Extreme values of a portfolio of Gaussian processes and a trend

Extreme values of a portfolio of Gaussian processes and a trend

Ju¨rg Hu¨sler_&Christoph M. Schmid

Received: 24 June 2003 / Revised: 1 December 2005 / Accepted: 2 December 2005

#Springer Science + Business Media, LLC 2006

Abstract We consider the extreme values of a portfolio of independent continuous Gaussian processes Pk

i¼1wiXiðtÞ (wi2R; k2N) which are asymptotically locally stationary, with expectationsE½XiðtÞ ¼0 and variancesVar½XiðtÞ ¼dit^2Hi ðdi2R^þ; 0<Hi<1Þ, and a trendctfor some constants;c>0 with >Hi. We derive the probabilityPfsupt>0Pk

i¼1w_iX_iðtÞ ct>ugfor u! 1, which may be interpreted as ruin probability.

Keywords Gaussian processes, Extreme values, Portfolio of assets, Tail behavior, Ruin probability, Large deviations

AMS 2000 Subject Classification Primary— 60G15, 62G32, 91B28 1. Introduction

The tail behavior of stochastic processes is important e.g., for calculating ruin probabilities in insurance or finance. In this context we consider in this paper particular Gaussian processes, to determine the probability that a Gaussian process YðtÞexceeds a certain boundaryu2Rin an intervalT2R

PðuÞ ¼P sup

t2T

YðtÞ>u

:

In general, it is almost impossible to find the distribution of this supremum.

Precise formulas are only known for a couple of stationary processes in a finite or infinite interval (cf. Adler (1990)). The best we can do in general, is to derive the asymptotic behavior of PðuÞ whenu! 1. This asymptotic behavior is sufficiently

DOI 10.1007/s10687-006-7966-9

J. Hu¨sler (*)

:

C. M. Schmid

Institut fu¨r Mathematische Statistik und Versicherungslehre, Universita¨t Bern, Sidlerstrasse 5,

3012 Bern, Switzerland

e-mail: juerg.huesler@stat.unibe.ch C. Schmid

e-mail: christoph.schmid@stat.unibe.ch

interesting for its own. However, we are going to derive the probability that the Gaussian processYðtÞ ¼XðtÞ ct, with trend, exceeds some boundaryu

P sup

t>0

ðXðtÞ ctÞ>u

;

asu! 1 withc; >0. In insurance we may note thatXðtÞ represents the sum of the claims up to timet,ctrepresents the sum of the premium payments up to timet anduthe initial reserve of the firm. The ruin occurs if at some timetthe sum of the claims is larger than the sum of premium payments and the reserve.

This problem is investigated for a class of Gaussian processesXðtÞ(including frac- tional Brownian motion and self-similar Gaussian processes) in Hu¨sler and Piterbarg (1999) and for integrated Gaussian processes in Debicki (2002) as well as Hu¨sler and Piterbarg (2004). In these cases the probability PðuÞ is approximated by exceedances ofYðtÞin a small neighborhood of a unique point where the boundary ðuþctÞ=ðtÞhas smallest value. This boundary is the result of usual standardization of the processXðtÞbyðtÞwhere²ðtÞdenotes its variance. Often, only the (unique) point of maximal variance plays the important role. Here the trend has to be considered also, which results in the mentioned minimal boundary value.

In this paper we deal with another particular class of Gaussian processes. We think that a portfolio consists of many different processes Xið

^Þ which can be modelled e.g., as fractional Brownian motions with E½XiðtÞ ¼0 and Var½XiðtÞ ¼ dit^2Hi and possibly different parameters Hi2 ð0;1Þ and di>0. Therefore, we consider XðtÞ ¼Pk

i¼1wiXiðtÞ as the portfolio of all risks at time t, with wið2RÞ some weights. As mentioned, the biggest liability of the firm after its start of economic activities at timet¼0 is then denoted by sup_t>0ðXðtÞ ctÞ. Thus

P sup

t>0

X^k

i¼1

wiXiðtÞ ct

!

>u

( )

will be investigated. We assume that the Xið

^Þ are independent processes. This probability is well-defined ifHi< for alli. As mentioned above, the minima of the boundary function ðuþctÞ=ðtÞ have to be analyzed together with the path behaviour ofXðtÞin the neighborhood of possible minima.

Note thatXðtÞ is thus a centered Gaussian process with variance Var½XðtÞ ¼X^k

i¼1

w²_idit^2Hi¼X^k

i¼1

Wit^2Hi

where W_i¼w²_id_i. Hence, we might set w.l.o.g. d_i¼1 or w_i¼1, since in the following only the Wi_s are used. It is not necessary to assume that the Gaussian processes Xið

^Þ are fractional Brownian motions. But certain regularity conditions will be assumed. E.g., we assume that the Gaussian processes are asymptotically locally stationary (see (8) in Condition (A1)) for largeu. The processes with largest H_iare important. Hence, let w.l.o.g.H¼H₁H₂. . .H_kand definemð1Þas largest index such thatHm¼H.

In the next section we introduce the sufficient conditions on the Gaussian process XðtÞand the main result which is proved in the third section. For its proof we need to investigate the local behaviour of the boundary function in the vicinity of the points with minimum value. This will be combined with the behavior of the weighted sumXðtÞof Gaussian processes in these vicinities.

