Marie Du

Roy De Chaumaray

LARGE DEVIATIONS FOR THE SQUARED RADIAL ORNSTEIN-UHLENBECK PROCESS

Keywords: Squared radial Ornstein-Uhlenbeck process, Maximum likelihood estimates, Large deviations

We establish large deviation principles for the couple of the maximum likelihood estimators of dimensional and drift coefficients in the generalised squared radial Ornstein-Uhlenbeck process. We focus our attention to the most tractable situation where the dimensional parameter a ą 2 and the drift parameter b ă 0. In contrast to the previous literature, we state large deviation principles when both dimensional and drift coefficient are estimated simultaneously.

Introduction

The generalized squared radial Ornstein-Uhlenbeck process, also known as the Cox-Ingersoll-Ross process, is the strong solution of the stochastic differential equation (1.1) dX t " pa `bX t qdt `2a X t dB t where the initial state X 0 " x ě 0, the dimensional parameter a ą 0, the drift coefficient b P R and pB t q is a standard Brownian motion. The behaviour of the process has been widely investigated and depends on the values of both coefficient a and b. We shall restrict ourself to the most tractable situation where a ą 2 and b ă 0. In this case, the process is ergodic and never reaches zero. We estimate parameters a and b at the same time using a trajectory of the process over the interval r0, T s. The maximum likelihood estimators (MLE) of a and b are given by:

(1.2) p a T " ş T 0 X t dt ş T 0 1 Xt dX t ´T X T ş T 0 X t dt ş T 0 1 Xt dt ´T 2 and p b T " X T ş T 0 1 Xt dt ´T ş T 0 1 Xt dX t ş T 0 X t dt ş T 0 1 Xt dt ´T 2 .
Overbeck [START_REF] Overbeck | Estimation for continuous branching processes[END_REF] has shown that p a T and p b T both converge almost surely to a and b. In addition, he has proven that ? T ˆp a T ´a p b T ´b˙L Ý Ñ N p0, 4C ´1q where C "

ˆ´b a´2 1 1 ´a b ˙.
Moderate deviation results for p a T and p b T are achieved in [START_REF] Gao | Moderate deviations for squared Ornstein-Uhlenbeck process[END_REF]. In addition, Zani [START_REF] Zani | Large deviations for squared radial ornsteinuhlenbeck processes[END_REF] established large deviation principles (LDP) for the MLE of a assuming b known and, conversely, for the MLE of b assuming a known. Our goal is to extend her results to the case where both parameters are estimated simultaneously. Our method is also different and we explain how we have simplified her approach at the beginning of Section 2.2 and Section 4, using a new strategy introduced by Bercu and Richou in [START_REF] Bercu | Large deviations for the ornstein-uhlenbeck process with shift[END_REF] for the study of the Ornstein-Uhlenbeck process with shift.

The paper is organised as follows. Section 2 is devoted to an LDP for the couple pp a T , p b T q, which is obtained via LDPs for two other couples of estimators constructed on the MLE. Before we prove those results, which is respectively the aim of Sections 5 and 6, we investigate in Section 3 LDPs for some useful functionals of the process and compute in Section 4 the normalized cumulant generating function of a given quadruplet, which is a keystone for every LDP we establish in this paper. Technical proofs are postponed to Appendix A to E.

Main results

We start by rewriting the estimators p a T and p b T in such a way that they are much easier to handle. We need to suppose the starting point x ą 0 to apply the well-known Itô's formula to log X T . We obtain that (2.1)

ż T 0 1 X t dX t " log X T ´log x `2 ż T 0 1 X t dt which leads to (2.2) p a T " S T p2 Σ T `LT q ´XT T V T and p b T " p X T T ´2q Σ T ´LT V T
where the denominator V T " S T Σ T ´1 with

S T " 1 T ż T 0 X t dt and Σ T " 1 T ż T 0 1 X t dt,
and

L T " log X T ´log x T .
For the remaining of the paper, we suppose the starting point x equal to 1. This assumption does not change the large deviation results because both estimators, with and without log x, are exponentially equivalent so that they share the same LDP. We first consider couples of simplified estimators constructed from pp a T , p b T q using the fact that L T and X T {T both tend to zero almost surely for T going to infinity. This will be useful for the study of the MLE pp a T , p b T q.

