem. Stochastic Process. Appl. 118 (11), 2014–2021. Piterbarg, V.I., 1996. Asymptotic Methods in the Theory of Gaussian Processes and Fields. In: Translations of Mathematical Monographs, vol. 148. American Mathematical Society, Providence, RI. Rolski, T., Schmidli, H., Schmidt, V., Teugels, J., 2009. Stochastic Processes for Insurance and Finance. John Wiley & Sons.
Contents lists available at ScienceDirect
Statistics and Probability Letters journal homepage: www.elsevier.com/locate/stapro
Parisian ruin of the Brownian motion risk model with constant force of interest Long Bai a,∗ , Li Luo a,b a
Department of Actuarial Science, University of Lausanne, UNIL-Dorigny, 1015 Lausanne, Switzerland
b
School of Mathematical Sciences and LMPC, Nankai University, Tianjin, 300071, PR China
article
abstract
info
Let B(t ), t ∈ R be a standard Brownian motion. Define a risk process
Article history: Received 23 June 2016 Received in revised form 13 September 2016 Accepted 15 September 2016 Available online 23 September 2016
Rδu (t ) = eδ t
u+c
t
e−δ s ds − σ
t
0
e−δ s dB(s) , t ≥ 0,
(0.1)
0
where u ≥ 0 is the initial reserve, δ ≥ 0 is the force of interest, c > 0 is the rate of premium and σ > 0 is a volatility factor. In this contribution we obtain an approximation of the Parisian ruin probability
MSC: primary 60G15 secondary 60G70
δ
KS (u, Tu ) := P
Keywords: Parisian ruin Ruin probability Ruin time Brownian motion
inf
sup
t ∈[0,S ] s∈[t ,t +Tu ]
δ
Ru (s) < 0 , S ≥ 0,
as u → ∞ where Tu is a bounded function. Further, we show that the Parisian ruin time of this risk process can be approximated by an exponential random variable. Our results are new even for the classical ruin probability and ruin time which correspond to Tu ≡ 0 in the Parisian setting. © 2016 Elsevier B.V. All rights reserved.
1. Introduction In a theoretical insurance model the surplus process Ru (t ) can be defined by Ru (t ) = u + ct − X (t ),
t ≥ 0,
see Embrechts et al. (1997), where u ≥ 0 is the initial reserve, c > 0 is the rate of premium and X (t ), t ≥ 0 denotes the aggregate claims process. More specifically, we assume that the aggregate claims process is a Brownian motion, i.e., X (t ) = σ B(t ), σ > 0. Due to the nature of the financial market, we shall consider a more general surplus process including interest rate, see Rolski et al. (2009), called a risk reserve process with constant force of interest, i.e., Rδu (t ), t ≥ 0, in (0.1). See Rolski et al. (2009), Dębicki et al. (2015a) and He and Hu (2007) for more studies on risk models with force of interest. During the time horizon [0, S ], S ∈ (0, ∞], the classical ruin probability is defined as below δ
ψS (u) := P
∗
δ
inf Ru (t ) < 0 ,
t ∈[0,S ]
Corresponding author. E-mail addresses: [email protected] (L. Bai), [email protected] (L. Luo).
http://dx.doi.org/10.1016/j.spl.2016.09.011 0167-7152/© 2016 Elsevier B.V. All rights reserved.
(1.1)
L. Bai, L. Luo / Statistics and Probability Letters 120 (2017) 34–44
35
see Embrechts et al. (1997), Hüsler and Piterbarg (1999), Hüsler and Piterbarg (2008) and Dieker (2005). In Emanuel et al. δ (1975) and Harrison (1977) the exact formula of ψ∞ (u) for δ > 0 is shown to be
δ ψ∞ (u) =
Ψ
2δ
2c 2
u + σ 2δ ,
σ2
u > 0,
2c 2
Ψ
σ 2δ
where Ψ (x) = 1 − Φ (x) with Φ (·) the distribution function of an N (0, 1) random variable. 0 For δ = 0, the exact value of ψ∞ (u) is well-known (cf. Deelstra, 1994) with 0 ψ∞ (u) = e
− 2cu 2 σ
,
u > 0.
In the literature, there are no results for the classical ruin probability in the case of finite time horizon, i.e., S ∈ (0, ∞). For S ∈ (0, ∞), with motivation from the recent contributions (Dębicki et al., 2015b, 2016) we shall investigate in this paper the Parisian ruin probability over the time period [0, S ] defined as
KSδ (u, Tu ) := P
inf
sup
t ∈[0,S ] s∈[t ,t +Tu ]
Rδu (s) < 0 ,
(1.2)
where Tu ≥ 0 models the pre-specified time. Our assumption on Tu is that lim Tu u2 = T ∈ [0, ∞)
u→∞
and thus ψSδ (u) is a special case of KSδ (u, Tu ) with Tu ≡ 0. Another quantity of interest is the conditional distribution of the ruin time for the surplus process Rδu (t ). The classical ruin time, e.g., Dębicki et al. (2015a), Hashorva and Ji (2015a) and Hüsler and Piterbarg (2008), is defined as
τ (u) = inf{t > 0 : Rδu (t ) < 0}.