2. Weighted sum of Gaussian processes and main result

The portfolioXðtÞ is modelled as weighted sum of centered independent Gaussian processes X_iðtÞ with Var½XiðtÞ ¼t^2Hi with the mentioned numeration H¼H₁ H2. . .Hk. Defining the standardized process

e

XðtÞ ¼XðtÞ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Var½XðtÞ

p ¼X^k

i¼1

wiXiðtÞ. ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X^k

i¼1

Wit^2Hi vu

ut ;

we analyze

P sup

t>0

ðXðtÞ ctÞ>u

¼P 9t>0 :XðtÞe >~ f_uðtÞ

n o

; where

~fuðtÞ ¼ uþct ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P^k

i¼1

W_it^2Hi

s ¼ uþct

ffiffiffiffiffiffiffiffiffiffiffi Wt^2H

p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1þP

j>m

W¹Wjt^2ðHjHÞ r

is the boundary function and

W¼X^m

i¼1

Wi: ð1Þ

We make an appropriate time transformation such that the points where the boundary values are minimal, remain finite asu! 1. Let for eachu

s¼W^2H1u¹t: ð2Þ

The transformed centered Gaussian processes, depending onu, are denoted by X_i^ðuÞðsÞ ¼X_iðu¹W^2H1sÞ; ik; ð3Þ X^ðuÞðsÞ ¼X^k

i¼1

wiX_i^ðuÞðsÞ ¼Xðu¹W^2H1sÞ: ð4Þ The time transformation results in the corresponding boundary functionfuðsÞ:

f_uðsÞ ¼~

f_uðW^2H1u¹sÞ ¼u^1HvðsÞð1þuðsÞÞ ð5Þ consisting of three factors where

vðsÞ ¼1þ~cs

s^H ð6Þ

and

uðsÞ ¼ 1þX

j>m

WjW^HjHu^2ðHjHÞs^2ðHjHÞ

!¹₂

1 ð7Þ

with ~c¼cW^2H. Note that uðsÞ !1 as u! 1. The boundary fuðsÞ may have several points with minimal value, depending onu. The smallest of these points is denoted by su*¼inffargminfuðsÞg. We will show that these points with minimal

value converge to the point of minimal value ofvðsÞ. Hence we have to investigate the approximation of the probability:

P sup

t>0

ðXðtÞ ctÞ>u

¼Pn9s>0 :Xe^ðuÞðsÞ>f_uðsÞo

;

whereXe^ðuÞðsÞ ¼X^ðuÞðsÞ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Var½X^ðuÞðsÞ

p denotes the standardized processX^ðuÞðsÞ.

We need the following assumptions for the main result:

(A1) We assume that each Xe^ðuÞ_i ðsÞ ¼X_i^ðuÞðsÞ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Var½X_i^ðuÞðsÞ q

is asymptotically locally stationary forik, i.e., there exists a functionK²_ið

Þ, regular varying (at 0) with parameteri2 ð0;2Þ, such that

lim

u!1

E½Xe^ðuÞ_i ðsÞ Xe^ðuÞ_i ðs⁰Þ²

K_i²ðjss⁰jÞ ¼Di ð8Þ uniformly for s;s⁰2Su¼ ½su*ðuÞ;s*uþðuÞ withDi>0, ðuÞ ¼u^H1logu and s_u* denotes the smallest point with minimal boundary value: s*_u¼ inffarg minfuðsÞg.

This condition implies that each Gaussian process Xið

^Þ has continuous paths in the crucial interval. We could assume this regularity condition to hold for alls, but this is not necessary.

For the variance of the increments ofXXe^ðuÞðsÞ we consider the weighted sum of the regularly varying functionsKið

Þ. By (A1) there existsK²ð

^{Þ, reg-}

ularly varying at 0 with index¼minfig, such that for some positiveWWe

lim

s!s⁰

P

im

WiDiK_i²ðjss⁰jÞ K²ðjss⁰jÞ ¼WWe:

We denote the inverse ofKð

^ÞbyK1ð

^{ÞwhereK1ðyÞ ¼inffs:KðsÞ yg.}

Since Kð

^Þ is regularly varying at 0 with index =2, K¹ð

^Þ is regularly varying at 0 with index 2=.

Note that for fractional Brownian motions with Hurst parameterHi, we havei¼2HiandK_i²ðhÞ ¼ jhj^2Hi.

(A2) Forj>m, letK²_jð

^Þbe such that lim sup_h#0KjðhÞ

KðhÞ<1, asu! 1.

This condition implies thatj also forj>m. Note that (A2) holds for jmalso by (A1). We use in the following Pickands constant defined by

H¼lim

T!1

1

TE exp max

0tT ðtÞ

;

where ðtÞ,t0, is a fractional Brownian motion with drift E½ðtÞ ¼ t and covariance functionCov½ðsÞ; ðtÞ ¼tþs jtsj, hence with Hurst parameterH¼=2. Note that here the fractional Brownianð

^{Þmotion has}

variance 2t.