2.1. Simplified estimators. A first strategy to propose simplified estimators of a and b is to remove the logarithmic term L T in the expression of p a T and p b T given by (2.2). This way, we obtain a new couple pr a T , r b T q defined by

(2.3) r a T " 2 S T Σ T ´XT T V T and r b T " p X T T ´2q Σ T V T .
It is clear that r a T and r b T converge almost surely to a and b. Moreover, we also have the same asymptotic normality

? T ˆr a T ´a r b T ´b˙L Ý Ñ N p0, 4C ´1q.
The proof of this result can be found in appendix E. We also state an LDP for the couple pr a T , r b T q assuming both parameters a and b unknown.

Theorem 2.1. The couple pr a T , r b T q satisfies an LDP with good rate function J a,b pα, βq " $ ' ' ' ' ' ' ' ' ' ' ' & ' ' ' ' ' ' ' ' ' ' ' % pa ´2q Proof. The proofs of this theorem and the two following corollaries are postponed to Section 5.

We give the shape of this rate function in Figure 1 below in the particular case pa, bq " p4, ´1q and over r3, 5s ˆr´4, ´0.5s. One can notice that the rate function reaches zero at point p4, ´1q. LDPs for each estimator r a T and r b T immediately follow from contraction principle [START_REF] Dembo | Large deviations techniques and applications[END_REF].

Corollary 2.1. The sequence pr a T q satisfies an LDP with good rate function

J a pαq " $ ' ' & ' ' % b 4 ´a ´6 ´apa ´2q 2 `16p2 ´αq ¯if α ď ℓ a , b 4 ˜a ´dα ˆpa ´2q 2 α ´2 `2˙¸i f α ě ℓ a ,
with ℓ a " 10 9 `1 9 a 64 `9pa ´2q 2 .

Corollary 2.2. The sequence p r b T q satisfies an LDP with good rate function

J b pβq " $ & % ´β 4 ˆ1 ´b β ˙2 if β ď b 3 , 2β ´b if β ě b 3 .
Remark 2.1. This rate function is quite similar to the one obtained by Zani [START_REF] Zani | Large deviations for squared radial ornsteinuhlenbeck processes[END_REF] for the MLE of b assuming a known. They only differ from a multiplicative factor.

Figure 3 displays rate functions for both estimators in the particular case where pa, bq " p4, ´1q.

A second strategy to propose simplified estimators of a and b is to remove the term X T {T in the expression of p a T and p b T given by (2.2). Then, we obtain a new couple pq a T , q b T q defined by (2.4)

q a T " S T p2 Σ T `LT q V T and q b T " ´2 Σ T ´LT V T .
As previously, q a T and q b T converge almost surely to a and b, and

? T ˆq a T ´a q b T ´b˙L Ý Ñ N p0, 4C ´1q.
The proof of this result is given in appendix E. We also establish an LDP for the couple pq a T , q b T q, and deduce as corollaries LDPs for both estimators, assuming a and b unknown.

Theorem 2.2. The couple pq a T , q b T q satisfies an LDP with good rate function

K a,b pα, βq " $ ' ' ' ' ' ' ' ' ' ' & ' ' ' ' ' ' ' ' ' ' % a 4 pb ´βq ´α 8β `b2 ´β2 ˘´β α ´?2 `aC α ¯2 if β ă 0, 0 ă α ď α a or if β ą 0, α ă 0, a 4 pb ´βq ´α 8β `b2 ´β2 ˘´β pa ´αq 2 8 pα ´2q if β ă 0, α ě α a , ´b C a if pα, βq " p0, 0q, ` 8 otherwise,
where C α " 1 8 pa ´αq 2 `2 ´α, C a " 1 4 `4 ´a `?a 2 `16 ˘and α a " ´2 3 `a 2 ´2 ´?a 2 ´2a `4˘.

One can observe that the rate functions J a,b and K a,b are equal over some domain of R 2 . It is possible to see it on Figure 2 below which is quite similar to the previous one and displays the rate function K a,b in the particular case where pa, bq " p4, ´1q.

Corollary 2.3. The sequence pq a T q satisfies an LDP with good rate function

K a pαq " $ ' ' ' ' & ' ' ' ' % ´b C a if α " 0, K a,b pα, β b q if α ă α a , α ‰ 0, b 4 ˜a ´dα ˆpa ´2q 2 α ´2 `2˙¸i f α ě α a with β b " bλ `16 ? 2 ? C α `a2 ´8λ `32
˘´1{2 and α a , C α and C a are defined in Theorem 2.2.