(1.3) δ
Here as in Dębicki et al. (2015b) we define the Parisian ruin time of the risk process Ru (t ) by
η(u) = inf{t ≥ Tu : t − κt ,u ≥ Tu , Rδu (t ) < 0},
with κt ,u = sup{s ∈ [0, t ] : Rδu (s) ≥ 0},
(1.4)
and τ (u) is a special case of η(u) with Tu ≡ 0. Brief organization of the rest of the paper: in Section 2 we first present our main results on the asymptotics of KSδ (u, Tu ) as u → ∞ and then we display the approximation of the Parisian ruin time. All the proofs are relegated to Section 3. 2. Main results Before giving the main results, we shall introduce a generalized Piterbarg constant as
(T ) = lim P (λ, T ), P λ→∞
T ≥ 0,
(2.1)
where for λ, T ≥ 0
(λ, T ) = E P
√
sup
inf e
2B(t −s)−|t −s|−(t −s)
t ∈[0,λ] s∈[0,T ]
.
(0) and P11 [0, ∞) = 2, see Dębicki and Mandjes (2003), Note further that the classical Piterbarg constant P11 [0, ∞) equals P Bai et al. (2016) and Hashorva and Ji (2015b). Throughout this paper ∼ means asymptotic equivalence when the argument tends to 0 or ∞. Recall that Ψ (·) denotes the tail distribution function of an N (0, 1) random variable and Ψ (u) ∼ √ 1 e− 2π u
u2 2
, u → ∞.
Theorem 2.1. For δ > 0, S > 0 and limu→∞ Tu u2 = T ∈ [0, ∞), we have
√ δ
(aT )Ψ K S ( u, T u ) ∼ P 2 −2δ S
where a := σ 22(1δ−ee−2δS )2 .
2δ u + δc (1 − e−δ S )
√ σ 1 − e−2δS
,
u → ∞,
(2.2)
36
L. Bai, L. Luo / Statistics and Probability Letters 120 (2017) 34–44
Remark 2.2. (a) When Tu ≡ 0, KSδ (u, Tu ) reduces to the classical ruin probability ψSδ (u), and by Theorem 2.1 with T = 0
√ δ
δ
KS (u, 0) = ψS (u) ∼ 2Ψ
2δ u + δc (1 − e−δ S )
√ σ 1 − e−2δS
,
u → ∞.
(b) If δ = 0
KS0 (u, Tu ) = P
inf
sup
t ∈[0,S ] s∈[t ,t +Tu ]
(bT )Ψ ∼P
(u + cs − σ B(s)) < 0
u + cS
√ σ S
,
u → ∞,
(2.3)
where b := 2σ 12 S 2 and we used the result of corollary 3.4(ii) in Dębicki et al. (2016). Further, if δ = 0 and Tu ≡ 0, by (2.3) with T = 0, we get the asymptotic result of the classical ruin probability
ψS0 (u) ∼ 2Ψ
u + cS
√
σ S
,
u → ∞.
(2.4)
In fact, Deelstra (1994) gave the exact result of ψS0 (u), u > 0, i.e.,
cS − u − 2cu + e σ2 Φ √ σ S σ S u + cS ∼ 2Ψ , u → ∞, √ σ S
ψS0 (u) = Ψ
u + cS
√
which follows from e lim
u→∞
− 2cu 2 σ
Ψ
Φ
cS√ −u σ S
u+ √cS σ S
− 2cu 2
= lim
− σ2c2 e
σ
Φ
u→∞
−σ
cS√ −u σ S
√1
2π S
−
√1
e
−
σ 2π S 2 − u+√cS /2
e
u+ √cS σ S
2
/2
= 1.
σ S
Our next result discusses the approximation of the conditional ruin time. Theorem 2.3. Let η(u) satisfy (1.4), under the assumptions of Theorem 2.1, we have for any x > 0 and δ ≥ 0,
exp (−ax) , P u (S + Tu − η(u)) > x η(u) ≤ S + Tu ∼ exp (−bx) ,
2
if δ > 0, if δ = 0,
u → ∞,
2 −2δ S
where a := σ 22(1δ−ee−2δS )2 and b := 2σ 12 S 2 . Remark 2.4. If Tu ≡ 0, then η(u) = τ (u) and by Theorem 2.3, we obtain as u → ∞
P u2 (S − τ (u)) > x | τ (u) ≤ S ∼
exp (−ax) , exp (−bx) ,
if δ > 0, if δ = 0.