Now we can state our main result on the extreme values of a portfolio of the Gaussian processesX_i^ðuÞðsÞði¼1;. . .;kÞas defined in (3). Condition (A1) restricts the behavior of these processes in the small interval S_u. For the

other time points s we suppose a quite common Ho¨lder condition for the increments. We assume that for eachik

lim sup

u!1

E Xh _i^ðuÞðsÞ X_i^ðuÞðs⁰Þi2

Gijss⁰jⁱ: ð9Þ holds for anys;s⁰withjss⁰j C, for someGi; i>0 and some largeC. For the asymptotic expression we use the constantsAandB:

A¼ H

~cðHÞ

^H

H

B¼ H

~cðHÞ

^Hþ2

H;

which arevðs0Þ and v⁰⁰ðs0Þ, respectively, withs0¼argminvðsÞ ¼ ðH=ð~cð HÞÞ¹⁼, the unique point of minima ofvð

^Þ(see Lemma 3.3 below). We can now state the main theorem.

Theorem 2.1: Let XiðtÞ;t>0;ði¼1;. . .;kÞ be independent centered continuous Gaussian processes with variance dit^2Hi and ct a trend where;c;di>0 and0<

Hk . . . H1<minf1; g.Let wi2R denote the weights. Assume the conditions (A1) and (A2) with0< i < 2and(9).Then, the tail behavior is given by

P sup

t>0

X^k

i¼1

wiXiðtÞ ct

!

>u

( )

ffiffiffiffi^We

W

q A

²

H2¹ exp ¹₂f_u²ðsu*Þ A ffiffiffiffiffiffiffiffi

pAB

u^22HK¹u^H1 ; as u! 1,where ¼min_imi,m the number of Hi¼H,fuðsÞ defined in (5)and su*¼inffargminfuðsÞg.

Remark 2.1: In some particular cases we have explicit expressions for su*and fuðs_uÞ. For example, ifk¼1, we have H¼H1,¼₁,Kð

Þ ¼K1ð

Þ,W¼W1¼d1w²₁, We

W¼W1D1,A¼ ðH=ð~cðHÞÞÞ^H=ðHÞ andB¼ ðH=ð~

cðHÞÞÞ^H2 H. K¹ðsÞ is regularly varying at 0 with index2=ð0< <2ÞandfuðsÞreduces tofuðsÞ ¼u^1HvðsÞ. Hence f_u²ðsu*Þ ¼u^22Hv²ðs0Þ ¼u^22HA². We note that^~c¼cW^2H ¼cðd1w²₁Þ^2H ¼c, by choosingd1¼w1¼1. Hence

P sup

t>0

XðtÞ ct

>u

ffiffiffiffiffiffi D1

p A

²

H2¹expn¹₂A²u^22Ho A ffiffiffiffiffiffiffiffi

pAB

u^22HK¹u^H1 ;

which is the result of Hu¨sler and Piterbarg (1999). But this result holds also for a more general Gaussian process, not only for a fractional Brownian motion, if the stated assumptions (A1) and (9) hold.

Remark 2.2 Further Examples: Let us consider some other simple examples withXðtÞsatisfying the conditions of the theorem. We do not need to assume that X_iðtÞare fractional Brownian motions.

We begin with two processes XiðtÞ;i¼1;2. Let H¼H1>H2. It implies that m¼1 andW¼W1. Also Kð

^{Þ ¼K}1ð

^{Þ with¼}1 andWWe ¼W1D1 and the same result holds as in Remark 2.1.

This simple situation holds also with more than two processes, ifH₁¼H₂¼. . .¼ Hm (k¼m2) and 1 < j;j>1. Then WWe ¼W1D1, ¼1 and K¼K1 in the asymptotic formula.

If in addition to Hj also some of thej_s are equal to 1¼, then the result depend on the possible domination of one of theKj_s. For example, letHj¼H1;for all jm¼k, and 1¼j for jm⁰m with 1< j for j>m⁰. In addition, assume that KjcjK1 for jm⁰ and some cj0, as h!0. Then WWe ¼ P

jm⁰WjDjcj with c1¼1, and the result of Theorem 2.1 holds with K¼K1, ¼ 1andW¼P

jWj.

If one of the Kj_s ðjm⁰Þdominates the others, by renumbering let this beK1, then this would mean thatcj¼0 for all 1<jm⁰. The result holds then also with suchc_j_s.

3. Proof

Idea of the proof:

Applying the time transformation (2), the original problem gets P sup

t>0

ðXðtÞ ctÞ>u

n o

¼P 9t>0 :XðtÞ>uþct

¼P 9t>0 :XðtÞe > ~ f_uðtÞ

n o

¼Pn9s>0 :XXe^ðuÞðsÞ>fuðsÞo :

In Proposition 3.2 we show that all minima of fuðsÞ occur in the interval

½su*ðuÞ;s0, wheresu*¼inffargminfuðsÞg. Therefore we splitPf9s>0 :XXe^ðuÞðsÞ>

fuðsÞginto the probabilities

Pn9s2S_u :XXe^ðuÞðsÞ>f_uðsÞo

and Pn9s62S_u :XXe^ðuÞðsÞ>f_uðsÞo :

whereSu¼ ½s*uðuÞ;su*þðuÞ. Then we will show that foru! 1 Pn9s62Su :XXe^ðuÞðsÞ>fuðsÞo

¼o P n9s2Su :XXe^ðuÞðsÞ>fuðsÞo :

We choose ðuÞ ¼u^H1logu, since we need later that ðuÞu^1H ! 1 ðu! 1Þ.

Hence it remains to analyze the asymptotic behavior of the leading probability term, wheres2S_u. For this proof we need to know the behavior of the portfolio process X^ðuÞðsÞorXXe^ðuÞðsÞand the boundary functionfuðsÞ.