Corollary 2.4. The sequence p q b T q satisfies an LDP with good rate function

K b pβq " inf ! K a,b pα, βq { α P R ) .
In particular, K b p0q " ´b C a with the notations of the Theorem 2.2.

Figure 3 displays the rate functions for both estimators in the particular case where pa, bq " p4, ´1q. 2.2. Large deviation results for the MLE. The next theorem gives a large deviation principle for the MLE pp a T , p b T q of the couple pa, bq. In contrast with the previous literature, we consider both parameters unknown and estimate them simultaneously. We also simplified the approach of the previous literature as our proofs only rely on the Gärtner-Ellis theorem and do not need, for example, accurate time-depending changes of probability.

Some results about the process

The aim of this section is to establish LDPs for S T , Σ T and V T , which will be useful for the proof of the main theorem. Proof. See appendix A.

The following result can be proven either directly with the same method or using the previous lemma together with the contraction principle [START_REF] Dembo | Large deviations techniques and applications[END_REF].

Theorem 3.1. The sequence pS T q satisfies an LDP with good rate function

Ipxq " " pa`bxq 2 8x if x ą 0 `8 if x ď 0.
In addition, the sequence pΣ T q satisfies an LDP with good rate function

Jpxq " # `pa´2qx`b ˘2 8x if x ą 0 `8 if x ď 0.
It is now easy to establish a LDP for V T . We recall that V T " h pS T , Σ T q where h is the function defined on R 2 by hpx, yq " xy ´1.

Ipx, yq

where I is defined in Lemma 3.1.

Cumulant generating function for the quadruplet

To establish LDPs for the estimators by the same method as above for V T , we need the normalized cumulant generating function of the quadruplet pX T {T, S T , Σ T , L T q. However, we can show that this does not lead to a steep function (see [START_REF] Dembo | Large deviations techniques and applications[END_REF] for the definition). Thus the Gärtner-Ellis theorem cannot be applied directly. In contrast with the previous literature, we will not search another method to obtain large deviation results. Our idea to overcome this difficulty is to consider instead the quadruplet ´aX T {T , S T , Σ T , L T ¯, where

(4.1) L T " ´c ´log X T T 1 X T ă1 `log X T T 1 X T ě1 . Lemma
¨2λ d´b 1 ∆ 1 2p1`f q d `4λ 2 dpd´bq 2 1 ∆ 1 d 2f `2γ 2 f p2f `a`2q 2 1 ∆ 2 2γ 2f `a`2 1 ∆ 2 ‹ ‹ ‹ ' ,
where ∆ 1 " ! pλ, µ, ν, γq P D Λ { λ ą 0 and γ ě 0 or γ ă 0, λ ą 0 and γ 2 λ 2 ă 2f `a`2 d´b ) and ∆ 2 " ! pλ, µ, ν, γq P D Λ { λ ď 0 and γ ă 0 or γ ă 0, λ ą 0 and γ 2 λ 2 ě 2f `a`2 d´b ) . Using that lim µÑ b 2 8 d " 0 and lim νÑ 1 8 pa´2q 2 f " 0, we easily obtain that the norm of (4.2) goes to infinity for any sequence in the interior of D Λ converging to a boundary point.

Proof. We want to find the limit as T Ñ `8 of

Λ T pλ, µ, ν, γq " 1 T log ˆE " exp ˆλ? T a X T `γ T L T `µ ż T 0 X t dt `ν ż T 0 1 X t dt ˙˙.
It follows from Theorem 5.10 in [START_REF] Craddock | The calculation of expectations for classes of diffusion processes by Lie symmetry methods[END_REF] (with a misprint pointed out in [START_REF] Ben Alaya | Asymptotic Behavior of The Maximum Likelihood Estimator For Ergodic and Nonergodic Square-Root Diffusions[END_REF]) that We have to establish the asymptotic behaviour of We need the four following lemmas, whose proofs are postponed to Appendix B.

(4.3) Λ T pλ, µ, ν,
Lemma 4.2. For all γ ă 0 and λ P R, one can find the following bounds for H T as T goes to infinity.

H T ď 2 1´f Γpf `1q ? πg ´3{2 |γ| ? T e |λ|`α T {T `βT { ? T β f T exp ˆγ2 T 4g ȧnd H T ě 2 ´1´f Γpf `1q ? πg ´3{2 |γ| ? T e ´|λ|´α T {T β f T exp ˆγ2 T 4g ˙,
where g " 2f `a`2 4 .