3. Proofs Hereafter we assume that Ci , i ∈ N are some positive constants. Proof of Theorem 2.1. For S > 0 and u large enough δ
KS (u, Tu ) = P
sup
inf
t ∈[0,S ] s∈[t ,t +Tu ]
=P
sup
sup
0
fu (S )
0
> fu (S )
inf
X (s)
inf
Xu (s) > fu (S ) ,
t ∈[0,S ] s∈[t ,t +Tu ]
=P
s s −δ z −δ z σ e dB(z ) − c e dz > u
t ∈[0,S ] s∈[t ,t +Tu ]
fu (s)
(2.5)
L. Bai, L. Luo / Statistics and Probability Letters 120 (2017) 34–44
37
with X ( s) = σ
X (s)
s
e−δ z dB(z ),
X (s) =
σX (s)
0
fu (s) =
,
u + δc (1 − e−δ s )
σX (s)
and
Xu (s) = X (s)
fu (S ) fu (s)
,
where σX2 (s) is the variance of X (s) with σX2 (s) = σ2δ (1 − e−2δ s ). 2 Set ρ(u) = lnuu and for any λ > 0, Bonferroni inequality yields 2
Π0 (u) := P
sup
inf
s∈[t ,t +Tu ] t ∈[S −λu−2 ,S ]
Xu (s) > fu (S )
≤ KSδ (u, Tu ) ≤ Π0 (u) + Π1 (u) + Π2 (u),
(3.1)
where
Π 1 ( u) = P
sup
inf
t ∈[0,S −ρ(u)] s∈[t ,t +Tu ]
Xu (s) > fu (S ) ,
Π2 (u) = P
sup
t ∈[S −ρ(u),S −λu−2 ]
inf
s∈[t ,t +Tu ]
Xu (s) > fu (S ) .
First we give some upper bounds of Π1 (u) and Π2 (u) which finally show that
Π1 (u) + Π2 (u) = o (Π0 (u)) ,
u → ∞.
(3.2)
For all u large
2 f ( S ) f ( S ) u u ) X (t1 ) E (Xu (t1 ) − Xu (t2 ))2 = E − X ( t 2 u + δc 1 − e−δ t1 u + δc 1 − e−δ t2 2 2 c c −δ S −δ S t2 u + u + ( 1 − e ) ( 1 − e ) δ δ ≤ C1 E e−δ z dB(z ) + C2 − u + δc (1 − e−δ t1 ) u + δc (1 − e−δ t2 ) t1 t1 < t2 , t1 , t2 ∈ (0, S ].
≤ C3 |t1 − t2 |, Moreover, sup
t ∈[0,S −ρ(u)]
Var (Xu (t )) =
sup
fu ( S )
=
fu (t )
t ∈[0,S −ρ(u)]
fu2 (S )
2 fu2
(S − ρ(u))
,
where we use the fact that fu (t ) is a decreasing function for t ∈ [0, S ] when u large enough. Therefore, by Theorem 8.1 in Piterbarg (1996), we obtain
Π 1 ( u) ≤ P
sup
t ∈[0,S −ρ(u)]
Xu (t ) > fu (S )
≤ C4 u2 Ψ (fu (S − ρ(u))) ,
(3.3)
and direct calculation yields that u Ψ (fu (S − ρ(u))) ≤ √
u2
2
−
2π fu (S )
2 −a(ln u)2
∼u e
e
fu2 (S ) 2
fu2 (S −ρ(u)) −1 fu2 (S )
e−
fu2 (S ) 2
Ψ (fu (S )) = o (Ψ (fu (S ))) ,
u → ∞,
2 −2δ S
where a = σ 22(1δ−ee−2δS )2 and we use the fact that
δ e−2δS t, 1 − e−2δ S
t → 0.
△k = kλu−2 , (k + 1)λu−2 ,
k ∈ N,
1−
fu (S ) fu (S − t )
∼
(3.4)
Set and N (u) = λ−1 ρ(u)u2 ,
where ⌊·⌋ stands for the ceiling function, then
Π 2 ( u) ≤ P
sup
t ∈[S −ρ(u),S −λu−2 ]
Xu (t ) > fu (S )
=P
sup
t ∈[λu−2 ,ρ(u)]
Xu (S − t ) > fu (S )
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L. Bai, L. Luo / Statistics and Probability Letters 120 (2017) 34–44
≤
N (u)
P sup Xu (S − t ) > fu (S ) t ∈∆k
k=1 N (u)
≤
P sup X (S − t ) > fu (S − kλu−2 )
t ∈∆0
k=1 N (u)
≤
P
k=1
sup X (S − u−2 t ) > fu (S − kλu−2 ) .