3.1. Properties of the portfolio process

By the definition of the portfolio process its behavior can be characterized as follows.

Lemma 3.1: For ik;the means and variances are given by E

X_i^ðuÞðsÞ

¼ E X^ðuÞðsÞ

¼0;

Var½X_i^ðuÞðsÞ ¼ diu^2HW¹s^2H for im Var½X_j^ðuÞðsÞ ¼ djs^2HjW^HjHu^2Hj ¼oðu^2HÞ for j>m

Var½X^ðuÞðsÞ ¼ u^2H=s^2Hð1þuðsÞÞ²¼u^2H=s^2H1þO u ^2ðHmþ1HÞ as u! 1.

Proof: The processes are obviously centered. For any im and j>m, the variances are simply

Var½X_i^ðuÞðsÞ ¼ diðu¹W^2H1sÞ^2Hi¼diu^2HW¹s^2H; for im Var½X_j^ðuÞðsÞ ¼ djðu¹W^2H1sÞ^2Hj¼dju^2Hj=W^Hj=Hs^2Hj ¼oðu^2HÞ;

asu! 1, and

Var½X^ðuÞðsÞ ¼ P

im

w²_iVar½X_i^ðuÞðsÞ þP

j>m

w²_jVar½X_j^ðuÞðsÞ

¼ u^2Hs^2H 1þP

j>m

WjW^HjHu^2ðHjHÞs^2ðHjHÞ

!

¼ u^2Hs^2Hð1þuðsÞÞ²

¼ u^2Hs^2H1þO u ^2ðHmþ1HÞ

The correlation of Xe^ðuÞðsÞ is given by the regularly varying function K²ð

^{Þ, in}

the intervalSu.

Í

Lemma 3.2: Assume the conditions (A1) and (A2). Then the correlation function of Xe^ðuÞðsÞ ¼X^ðuÞðsÞ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Var½X^ðuÞðsÞ

p is for small lags, for s;s⁰2Su

1Corr½Xe^ðuÞðsÞ;Xe^ðuÞðs⁰Þ ¼1

2E½Xe^ðuÞðsÞ Xe^ðuÞðs⁰Þ² WWe

2WK²ðjss⁰jÞ:

Proof: Using the fact that Xe^ðuÞðsÞandXe^ðuÞ_i ðsÞare standardized, we get

EhXe^ðuÞðsÞ Xe^ðuÞðs⁰Þi2

¼E P^k

i¼1

wiX_i^ðuÞðsÞ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Var½X^ðuÞðsÞ

p

P^k

i¼1

wiX_i^ðuÞðs⁰Þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Var½X^ðuÞðs⁰Þ p

2 66 64

3 77 75

2

¼E X^k

i¼1

w²_i

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Var½X_i^ðuÞðsÞ Var½X^ðuÞðsÞ s

e X^ðuÞ_i ðsÞ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Var½X_i^ðuÞðs⁰Þ Var½X^ðuÞðs⁰Þ s

e X^ðuÞ_i ðs⁰Þ 0

@

1 A 2 2

64

3 75:

To derive the claim, we split the sumPk

i¼1 intoP

im andP

j>m. Using Lemma 3.1, we see that forimandu! 1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Var½X_i^ðuÞðsÞ Var½X^ðuÞðsÞ s

¼ ffiffiffiffiffiffi

di

W r

ð1þuðsÞÞ

and get

E

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Var½X_i^ðuÞðsÞ Var½X^ðuÞðsÞ s

e X^ðuÞ_i ðsÞ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Var½X_i^ðuÞðs⁰Þ Var½X^ðuÞðs⁰Þ s

e X^ðuÞ_i ðs⁰Þ 2

4

3 5

2

¼ di

WE ð1þuðsÞÞXe^ðuÞ_i ðsÞ Xe^ðuÞ_i ðs⁰Þ

þ ðuðsÞ uðs⁰ÞÞXe^ðuÞ_i ðs⁰Þ

h i2

¼ di

Wð1þuðsÞÞ²EhXe^ðuÞ_i ðsÞ Xe^ðuÞ_i ðs⁰Þi2

þdi

WðuðsÞ uðs⁰ÞÞ² þ2d_i

Wð1þuðsÞÞðuðsÞ uðs⁰ÞÞE Xe^ðuÞ_i ðsÞ Xe^ðuÞ_i ðs⁰Þ e X^ðuÞ_i ðs⁰Þ

h i

di

WEhXe^ðuÞ_i ðsÞ Xe^ðuÞ_i ðs⁰Þi2

;

where for the last step we used that Xe_i^ðuÞðsÞis asymptotically locally stationary with i2 ð0;2Þ and that uðsÞ uðs⁰Þ ¼ ðss⁰Þu0ðÞ ¼ ðss⁰ÞO u ^2ðHmþ1HÞ

for some 2 ðs⁰;sÞ.

Forj>mwe use Lemma 3.1 again to derive ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Var½X_j^ðuÞðsÞ Var½X^ðuÞðsÞ vu

ut ¼ ffiffiffiffi

dj

q

s^HjW^Hj2Hu^Hj=.