Lemma 4.3. For all γ ě 0 and λ P R,

H T ď pβ T q f Γpf `1q2 f exp ´|λ| ? T `βT ānd H T ě pβ T q f ε T 2 f Γ p1 `f q exp ´´γ ? T a log T ´g log T ¯,
where ε T " e ´αT ´ε `e´|λ| g ´1 ´γ? T 2g ? log T `γ?

T ¯¯and ε " exp `λ1 λě0 `λ? T 1 λă0 ˘.

Lemma 4.4. For all λ ď 0 and γ P R and T tending to infinity, K T " O ´pβ T q f ¯.

Moreover if γ ě 0,

K T ě 2 1´f pβ T q f Γp1 `f q e ´αT 2α T 1 1 ´λ? T 2α T exp ´λ? T ¯.
Lemma 4.5. For all λ ą 0 and γ P R,

K T ď 2 2´f ? 2π pβ T q f Γp1 `f q ´βT `λ? T ¯2γ`2g´1 exp ˆpλ ? T `βT q 2 4α T ȧnd K T ě 2 1´f c π d ´b pβ T q f Γp1 `f q m γ,λ,T exp ˆλ2 T 4α T ˙.
where m γ,λ,T " min " ´λ?

T α T ¯2pγ`g´1q ; λ ? T α T * .
It is clear with those lemmas that the asymptotic behaviour of J T depends on the sign of λ and γ: ' For all λ ą 0 and γ ě 0: we directly deduce from Lemmas 4.5 and 4.3 that, for T large enough,

H T `KT ď 2 2´f ? 2π pβ T q f Γp1 `f q ´βT `λ? T ¯2γ`2g´1 ˆepλ ? T `βT q 2 4α T `e|λ| ? T `βT ȧnd thus (4.11) lim T Ñ8 1 T log J T ď f lim T Ñ8 1 T log β T `lim T Ñ8 1 T pλ ? T `βT q 2 4α T " ´f d 2 `λ2 d ´b.
We show alike by using the lower bounds of Lemmas 4.5 and 4.3 that (4.12) lim

T Ñ8 1 T log J T ě ´f d 2 `λ2
d ´b and we finally obtain (4.13) lim

T Ñ8 1 T log J T " ´f d 2 `λ2 d ´b .
' For all λ ď 0 and γ ă 0: With Lemma 4.4, we know that

K T " O ´pβ T q f ¯.
Thus,

H T `KT " H T `O ´pβ T q f ¯. Lemma 4.2 gives bounds for H T , which lead to lim T Ñ8 1 T log J T ď lim T Ñ8 1 T log ˆpβ T q f h T,λ,γ exp ˆγ2 T 4g ˙˙`lim T Ñ8 1 T log ˆ1 `Ch ´1 T,λ,γ e ´γ2 T 4g
ẇhere C is some positive constant and

h T,λ,γ " 2 1´f Γpf `1q ? πg ´3{2 |γ| ? T e |λ|`α T {T `βT { ? T .
Using the fact that h ´1 T,λ,γ e ´γ2 T 4g tends to zero as T goes to infinity, we obtain

(4.14) lim T Ñ8 1 T log J T ď ´f d 2 `γ2 4g .
We obtain the same lower bound by using the lower bound in Lemma 4.2.

' For all λ ď 0 and γ ě 0: Lemma 4.3 gives

H T " O ´pβ T q f exp `|λ| ? T ˘¯and by Lemma 4.4, we know that K T " O ´pβ T q f ¯. Consequently, (4.15) lim T Ñ8 1 T log J T ď f lim T Ñ8 1 T log β T " ´f d 2 .
And the lower bounds given in Lemmas 4.3 and 4.4 lead to (4.16) lim

T Ñ8 1 T log J T ě f lim T Ñ8 1 T log β T " ´f d 2 .
' For all λ ą 0 and γ ă 0: using Lemmas 4.5 and 4.2, we show that

H T `KT ď C pβ T q f ˆhT,λ exp ˆγ2 T 4g ˙`´β T `λ? T ¯2γ`2g´1 exp ˆpλ ? T `βT q 2 4α T ˙ẇhere h T,λ " ? T e |λ|`α T {T `βT { ?
T and C is some positive constant. Thus lim

T Ñ8 1 T log J T ď lim T Ñ8 1 T log ˜pβ T q f e T ˆmax ˆγ2 4g ; λ 2 d´b ˙¸" ´f d 2 `max ˆγ2 4g ; λ 2 d ´b˙.
We also show that