(3.5)
t ∈[0,λ]
Clearly, inf
1≤k≤N (u)
fu (S − kλu−2 ) → ∞,
u → ∞,
(3.6)
and for t1 < t2 , t1 , t2 ∈ [0, S ],
rX (t1 , t2 ) := E X (t1 )X (t2 ) =
1 − e−2δ t1 1 − e−2δ t2
.
Further,
−2 −2 2 −2 Var X (S − u t1 ) − X (S − u t2 ) − 1 lim sup sup fu (S − kλu ) u→∞ 1≤k≤N (u) t ̸=t , 2a|t1 − t2 | 1 2 t1 ,t2 ∈[0,λ] 2 − 2rX (S − u−2 t1 , S − u−2 t2 ) = lim sup sup fu2 (S − kλu−2 ) − 1 u→∞ 1≤k≤N (u) t ̸=t , 2a|t1 − t2 | 1 2 t1 ,t2 ∈[0,λ]
= 0,
(3.7)
and sup
sup
1≤k≤N (u) |t1 −t2 |<ε t1 ,t2 ∈[0,λ]
≤ C5 u2
fu2 (S − kλu−2 )E
X (S − u−2 t1 ) − X (S − u−2 t2 ) X (S )
rX (S − u−2 t1 , S ) − rX (S − u−2 t2 , S )
sup |t1 −t2 |<ε t1 ,t2 ∈[0,λ]
≤ C6 u2
sup |t1 −t2 |<ε t1 ,t2 ∈[0,λ]
≤ C7
−2 −2 1 − e−2δ(S −u t1 ) − 1 − e−2δ(S −u t2 )
|t1 − t2 | → 0,
sup
u → ∞, ε → 0.
(3.8)
|t1 −t2 |<ε t1 ,t2 ∈[0,λ]
According to (3.6), (3.7), (3.8) and Lemma 5.3 of De¸bicki et al. (2015), (3.5) is followed by
Π2 (u) ≤ C8 λ
N (u)
∞ Ψ fu (S − kλu−2 ) ≤ C9 Ψ (fu (S ))λ e−C10 kλ = o (Ψ (fu (S ))) ,
k =1
u → ∞, λ → ∞,
(3.9)
k=1
where the last inequality follows from (3.4). Next we give the asymptotic behavior of Π0 (u) as u → ∞ based on an appropriate application of the Appendix in Dębicki et al. (2016). For any ε1 > 0 and u large enough
Π0 (u) = P
sup
t ∈[S −λu−2 ,S ]
inf
s∈[t ,t +Tu ]
Xu (s) > fu (S )
≤P
sup
inf
t ∈[S −λu−2 ,S ] s∈[t ,t +(1−ε1 )Tu
−2 ]
Xu (s) > fu (S )
= P sup inf Xu (S + u s − u t ) > fu (S ) s∈[0,(1−ε1 )T ] t ∈[0,λ] = P sup inf Yu (t , s) > fu (S ) −2
t ∈[0,λ] s∈[0,(1−ε1 )T ]
=: Π0+ (u)
−2
L. Bai, L. Luo / Statistics and Probability Letters 120 (2017) 34–44
39
and
Π 0 ( u) ≥ P
sup
inf
t ∈[0,λ] s∈[0,(1+ε1 )T ]
Y u ( t , s) > f u ( S )
=: Π0− (u),
where Yu (t , s) := Xu (S + u−2 s − u−2 t ), for (t , s) ∈ [0, λ] × [0, (1 + ε1 )T ]. Since
σYu (t , s) :=
Var (Yu (t , s)) =
Var(Xu (S + u−2 s − u−2 t )) =
fu (S ) fu (S + u−2 s − u−2 t )
−2δ S
and (3.4), there exists d(t , s) = 1δ−ee−2δS (t − s) such that lim
sup
u→∞ (t ,s)∈[0,λ]×[0,(1+ε )T ] 1
2 u (1 − σY (t , s)) − d(t , s) = 0. u
(3.10)
Moreover, for (t1 , s1 ), (t2 , s2 ) ∈ [0, λ] × [0, (1 + ε1 )T ] and s1 − t1 > s2 − t2 , Var(Yu (t1 , s1 ) − Yu (t2 , s2 ))
X (S + u−2 s1 − u−2 t1 )
=
fu2
(S )E
=
fu2
(S )(J1 (u) + J2 (u) + J3 (u)),
u + δc (1 − e−δ(S +u
−2 s −u−2 t ) 1 1
)
−
2
X (S + u−2 s2 − u−2 t2 ) u + δc (1 − e−δ(S +u
−2 s −u−2 t ) 2 2
)
where
J1 (u) = E
X (S + u−2 s1 − u−2 t1 ) − X (S + u−2 s2 − u−2 t2 ) −2 s −u−2 t ) 1 1
u + δc (1 − e−δ(S +u