ðu^H=s^Hð1þuðsÞÞ¹Þ

¼ ffiffiffiffi dj

q

W^2HHju^1ðHjHÞs^HjHð1þuðsÞÞ ¼:gjðu;sÞ and

E

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Var½X_j^ðuÞðsÞ Var½X^ðuÞðsÞ vu

ut Xe^ðuÞ_j ðsÞ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Var½X_j^ðuÞðs⁰Þ Var½X^ðuÞðs⁰Þ vu

ut Xe^ðuÞ_j ðs⁰Þ 2

64

3 75

2

¼E gjðu;sÞXe^ðuÞ_j ðsÞ Xe^ðuÞ_j ðs⁰Þ

þgjðu;sÞ gjðu;s⁰ÞXe^ðuÞ_j ðs⁰Þ

h i2

¼g²_jðu;sÞEhXe^ðuÞ_j ðsÞ Xe^ðuÞ_j ðs⁰Þi2

þgjðu;sÞ gjðu;s⁰Þ2

þ2gjðu;sÞ gjðu;sÞ gjðu;s⁰Þ

E Xe^ðuÞ_j ðsÞ Xe^ðuÞ_j ðs⁰Þ e X^ðuÞ_j ðs⁰Þ

h i

¼O g ²_jðu;sÞEhXe^ðuÞ_j ðsÞ Xe^ðuÞ_j ðs⁰Þi2

¼O u ^2ðHjHÞK_j²ðjss⁰jÞ

;

as u! 1, where we used that gjðu;sÞ ¼Oðu^1ðHjHÞÞ and gjðu;sÞ gjðu;s⁰Þ ¼ ðss⁰Þgj0ðu; Þ ¼ ðss⁰ÞOðu^1ðHjHÞÞ for some2 ðs⁰;sÞ, as well as

2gjðu;sÞðgjðu;sÞ gjðu;s⁰ÞÞE½ðXe^ðuÞ_j ðsÞ Xe^ðuÞ_j ðs⁰ÞÞXe^ðuÞ_j ðs⁰Þ

¼O u ^2ðHjHÞjss⁰jK_j²ðjss⁰jÞ Putting the various terms together results in

EhXe^ðuÞðsÞ Xe^ðuÞðs⁰Þi2

¼ ð1þoð1ÞÞX

im

w²_i di

WEhXe^ðuÞ_i ðsÞ Xe^ðuÞ_i ðs⁰Þi2

þX

j>m

O u ^2ðHjHÞK²_jðjss⁰jÞ WWe

WK²ðjss⁰jÞ;

using that for allj>m

O u ^2ðHjHÞK_j²ðjss⁰jÞ

¼o K ²ðjss⁰jÞ sinceHj<HandKjðhÞ=KðhÞ ¼Oð1Þforhsmall.

Hence forularge, the correlation function ofXe^ðuÞðsÞbehaves fors!s⁰as 1Corr½Xe^ðuÞðsÞ;Xe^ðuÞðs⁰Þ ¼1

2EhXe^ðuÞðsÞ Xe^ðuÞðs⁰Þi2

WWe

2WK²ðjss⁰jÞ:

3.2. Behavior of the boundary function

Í

We need to analyze the behavior of the boundary function. Examining uðsÞ, we see that lim_s!0uðsÞ ¼ 1, lim_s!1uðsÞ ¼0 ands^2ðHjHÞstrictly decreases to zero, ass! 1, since Hj<Hforj>m. HenceuðsÞstrictly increases from1 to 0. With s0¼argmin vðsÞ, it is straightforward to prove the following lemma (given also in Hu¨sler and Piterbarg (2004)) which implies the next proposition by (5).

Lemma 3.3: We get s0¼ ðH

ð~cðHÞÞÞ¹ as well as v⁰ðsÞ<0 for s<s0 and v⁰ðsÞ>0for s>s0. Hence s0is unique as point of minimal value of vð

Þ.Further

vðs0Þ ¼ H

~cðHÞ

^H

H¼:A and

v⁰⁰ðs0Þ ¼ H

~cðHÞ

^Hþ2

H¼:B:

Proposition 3.1: For u! 1we get

fuðs0Þ ¼u^1HAð1þuðs0ÞÞ ¼u^1HA1þO u ^2ðHmþ1HÞ

; f_u⁰⁰ðs0Þ ¼u^1HB1þO u ^2ðHmþ1HÞ

; taking the derivative w.r.t. s.

Proof: Taking derivatives (w.r.t. s), we get ð1þuðsÞÞ⁰¼Oðu^2ðHmþ1HÞÞ and ð1þuðsÞÞ⁰⁰¼Oðu^2ðHmþ1HÞÞforson a compact interval ð0;1Þ. Thus

f_u⁰⁰ðsÞ ¼u^1H½v⁰⁰ðsÞð1þuðsÞÞ þ2v⁰ðsÞð1þuðsÞÞ⁰þvðsÞð1þuðsÞÞ⁰⁰

¼u^1Hv⁰⁰ðsÞh1þO u ^2ðHmþ1HÞi

;

asu! 1, fors2 ðs0;^{^} s0þÞ^{^} with >^{^} 0 such thatv⁰⁰ðsÞ>0. Together with Lem-

ma 3.3 the statements follows.

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Remark 3.1: The boundary function f_uðsÞ is continuous, has the limits lim_s!0 fuðsÞ ¼lims!1fuðsÞ ¼ 1 and at least one minimum, tending to 1, as u! 1.