H T `KT ě C pβ T q f ˆhT,λ exp ˆγ2 T 4g ˙`m γ,λ,T exp ˆλ2 T 4α T
˙ẇhere C is still some positive constant and h T,λ " ? T e ´|λ|´α T {T . It leads to (4.17) lim

T Ñ8 1 T log J T ě ´f d 2 `max ˆγ2 4g ; λ 2 d ´b˙.
5. Proofs of the LDPs for the couples of simplified estimators 5.1. Proof of Theorem 2.1. We will now establish an LDP for the first couple of simplified estimators. We notice that pr a T , r b T q " f p a X T {T , S T , Σ T q where f is the function defined on tpx, y, zq P R 3 |yz ‰ 1u by

f px, y, zq " ˆ2zy ´x2 yz ´1 , px 2 ´2qz yz ´1 ˙.
Thus, we first compute an LDP for the triplet ´aX T {T , S T , Σ T ¯and then apply the contraction principle to the obtained rate function.

Lemma 5.1. The sequence !´a X T {T , S T , Σ T ¯) satisfies an LDP with good rate function:

Ipx, y, zq " $ & % ab 4 `b2 8 y `pa ´2q 2 8 z ´b 4 x 2 `px 2 `2q 2 z 8pyz ´1q if x ě 0, y, z, yz ´1 ą 0 `8 otherwise.
Proof. See appendix C.

We apply the contraction principle to the rate function I of Lemma 5.1. We obtain that pr a T , r b T q satisfies an LDP with good rate function J a,b given by (5.1) J a,b pα, βq " inf tIpx, y, zq |f px, y, zq " pα, βqu which reduces to

J a,b pα, βq " inf D I " ab 4 `b2 8 y `pa ´2q 2 8 z ´b 4 x 2 `px 2 `2q 2 z 8pyz ´1q
|f px, y, zq " pα, βq * where D I " tpx, y, zq P R 3 |x ě 0, y ą 0, z ą 0, yz ´1 ą 0u and the infimum over the empty set is equal to the infinity. If pα, βq " p2, 0q, as z ą 0 on D I , the only condition to satisfy f px, y, zq " p2, 0q is x 2 " 2. Therefore

(5.2) J a,b p2, 0q " inf yą0,zą0,yz´1ą0 " ab 4 `b2 8 y `pa ´2q 2 8 z ´b 2 `2z yz ´1 * .
We look for critical points. We find that the derivative equals zero for z " ´b a´2 and y " ´a`2 b , which, replaced into (5.2), leads to (5.3) J a,b p2, 0q " ´b.

It is easy to see that J a,b pα, βq " `8 when α " 2 and β ‰ 0 and when β " 0 and α ‰ 2 since we take the infimum over the empty set.

Otherwise, as f px, y, zq " pα, βq, we have

z " β 2 ´α , yz ´1 " βy `α ´2 2 ´α and x 2 " βy `α.
This gives us some conditions on the parameters. First of all, z ą 0 implies β 2´α ą 0 and yz ą 1 implies y ą ' First case: β ă 0. If α ă 2 then β 2´α ă 0 and J a,b pα, βq " `8. We now suppose α ą 2. As β ă 0, the condition βy `α ě 0 implies y ď ´α β which is in the domain. Therefore,

J a,b pα, βq " inf 2´α β ăyď´α β gpyq.
The derivative g 1 pyq vanishes at 2´α β ´4 b´β if β ‰ b and at 2´α β if β " b. Depending on the value of β, the function g will or not vanish on the domain. So the infimum will sometimes be at the boundary of the domain. It is the case when β ď b, but g tends to infinity when y tends to 2´α β , so necessarily

J a,b pα, βq " g ˆ´α β ˙" pa ´2q 2 β 8p2 ´αq ˆ1 `p2 ´αqb βpa ´2q ˙2 ´β 4 ˆ1 ´b β ˙2 .
For β ą b, the condition

J b pβq " inf αPR I a,b pα, βq.
We have directly that J b p0q " J a,b p2, 0q " ´b and that, for β ‰ 0,

J b pβq " J a,b p2 `β a ´2 b , βq.
This leads to the result noticing that it is continuous at point zero.

Proof of Corollary 2.1. With the contraction principle again, we have

J a pαq " inf βPR J a,b pα, βq.
' First case: α " 2. We easily show that J a p2q " J a,b p2, 0q " ´b. ' Second case: α ă 2.