2
)
− e−δ(S +u s2 −u t2 ) ) J2 (u) = 2 − 2 − 2 −2 −2 u + δc (1 − e−δ(S +u s1 −u t1 ) ) u + δc (1 − e−δ(S +u s2 −u t2 ) ) X (S + u−2 s1 − u−2 t1 ) − X (S + u−2 s2 − u−2 t2 ) −2 −2 X (S + u s2 − u t2 ) = 0, ×E −2 −2 u + δc (1 − e−δ(S +u s1 −u t1 ) ) 2 −2 −2 −2 −2 c 2 (e−δ(S +u s1 −u t1 ) − e−δ(S +u s2 −u t2 ) ) δ E X (S + u−2 s2 − u−2 t2 ) . J3 (u) = − 2 − 2 − 2 − 2 c c u + δ (1 − e−δ(S +u s1 −u t1 ) ) u + δ (1 − e−δ(S +u s2 −u t2 ) ) c
δ
(e−δ(S +u
−2 s −u−2 t ) 1 1
,
−2
−2
Since
2
lim u2 fu2 (S )J1 (u) = lim fu2 (S )E X (S + u−2 s1 − u−2 t1 ) − X (S + u−2 s2 − u−2 t2 )
u→∞
u→∞
= lim = =
2δ e
1 − e−2δ S 2δ e−2δ S 1 − e−2δ S
σ 2 −2δ(S +u−2 s2 −u−2 t2 ) −2 −2 (e − e−2δ(S +u s1 −u t1 ) ) − e−2δS ) 2δ u2
u→∞ σ 2 (1 2δ −2δ S
((s1 − s2 ) − (t1 − t2 )) Var (B(s1 − t1 ) − B(s2 − t2 )) , −2 s −u−2 t ) 1 1
lim u2 fu2 (S )J3 (u) ≤ lim C11 (e−δ(S +u
u→∞
u→∞
−2 s −u−2 t ) 2 2
− e−δ(S +u
2 )E X (S + u−2 s2 − u−2 t2 ) = 0,
thus lim u2 Var(Yu (t1 , s1 ) − Yu (t2 , s2 )) =
u→∞
2δ e−2δ S 1 − e−2δ S
Var (B(s1 − t1 ) − B(s2 − t2 )) .
(3.11)
Further, there exist some constants G, u0 > 0, such that for any u > u0 u2 Var(Yu (t1 , s1 ) − Yu (t2 , s2 )) ≤ G(|t1 − t2 | + |s1 − s2 |)
(3.12)
holds uniformly with respect to (t1 , s1 ), (t2 , s2 ) ∈ [0, λ] × [0, (1 + ε1 )T ]. By (3.10), (3.11), (3.12), Lemma 5.1 in Dębicki et al. (2016) and limu→∞ fu (S )/u = 1/σX (S ), we obtain
(aλ, a(1 + ε1 )T )Ψ (fu (S )), Π0− (u) ∼ P
u → ∞.
(3.13)
40
L. Bai, L. Luo / Statistics and Probability Letters 120 (2017) 34–44
Similarly
(aλ, a(1 − ε1 )T )Ψ (fu (S )), Π0+ (u) ∼ P
u → ∞.
Letting ε1 → 0 and λ → ∞, we have
(aT )Ψ (fu (S )), Π0 (u) ∼ P
u → ∞.
The above combined with (3.3) and (3.9) derives (3.2), therefore by (3.1) the proof is complete.
Proof of Theorem 2.3. Case 1 δ > 0: According to the definition of conditional probability, for any x, u > 0
P u2 (S + Tu − η(u)) > x | η(u) ≤ S + Tu
sup
P
inf
t ∈[0,S −xu−2 ]
=
s∈[t ,t +Tu ]
P
sup
inf
t ∈[0,S ] s∈[t ,t +Tu ]
s s −δz σ 0 e dB(z ) − c 0 e−δz dz > u
s s σ 0 e−δz dB(z ) − c 0 e−δz dz > u
.
(3.14)
Using the same notation of X (s), X (s), fu (s), Xu (s), σX (s) as in the proof of Theorem 2.1, we have for u large enough
sup
P
inf
s∈[t ,t +Tu ] t ∈[0,S −xu−2 ]
s s −δ z −δ z σ e dB(z ) − c e dz > u 0
0
=P
sup
inf
s∈[t ,t +Tu ] t ∈[0,S −xu−2 ]
X (s)
fu (S ) fu (s)
> fu (S )
=P Set ρ(u) =
sup
inf
t ∈[0,S −xu−2 ]
ln u 2 u
s∈[t ,t +Tu ]
Xu (s) > fu (S ) .