Unfortunately,fuðsÞ is a non-algebraic function and thus explicit solutions for the minimum points offuðsÞdo not exist in the general casek>1. Further it is unclear whether the global minimum is unique.

Because of Remark 3.1 we need an upper and a lower simple approximation function offuðsÞ. Let us define

u ¼X

j>m

W_jW^HjHu^2ðHjHÞ: ForðuÞ>0, we introduce the functions

f_u^þðsÞ ¼u^1HvðsÞð1þuðs0þðuÞÞÞ ðs>0Þ and for some smalls1<minf1;s0ðuÞg

f_uðsÞ ¼

f_u;1ðsÞ ¼u^1HvðsÞð1þ us^2ðHkHÞÞ¹² ð0<ss1Þ f_u;2ðsÞ ¼u^1HvðsÞð1þuðs1ÞÞ ðs1<ss₀ðuÞÞ f_u;3ðsÞ ¼u^1HvðsÞð1þuðs0ðuÞÞÞ ðs>s0ðuÞÞ:

8>

>>

<

>>

>:

The constant s₁ is chosen such that f_u;1ðsÞ is strictly decreasing in ð0;s₁Þ. This holds because

@

@sf_u;1ðsÞ ¼ u^1H sðs^2Hþ us^2HkÞ³²

~cs^þ2Hþ~c _us^þ2HkHs^2HH_{k u}s^2Hk~cHs^þ2H~cH_{k u}s^þ2Hk is negative if~cðHÞs^þ2ðHHkÞ þ ~c uðHkÞsHs^2ðHHkÞHk u < 0, which is true ifs<s1 for somes1>0 small enough.

Since 1þuðsÞis strictly increasing, we have

f_uðsÞ < fuðsÞ < f_u^þðsÞ if s2 ð0;s0þðuÞÞ ð10Þ for anyðuÞ 0, and

fuðsÞ f_u^þðsÞ>f_uðsÞ if s2 ½s0þ 2 ðuÞ;1Þ; ð11Þ with equality fors¼s0þðuÞ.

Lemma 3.4: The global minimum/minima of fuðsÞis/are in the interval sð 1;s0. Proof: Since fuðsÞis strictly increasing for s>s0, we have mins>s0fuðsÞ ¼fuðs0Þ hence the minima are smaller thans0. Nowf_u;1ðsÞis strictly decreasing inð0;s1Þand f_u;1ðsÞ ! 1fors!0. Since s1<s0 we have with (10)

min

ss0

f_uðsÞ f g min

ss1

f_u;1ðsÞ

n o

< min

ss1

f_uðsÞ

f g

for anyu, which finishes the proof.

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Even if there are more than one point with minimal boundary value, they all converge tos0 which is shown next since this holds for the smallest of these points, denoted bys*_u.

Proposition 3.2: We have su*!s₀for u! 1.Hence the global minimum/minima points are inðsu*;s0Þ.

Proof: Lemma 3.4 guarantees that s_u*s₀. It remains to show for any >0 that if there exists a sequence uðiÞ ! 1such thats_uðiÞ* < s0leads to a contra- diction. Because s0 minimizes vðsÞ and sinces*_uðiÞ< s0 we havevðs*_uðiÞÞ=vðs0Þ>

1þDðÞfor some DðÞ>0. Further we know that ð1þ_uðiÞðs*_uðiÞÞÞ=ð1þ_uðiÞðs0ÞÞ ¼ 1þoð1Þsince_uðiÞð

Þ ¼oð1Þ, asuðiÞ ! 1. Hence

f_uðiÞðs*_uðiÞÞ ¼f_uðiÞðs0Þvðs*uðiÞÞð1þ_uðiÞðs*uðiÞÞÞ vðs0Þð1þ_uðiÞðs0ÞÞ

>f_uðiÞðs0Þð1þDðÞÞð1þoð1ÞÞ

>f_uðiÞðs0Þ:

So we get f_uðiÞðs*uðiÞÞ>f_uðiÞðs0Þ, which cannot be true sinces*_uðiÞis the smallest of

the possible minimal points.

Í

Combining the proof of Proposition 3.1 with the above Proposition 3.2 gives Proposition 3.3: For u! 1we get

fuðsu*Þ u^1HA f_u⁰⁰ðsu*Þ u^1HB:

3.3. Tail behavior ofX^ðuÞðsÞfors2Su

We derive the probability thatX^ðuÞðsÞexceedsfuðsÞfors2Suby applying a result of Bra¨ker (1993a), given also in Bra¨ker (1993b), similar to Hu¨sler and Piterbarg (1999).

This probability will be the major contribution to the investigated probability, asymptotically. Bra¨ker’s result (formulated below) is given for locally stationary Gaussian processes, being not dependent onu. So a further approximation step is necessary sinceX^ðuÞðsÞdepends onu.

Proposition 3.4: Assume (A1) and (A2). Then with the correlation function Kð

^{Þof XðuÞðsÞ}we have for u! 1

Pn9s2Su:XXe^ðuÞðsÞ>fuðsÞo

ffiffiffiffi^We

W W W

q A

²

H2¹exp¹₂f_u²ðs*uÞ A ffiffiffiffiffiffiffiffi

pAB

u^22HK¹u^H1 :

Proof: To apply Bra¨ker’s Theorem (Bra¨ker, 1993a), we need to approximate e

X^ðuÞðsÞ by Gaussian processes U_þðsÞ and UðsÞ which are independent of u. The original probability will then be estimated applying Slepian’s inequality (Adler,1990).