J a pαq " inf βą0 # pa ´2q 2 β 8p2 ´αq ˆ1 `p2 ´αqb βpa ´2q ˙2 `2β ´b+ .
We investigate critical points β 0 . We get β 0 " ´p2 ´αq b a pa ´2q 2 `16 p2 ´αq which leads to J a pαq " J a,b pα, β 0 q " b 4 ´a ´6 ´apa ´2q 2 `16 p2 ´αq ¯.

' Third case: α ą 2. J a,b pα, βq " minpI 1 , I 2 q where (5.5) (5.7)

I 1 " inf βď b 3 # pa ´2q
I 2 " J a,b pα, β 0 q " b 4 ´a ´6 ´apa ´2q 2 `16p2 ´αq ¯.
Otherwise, for α ě ℓ a , the derivative never vanishes on the domain and the minimum is reached at one of the boundaries. When β goes to zero, the function goes to infinity. Consequently, (5.8)

I 2 " J a,b pα, b 3 q " b 3 ˜ppa ´2q `3p2 ´αqq 2 8p2 ´αq ´1¸.
For I 1 , the idea is similar. The derivative equals zero for β 1 satisfying

β 2 1 " α b 2 pα ´2q pa ´2q 2 `2 pα ´2q .
This time the domain is

β ă b 3 ( . So β 1 is inside the domain if β 1 ă 0 and β 2 1 ě b 2 9 . It leads us to β 1 " b d α pα ´2q pa ´2q 2 `2 pα ´2q
with the condition 9αpα ´2q ą pa ´2q 2 `2pα ´2q on α which gives the same limit value ℓ a . We get (5.9)

I 1 " $ ' ' ' ' & ' ' ' ' % J a,b pα, b 3 q " b 3 ˜ppa ´2q `3p2 ´αqq 2 8p2 ´αq ´1¸i f 2 ă α ă ℓ a J a,b pα, β 1 q " b 4 ˜a ´dα ˆpa ´2q 2 α ´2 `2˙¸i f α ě ℓ a .
We now come back to J a,b pα, βq. Combining (5.7), (5.8) and (5.9), we obtain

J a,b pα, βq " minpI 1 , I 2 q " " min `Ja,b pα, β 0 q, J a,b pα, b 3 q ˘if 2 ă α ď ℓ a min `Ja,b pα, β 1 q, J a,b pα, b
3 q ˘if α ą ℓ a , and it is easy to deduce that (5.10) J a,b pα, βq "

" J a,b pα, β 0 q if 2 ă α ď ℓ a J a,b pα, β 1 q if α ą ℓ a .
This leads to the conclusion, noticing that it is continuous at the point α " 2.

5.3. Proof of Theorem 2.2. We consider the second couple of simplified estimators defined by

q a T " S T p2 Σ T `LT q V T and q b T " ´2 Σ T ´LT V T .
We notice that ´q a T , q b T ¯" hpS T , Σ T , L T q, where h is the function defined on tpy, z, tq P R 3 |yz ´1 ‰ 0u by hpy, z, tq " ˆy p2z ´t2 1 tď0 `t1 tą0 q yz ´1 , t 2 1 tď0 ´t1 tą0 ´2z yz ´1

˙.

Once again, we start by computing an LDP for the triplet pS T , Σ T , L T q and then we deduce an LDP for the couple of estimators applying the contraction principle to the obtained rate function.

Lemma * where D α,β " tpy, z, tq P R 3 |y ą 0, z ą 0, yz ´1 ą 0, t ď 0 and hpy, z, tq " pα, βqu and the infimum over the empty set is equal to infinity. The condition hpy, z, tq " pα, βq implies that βy " ´α and t 2 " p2 ´αqz ´β. It gives us some additional conditions on the parameters. First of all, we notice that for α or β equal to zero, D α,β is not empty if and only if the other one is also zero. Thus, K a,b pα, βq " `8 over t0u ˆR`a nd R `ˆt0u. If α " β " 0, t 2 " 2z. We replace it into the infimum and look for critical points. We obtain z 0 " ´b ? a 2 `16 and y 0 " 4`?a 2 `16 ´b . This leads to

K a,b p0, 0q " ´b 4
´4 ´a `?a 2 `16 ¯.