. For any λ > 0, Bonferroni inequality yields
Π0 (u) ≤ P ∗
sup
inf
s∈[t ,t +Tu ] t ∈[0,S −xu−2 ]
Xu (s) > fu (S )
≤ Π0∗ (u) + Π1∗ (u) + Π2∗ (u),
(3.15)
where
Π0 (u) = P
∗
sup
inf
t ∈[S −xu−2 −λu−2 ,S −xu−2 ]
s∈[t ,t +Tu ]
Π1 (u) = P ∗
Xu (s) > fu (S ) ,
sup
inf
t ∈[0,S −ρ(u)] s∈[t ,t +Tu ]
Xu (s) > fu (S ) ,
Π2 (u) = P ∗
sup
inf
t ∈[S −ρ(u),S −xu−2 −λu−2 ]
s∈[t ,t +Tu ]
Xu (s) > fu (S ) .
By (3.3) and (3.9) in the proof of Theorem 2.1, we know
Π1∗ (u) = o (Ψ (fu (S ))) ,
u → ∞,
(3.16)
and
Π2 (u) ≤ P ∗
sup
t ∈[S −ρ(u),S −λu−2 ]
inf
s∈[t ,t +Tu ]
Xu (s) > fu (S )
= o (Ψ (fu (S ))) ,
u → ∞, λ → ∞.
Next we give the asymptotic behavior of Π0∗ (u) as u → ∞. For any ε1 > 0 and u large enough
Π0 (u) = P ∗
sup t ∈[S −xu−2 −λu−2 ,S −xu−2 ]
inf
s∈[t ,t +Tu ]
X (s)
=P
sup
inf
s∈[t ,t +Tu ] t ∈[S −xu−2 −λu−2 ,S −xu−2 ]
X (s)
fu (S ) f u ( s)
> fu (S )
fu (S − xu−2 ) fu (s)
> fu (S − xu ) −2
(3.17)
L. Bai, L. Luo / Statistics and Probability Letters 120 (2017) 34–44
≤P
sup
inf
t ∈[S −xu−2 −λu−2 ,S −xu−2 ] s∈[t ,t +(1−ε1 )Tu
=P
sup
fu (s)
X ( S + u− 2 s − u− 2 t − u− 2 x )
inf
Yu∗ (t , s) > fu (S − xu−2 )
inf
Yu∗ (t , s) > fu (S − xu−2 )
t ∈[0,λ] s∈[0,(1−ε1 )T ]
sup
fu (S − xu−2 )
inf
=P
−2 ]
X (s)
t ∈[0,λ] s∈[0,(1−ε1 )T ]
41
> fu (S − xu−2 )
fu (S − xu−2 ) fu (S + u−2 s − u−2 t − u−2 x)
> fu (S − xu−2 )
=: Π0∗+ (u), and
Π0∗ (u) ≥ P
sup
t ∈[0,λ] s∈[0,(1+ε1 )T ]
=: Π0∗− (u),
f (S −xu−2 ) where Yu∗ (t , s) := X (S + u−2 s − u−2 t − u−2 x) f (S +u−u 2 s−u−2 t −u−2 x) , (t , s) ∈ [0, λ] × [0, (1 + ε1 )T ] and σY2∗ (t , s) := u u
Var(Yu∗ (t , s)) =
fu (S −xu−2 ) fu (S +u−2 s−u−2 t −u−2 x)
2
.
Using the similar argumentation as (3.10) in the proof of Theorem 2.1, we have lim
sup
u→∞ (t ,s)∈[0,λ]×[0,(1+ε )T ] 1
2 u (1 − σY ∗ (t , s)) − d(t , s) = 0,
(3.18)
u
−2δ S
with d(t , s) = 1δ−ee−2δS (t − s). Moreover, (3.11), (3.12) still hold for Yu∗ (t , s) and (t1 , s1 ), (t2 , s2 ) ∈ [0, λ] × [0, (1 + ε1 )T ]. By Lemma 5.1 in Dębicki et al. (2016) and limu→∞ fu (S )/u = 1/σX (S ), we obtain
(aλ, a(1 + ε1 )T )Ψ (fu (S − xu−2 )) ∼ e−ax P (aλ, a(1 + ε1 )T )Ψ (fu (S )), Π0∗− (u) ∼ P
u → ∞.
Similarly,
(aλ, a(1 − ε1 )T )Ψ (fu (S )), Π0∗+ (u) ∼ e−ax P
u → ∞.
Letting ε1 → 0 and λ → ∞, we have
(aT )Ψ (fu (S )), Π0∗ (u) ∼ e−ax P
u → ∞.