As in Hu¨sler and Piterbarg (1999), standardized Gaussian processes U_þðsÞ and UðsÞ exist by the assumptions in Lemma 3.2 such that

lim

s!s⁰

E½UðsÞ Uðs⁰Þ² K²ðjss⁰jÞ

" #

¼WWe Wð1Þ for any >0, with correlation function given by

1Corr½UðsÞ;Uðs⁰Þ WWe

2Wð1ÞK²ðjss⁰jÞ;

fors!s⁰. By construction we have fors;s⁰2S_u andularge Corr½UðsÞ;Uðs⁰ÞyCorr½Xe^ðuÞðsÞ;Xe^ðuÞðs⁰Þ

for any >0, sinceðuÞ !0 foru! 1. Applying Slepian’s inequality we get Pn9s2S_u :Xe^ðuÞðsÞ>f_uðsÞo

Pf9s2S_u :U_þðsÞ>f_uðsÞg Pn9s2Su :Xe^ðuÞðsÞ>fuðsÞo

Pf9s2Su :UðsÞ>fuðsÞg:

We now calculate the two probabilities

PðuÞ ¼Pf9s2Su :UðsÞ>fuðsÞg

for s;s⁰2Su and show that P_þðuÞ ¼ ð1þOðÞÞPðuÞ for u! 1. Hence the probabilityPf9s2Su :Xe^ðuÞðsÞ>fuðsÞgis asymptotically equal to P_þðuÞ or PðuÞ,

letting!0.

Í

3.3.1. The calculation of P_þðuÞ andPðuÞ

We have to verify thatfuðsÞ satisfies the assumptions (f1), . . ., (f5) of Bra¨ker’s Theorem (Bra¨ker,1993a) (or Bra¨ker,1993b), for the derivations ofP_þðuÞandPðuÞ.

We consider onlyP_þðuÞ, since the other term is derived in the same way. Bra¨ker’s result states that if the following conditions (f1), (f2), (f3), (f4) and (f5) hold, then thePþðuÞcan be asymptotically approximated by the expression given bellow in 12.

(f1): Being an elementary function,fuðsÞis continuous, which is the condition (f1).

(f2): Since lim_u!1fuðsÞ ¼ 1for anys>0, we have lim_u!1inf_s2SufuðsÞ ¼ 1which is condition (f2).

(f3): SetGðxÞ ¼K¹ð1=xÞ; ðx>0Þ, and

DuðsÞ ¼G

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi We Wð1þÞ

W s

fuðsÞ 0

@

1

A for s2Su:

With ðxÞ ¼expðx²=2Þ=pffiffiffiffiffiffi2

xYðxÞ ¼1FðxÞ as x! 1, condition (f3) assumes that for any >0

Z

S_u

ðfuðsÞÞ DuðsÞ ds!0

foru! 1. BecauseSuis bounded, (f2) holds andK¹ð

^Þis regularly varying at 0 with index 2=, as mentioned, hence (f3) follows.

(f4): Withuðs; Þ ¼ ½fuðsþDuðsÞÞ fuðsÞfuðsÞ, condition (f4) states that uðs; Þ converges uniformly to a function ðs; Þ for s2Su and , some >0.

This holds since we show that foru! 1;jj ;0 < 1ands2Su

juðs; Þj!0:

For some2 ðs;sþDuðsÞÞwe have

fu0ðÞ ¼ f_uðsþDuðsÞÞ f_uðsÞ DuðsÞ

and thusuðs; Þ ¼DuðsÞfu0ðÞfuðsÞ:Now forDuðsÞwe get DuðsÞ ¼ W

We Wð1þÞ

!² 1 fuðsÞ

²

e L L

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi W We Wð1þÞ

s 1

fuðsÞ

!

¼O u ^2ðH1Þð1þuðsÞÞ²LLeð1=fuðsÞÞ

; asu! 1withLLðe

^Þa slowly varying function.

To estimatefu0ðÞwe use thatfu0ðs*uÞ ¼0 and Proposition 3.3:

f_u⁰ðÞ ¼f_u⁰ðsu*Þ þf_u⁰⁰ðsu*Þðs_u*Þ þoðs_u*Þ

¼O u ^1Hðsu*Þ

¼OðloguÞ;

sincesu*ðuÞfor2Su, by the choice ofðuÞ.

Putting together the estimations ofDuðsÞandfu0ðsÞwe have juðs; Þj ¼OðjDuðsÞfu0ðÞfuðsÞjÞ

¼O u ^ð1HÞð12Þð1þuðsÞÞ¹²LLeð1=fuðsÞÞlogu

¼oð1Þ:

since <2 and >H. Hence the sequence uðs; Þ converges to ðs; Þ ¼0 foru! 1, for any bounded.