Moreover, for β ‰ 0, as y " ´α β , D α,β is empty as soon as α and β have the same sign. So K a,b pα, βq " `8 over R `ˆR `and R ´ˆR ´. Besides, both expressions will give us boundaries for z depending on the sign of α, β and 2 ´α, because t 2 must be positive and z must be greater than αz`β over this domain. We find that critical points z 0 satisfy

pαz 0 `βq 2 " 2β 2 C α
We notice that for α negative C α is always positive. So the only critical point that remains in the domain is z 0 " ´β α ´1 `b 2 Cα ¯. As the function tends to infinity on the boundaries of D z , it actually reaches the infimum we were looking for at this critical point z 0 . Replacing it into (5.12), we find

(5.13) K a,b pα, βq " A α,β `Cα z 0 ´2βz 0 αz 0 `β " a 4 pb ´βq ´α 8β `b2 ´β2 ˘´β α ´?2 `aC α ¯2 .
' Second case: 0 ă α ď 2. For β ą 0, K a,b pα, βq " `8. For β ă 0, the condition p2 ´αqz ´β ě 0 is always verified. Thus D z " z ą ´β α ( . We do exactly the same than in the first case and find the same critical point and the infimum given by formula (5.13).

' Third case: α ą 2. The case β ą 0 has already been seen. We consider β ă 0. We investigate the critical points of A α,β `Cα z ´2βz αz`β over the domain D z , given in this case by D z " ´β α ă z ď β 2´α (

. We need to distinguish cases depending on the sign of C α .

If α ă a `4 ´2? a (which is greater than 2 because a ą 2), C α is positive and we find the same critical points than in the first case. The condition z ą ´β α is still only verified by

z 0 " ´β α ˆ1 `c 2 C α ˙.
But, the condition z 0 ď β 2´α is not satisfied for all α in r2, a `4 ´2? ar. Indeed, (5.14)

z 0 ą β 2 ´α if and only if 2 C α pα ´2q 2 ą 4
which leads to the following condition on α:

3α 2 `2pa ´4qα ´pa 2 `8q ą 0.
One of the roots is negative. The other one is inside r2, a `4 ´2? ar:

(5.15) α a " ´2 3 ´a 2 ´2 ´?a 2 ´2a `4¯.

Thus, for α ď α a , the infimum is reached at z 0 and is given by (5.13), while for α ą α a , z 0 is not inside the domain D z so the derivative does not vanish and the infimum is reached at one of the boundaries. We notice that for z tending to ´β α , A α,β `Cα z ´2βz αz`β tends to the infinity. Thus, the infimum is reached at the other boundary of the domain: z 0 " β 2´α . Replacing it into (5.12), we obtain ' First case: α " 0. We easily show that K a p0q " K a,b p0, 0q " ´bC a . ' Second case: α ă 0. We rewrite

(
K a pαq " inf βą0 " A α,β ´β α ´?2 `aC α ¯2* .
The critical points β 0 satisfy (5.17)

β 2 0 " α 2 b 2 16 ? 2 ? C α `a2 ´8α `32 ,
which is clearly well defined for all α ă 0. Using the fact that α ă 0 and b ă 0 and as β 0 must be positive, we obtain (5.18)

β 0 " bα ´16 ? 2 a C α `a2 ´8α `32 ¯´1{2 and (5.19) K a pαq " K a,b pα, β 0 q.
' Third case: 0 ă α ď α a . This time

K a pαq " inf βă0 " A α,β ´β α ´?2 `aC α ¯2* .
The critical points β 0 are still given by (5.17). It is well defined for all α ă α a . Namely, 16 ? 2 ? C α `a2 ´8α `32 is a decreasing function on α over this domain and is positive at the point α a . Indeed, we know from (5.14) and (5.15) that α a ą 2 satisfies pα a ´2q 2 " 2C αa , which leads to 16 ? 2 a C αa `a2 ´8α a `32 " 8α a `a2 ą 0. This time α ą 0 and β 0 must be negative but we obtain anyway the same critical point β 0 given by (5.18) and the infimum K a by (5.19). ' Fourth case: α ě α a . This time

K a pαq " inf βă0 # pa ´2q 2 β 8p2 ´αq ˆ1 `p2 ´αqb βpa ´2q ˙2 ´β 4 ˆ1 ´b β ˙2+ .
Using the results of the third case of the proof of Corollary 2.1, we obtain the same critical point and the same infimum.