The above combined with (3.15)–(3.17) derives that
sup
P
Xu (s) > fu (S )
inf
t ∈[0,S −xu−2 ]
s∈[t ,t +Tu ]
(aT )Ψ (fu (S )), ∼ e−ax P
u → ∞.
Thus, the claim follows by using the results of Theorem 2.1 and (3.14). Case 2 δ = 0:
sup
P
inf
s∈[t ,t +Tu ] t ∈[0,S −xu−2 ]
P u2 (S + Tu − η(u)) > xη(u) ≤ S + Tu =
P
sup
inf
(σ B(s) − cs) > u
(σ B(s) − cs) > u
.
t ∈[0,S ] s∈[t ,t +Tu ]
For u large enough
P
sup
inf
t ∈[0,S −xu−2 ]
(σ B(s) − cs) > u = P
s∈[t ,t +Tu ]
sup
inf
s∈[t ,t +Tu ] t ∈[0,S −xu−2 ]
Xu (s) > fu (S ) ,
with X (s) = σ B(s), Set ρ(u) =
ln u 2 u
B(s) X ( s) = √ , s
u + cs
√
σ s
and Xu (s) = X (s)
fu (S ) fu (s)
.
. For any λ > 0, Bonferroni inequality yields
0 (u) ≤ P Π
fu (s) =
sup
inf
s∈[t ,t +Tu ] t ∈[0,S −xu−2 ]
0 (u) + Π 1 (u) + Π 2 (u), Xu (s) > fu (S ) ≤ Π
(3.19)
42
L. Bai, L. Luo / Statistics and Probability Letters 120 (2017) 34–44
where
0 (u) = P Π
sup
inf
t ∈[S −xu−2 −λu−2 ,S −xu−2 ]
1 (u) = P Π
sup
inf
t ∈[0,S −ρ(u)] s∈[t ,t +Tu ]
s∈[t ,t +Tu ]
Xu (s) > fu (S ) ,
Xu (s) > fu (S ) ,
2 (u) = P Π
sup
inf
t ∈[S −ρ(u),S −xu−2 −λu−2 ]
s∈[t ,t +Tu ]
Xu (s) > fu (S ) .
Notice that for u large enough
1 E ( Xu (t1 ) − Xu (t2 ))2 = E
u + cS u + ct1
S
B(t1 ) −
≤ C12 E (B(t1 ) − B(t2 ))
≤ C14 |t1 − t2 |,
2
u + cS
B(t2 )
u + ct2
+ C13
2
u + cS u + ct1
−
u + cS
2
u + ct2
t1 < t2 , t1 , t2 ∈ (0, S ],
and Var Xu (t ) =
sup
t ∈[0,S −ρ(u)]
sup
fu (S ) fu (t )
t ∈[0,S −ρ(u)]
fu2 (S )
2 =
fu2
(S − ρ(u))
,
where we use the fact that fu (t ) is a decreasing function for t ∈ [0, S ] when u large enough. Moreover, 1−
fu ( S )
∼
fu (S − t )
inf
1≤k≤N (u)
1 2S
t,
t → 0,
fu (S − kλu−2 ) → ∞,
u → ∞,
and for t1 < t2 , t1 , t2 ∈ [0, S ], r X (t1 , t2 ) := E X (t1 )X (t2 ) =
t1 t2
.
Then lim
sup
u→∞ 1≤k≤N (u)
= lim
sup t1 ̸=t2 , t1 ,t2 ∈[0,λ]
sup
u→∞ 1≤k≤N (u)
−2 −2 2 −2 Var X (S − u t1 ) − X (S − u t2 ) − 1 fu (S − kλu ) 2b|t1 − t2 |
sup t1 ̸=t2 , t1 ,t2 ∈[0,λ]
−2 −2 2 f (S − kλu−2 ) 2 − 2rX (S − u t1 , S − u t2 ) − 1 = 0, u 2b|t − t | 1
(3.20)
2
where b = 2σ 12 S 2 , and sup
sup
1≤k≤N (u) |t1 −t2 |<ε t1 ,t2 ∈[0,λ]
≤ C15 u2
fu2 (S − kλu−2 )E
sup |t1 −t2 |<ε t1 ,t2 ∈[0,λ]
≤ C16 u2
sup |t1 −t2 |<ε t1 ,t2 ∈[0,λ]
≤ C17
sup
X (S − u−2 t1 ) − X (S − u−2 t2 ) X (S )
r(S − u−2 t1 , S ) − r(S − u−2 t2 , S ) X
X
S − u−2 t1 − S − u−2 t2
|t1 − t2 | → 0,
u → ∞, ε → 0.
(3.21)
|t1 −t2 |<ε t1 ,t2 ∈[0,λ]
By Theorem 8.1 in Piterbarg (1996) and Lemma 5.3 in De¸bicki et al. (2015), using the similar argumentation as in the proof of Theorem 2.1, we derive
1 (u) + Π 2 (u) = o (Ψ (fu (S ))) , Π
u → ∞, λ → ∞.