(f5): (f4) implies that the juðs; Þj !0, hence is finite for all 2R, which is condition (f5): sup_sjðs; Þj < 1for any:

Thus we can apply Bra¨ker’s Theorem since the considered stochastic process U_þðsÞ is a locally stationary Gaussian process with index2 ð0;2Þ, is independent of the parameter u and the sequence of boundary functions fuðsÞ satisfies the conditions (f1),. . .,(f5). Bra¨kers’ result states that

lim

u!1

1 Lu

Pf9s2Su :UþðsÞ>fuðsÞg ¼lim

u!1

P_þðuÞ Lu

¼1; ð12Þ

whereLu¼R

SuuðsÞdssincegðs; Þ ¼0 and where fors2Su

uðsÞ ¼ H2¹ ðfuðsÞÞ G

ffiffiffiffiffiffiffiffiffiffiffiffiffi_~

WWð1þÞ W

q

fuðsÞ

¼ H2¹ ðfuðsÞÞ

K¹ 1 ffiffiffiffiffiffiffiffiffiffiffiffiffi_~

WWð1þÞ W

q

fuðsÞ

:

We derive the behavior of the integral Lu asu! 1. Since K¹ð

Þ is regularly varying with index 2=, it follows uniformly fors2S_u

1=K¹ 1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e W Wð1þÞ

W s

fuðsÞ 0

@

1 A 0

@

1

A

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e WWð1þÞ

W s

A 0

@

1 A

2 .

K¹ðu^H1Þ:

Since fuðsÞ=fuðs0Þ !1, asu! 1, uniformly for s2Su, we derive with Proposi- tion 3.1

Z

S_u

uðsÞds¼H2¹

Z s*_uþðuÞ s*_uðuÞ

ðfuðsÞÞ=K¹ 1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e

WWð1þÞ=W q

f_uðsÞ

ds

H2¹ ffiffiffiffiffiffiffiffiffiffiffiffiffi e

WWð1þÞ W

r

A² ffiffiffiffiffiffi

p2

Au^1HK¹ðu^H1Þ

Z s*uþðuÞ s*_uðuÞ

e^12fu2ðsÞds:

We expand the exponent of the integrand fors!su* f_u²ðsÞ ¼ f_uðs*uÞ þf_u⁰ðs*uÞðss*_uÞ þ1

2f_u⁰⁰ðs*uÞðss*_uÞ²þoðss*_uÞ²

2

¼f_u²ðs*uÞ þfuðs*uÞfu00ðs*uÞðss*uÞ²ð1þoð1ÞÞ:

Then we change the variable x¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi fuðsu*Þfu00ðsu*Þ

p ðssu*Þ. The bounds su*ðuÞ are replaced by

ðuÞ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi fuðsu*Þfu00ðsu*Þ q

ðloguÞ ffiffiffiffiffiffiffiffi pAB

! 1 ðu! 1Þ:

Hence, the integral is by this transformation Z s*u_þðuÞ

s*_uðuÞ e^12fu2ðsÞds e^12fu2ðs*^uÞ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi fuðsu*Þfu00ðsu*Þ p

Z 1 1

e^{12x2ð1þoð1ÞÞ}dx ffiffiffiffiffiffi p2

e^12fu2ðs*^uÞ

ffiffiffiffiffiffiffiffi pAB

u^1H ;

and we get forP_þðuÞthe approximations

P_þðuÞ Z

S_u

uðsÞds

H2¹ ffiffiffiffiffiffiffiffiffiffiffiffiffi~ WWð1þÞ

W

q

A² e^12fu2ðs*^uÞ

A ffiffiffiffiffiffiffiffi pAB

u^22HK¹ðu^H1Þ :

In an analogous way we get the same approximation forPðuÞ, replacingþby . Taking the limit!0, finishes the proof of Proposition 3.4.

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3.4. Tail behavior ofX^ðuÞðsÞfors62Su

The probability of an exceedance outside of S_u is bounded by the sum of the following four terms:

Pn9s62Su :XXe^ðuÞðsÞ>fuðsÞo

Pn9s2 ð0;s1:XXe^ðuÞðsÞ>fuðsÞo

þPn9s2 ðs1;su*ðuÞÞ:XXe^ðuÞðsÞ>fuðsÞo þPn9s2 ðsu*þðuÞ;s₂Þ:XXe^ðuÞðsÞ>f_uðsÞo

þPn9ss₂:XXe^ðuÞðsÞ>f_uðsÞo :

for some larges2.

Now we make use of the Ho¨lder condition (9) which holds for eachX_i^ðuÞð

^{Þ. This}

implies by applying similar derivations as in the proof of Lemma 3.2 that also lim sup

u!1E Xh ^ðuÞðsÞ X^ðuÞðs⁰Þi2

Gjss⁰j;

for anys;s⁰withjss⁰j Cwhere¼minfig>0 andG>0 a suitable constant. It means that the Ho¨lder condition (9) holds also forX^ðuÞð

^Þ.

Proposition 3.5: For s1 and the lower bound f_u;1ðsÞ of the boundary function introduced in Section3.2,we get

Pn9s2 ð0;s1:XXe^ðuÞðsÞ>fuðsÞo

C0s1u^ð1HÞ2Yf_u;1ðs1Þ

;

as u! 1with C0 only depending on and G.

Proof: For 0<ss1 we have minffuðsÞg>minff_u;1ðsÞg ¼f_u;1ðs1Þ sincef_u;1ðsÞ is strictly decreasing onð0;s₁. By Theorem 8.1 of Piterbarg (1996) we get

Pn9s2 ð0;s1:XXe^ðuÞðsÞ>fuðsÞo

P sup

s2ð0;s1

e X

X^ðuÞðsÞ>fuðs1Þ

( )

C₀s₁u^ð1HÞ2Yf_u;1ðs1Þ

for someC0only depending on andG.

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