(3.22)
L. Bai, L. Luo / Statistics and Probability Letters 120 (2017) 34–44
43
0 (u) as u → ∞. For any ε1 > 0 and u large enough Next we give the asymptotic behavior of Π 0 (u) = P Π
sup
inf
s∈[t ,t +Tu ] t ∈[S −xu−2 −λu−2 ,S −xu−2 ]
X ( s)
=P
sup
inf
s∈[t ,t +Tu ] t ∈[S −xu−2 −λu−2 ,S −xu−2 ]
sup
fu (s)
> fu (S )
fu (S − xu−2 ) fu (s)
> fu (S − xu−2 )
inf
Yu (t , s) > fu (S − xu−2 )
inf
0− (u), Yu (t , s) > fu (S − xu−2 ) =: Π
≤P
X ( s)
fu (S )
t ∈[0,λ] s∈[0,(1−ε1 )T ]
0+ (u) =: Π and
0 (u) ≥ P Π
sup
t ∈[0,λ] s∈[0,(1+ε1 )T ]
f (S −xu−2 )
where Yu (t , s) := X (S + u−2 s − u−2 t − u−2 x) f (S +u−u 2 s−u−2 t −u−2 x) , for (t , s) ∈ [0, λ] × [0, (1 + ε1 )T ]. u Using the similar argumentation as (3.10)–(3.12) in the proof of Theorem 2.1, we obtain that lim
sup
u→∞ (t ,s)∈[0,λ]×[0,(1+ε )T ] 1
with d(t , s) =
1 2S
2 u (1 − σ (t , s)) − d(t , s) = 0, Y
(t − s) and σYu (t , s) :=
lim u2 Var( Yu (t1 , s1 ) − Yu (t2 , s2 )) =
u→∞
(3.23)
u
Var( Yu (t , s)), 1 S
Var (B(s1 − t1 ) − B(s2 − t2 )) ,
and for some constant G and all u large enough u2 Var( Yu (t1 , s1 ) − Yu (t2 , s2 )) ≤ G(|t1 − t2 | + |s1 − s2 |) uniformly for (t1 , s1 ), (t2 , s2 ) ∈ [0, λ] × [0, (1 + ε1 )T ]. By Lemma 5.1 in Dębicki et al. (2016) and limu→∞ fu (S )/u =
1 √ , σ S
we obtain
0− (u) ∼ P (bλ, b(1 + ε1 )T )Ψ (fu (S − xu−2 )) ∼ e−bx P (bλ, b(1 + ε1 )T )Ψ (fu (S )), Π
u → ∞.
Similarly,
0+ (u) ∼ e−bx P (bλ, b(1 − ε1 )T )Ψ (fu (S )), Π
u → ∞.
Letting ε1 → 0 and λ → ∞, we have
0 (u) ∼ e−bx P (bT )Ψ (fu (S )), Π
u → ∞.
The above combined with (3.19) and (3.22) leads to
P
sup t ∈[0,S −xu−2 ]
inf
s∈[t ,t +Tu ]
(bT )Ψ (fu (S )), Xu (s) > fu (Sx (u)) ∼ e−bx P
u → ∞.
Using the above asymptotic equality and (b) of Remark 2.2, we obtain the results.
Acknowledgment Thanks to Swiss National Science Foundation Grant no. 200021-166274. References Bai, L., De¸bicki, K., Hashorva, E., Luo, L., 2016. On generalised Piterbarg constants. Manuscript. Dębicki, K., Hashorva, E., Ji, L., 2015a. Gaussian risk model with financial constraints. Scand. Actuar. J. 2015 (6), 469–481. Dębicki, K., Hashorva, E., Ji, L., 2015b. Parisian ruin of self-similar Gaussian risk processes. J. Appl. Probab. 52 (3), 688–702. Dębicki, K., Hashorva, E., Ji, L., 2016. On Parisian ruin over a finite-time horizon. Sci. China Math. 59 (3), 557–572. De¸bicki, K., Hashorva, E., Liu, P., 2015. Ruin probabilities and passage times of γ -reflected Gaussian processes with stationary increments. http://arXiv.org/ abs/1511.09234. Dębicki, K., Mandjes, M., 2003. Exact overflow asymptotics for queues with many Gaussian inputs. J. Appl. Probab. 40 (3), 704–720. Deelstra, G., 1994. Remarks on boundary crossing result for Brownian motion. Blätter der DGVFM 21 (4), 449–456. Dieker, A.B., 2005. Extremes of Gaussian processes over an infinite horizon. Stochastic Process. Appl. 115 (2), 207–248.
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