6. Proof of Theorem 2.3

6.1. Existence of an LDP.

Lemma 6.1. The quadruplet p a X T {T , S T , Σ T , L T q satisfies an LDP with good rate function Λ ˚given by (6.1) Λ ˚px, y, z, tq " `8 for x ă 0, t ą 0, y ď 0, z ď 0 or yz ´1 ď 0.

and, otherwise, (

Λ ˚px, y, z, tq " sup

D d,f
hpd, f q where D d,f " td ą 0, f ą 0u and, with ϕpf q " 2f `a `2, hpd, f q " 1 4

´ta ϕpf q ´x? d ´b¯2 `y b 2 ´d2 8 `pa ´2q 2 ´4f 2 8 z `d 2 p1 `f q `ab 4 .
Proof. Using Gärtner-Ellis theorem, we have to compute the Fenchel-Legendre transform Λ ˚of the cumulant generating function Λ defined in Theorem 4.1: (6.3) Λ ˚px, y, z, tq " sup D txλ `yµ `zν `tγ ´Λpx, y, z, tqu where D "

! λ P R, γ P R, µ ă b 2 8 , ν ă pa´2q 2 8
)

. We show with the same arguments than for the other LDP proofs that (6.4) Λ ˚px, y, z, tq " `8 for x ă 0, t ą 0, y ď 0, z ď 0 or yz ´1 ď 0.

Besides, for x ě 0, the part involving λ in the function we want to optimize is always negative for λ ď 0 and sometimes positive for λ ą 0. Thus the supremum is necessarily reached for some λ ą 0. With the same argument for t ď 0, we show that we only have to consider γ ă 0. Replacing µ and ν by their expression in d and f , the domain D over which we optimize reduces to D " tλ ą 0, γ ă 0, d ą 0, f ą 0u. Replacing Λ by its value leads to Λ ˚px, y, z, tq " maxpS 1 , S 2 q, where

S 1 " sup DX ! γ 2 λ 2 ď ϕpf q d´b ) " xλ `y b 2 ´d2 8 `z pa ´2q 2 ´4f 2 8 `tγ `d 2 p1 `f q `ab 4 ´λ2 d ´b* and S 2 " sup DX ! γ 2 λ 2 ě ϕpf q d´b ) " xλ `y b 2 ´d2 8 `z pa ´2q 2 ´4f 2 8 `tγ `d 2 p1 `f q `ab 4 ´γ2 ϕpf q * .
We first consider S 1 . The domain over which we take the supremum is given by

D X " γ 2 λ 2 ď ϕpf q d ´b* " # λ ă 0, d ą 0, f ą 0, 0 ą γ ě ´c ϕpf q d ´bλ + .
Over this domain, as t ď 0, 0 ď tγ ď ´tb ϕpf q d´b λ, so that the supremum of tγ is equal to ´tλ b ϕpf q d´b . Thus, if we set D 1 " tλ ă 0, d ą 0, f ą 0u,

S 1 " sup D 1 # xλ `y b 2 ´d2 8 `z pa ´2q 2 ´4f 2 8 ´tλ c ϕpf q d ´b `d 2 p1 `f q `ab 4 ´λ2 d ´b+ .
The supremum over λ is easy to compute. Indeed, the function is concave on λ and the critical point is given by

λ " d ´b 2 ˜x ´tc ϕpf q d ´b¸.
Finally, with D d,f " td ą 0, f ą 0u, we obtain

S 1 " sup D d,f # 1 4 ´ta ϕpf q ´x? d ´b¯2 `y b 2 ´d2 8 `pa ´2q 2 ´4f 2 8 z `d 2 p1 `f q `ab 4
+ .
We do the same thing with S 2 , computing first the supremum over λ and then over γ. We obtain S 1 " S 2 , so that (6.5) Λ ˚px, y, z, tq " S 1 " sup

D d,f
hpd, f q where hpd, f q " 1 4

´ta ϕpf q ´x? d ´b¯2 `y b 2 ´d2 8 `pa ´2q 2 ´4f 2 8 z `d 2 p1 `f q `ab 4 .
Remark 6.1. This supremum is not explicitly computable but, as the function h is concave, it is reached for some pd ˚, f ˚q and this gives the rate function of the LDP satisfied by the quadruplet p a X T {T , S T , Σ T , L T q.

Lemma 6.2. The couple pp a T , p b T q satisfies an LDP with good rate function I a,b given over R 2 by

I a,b pα, βq " $ ' ' ' ' ' ' & ' ' ' ' ' ' % K a,b p0, 0q if pα, βq " p0, 0q, J a,b p2, 0q if pα, βq " p2, 0q, inf Dx,t sup D d,f Hpx, t, d, f q if pα, βq P D 1 Y D 2 Y D 3 , `8 otherwise. where D 1 " R ´ˆR `, D 2 "s0, 2rˆR, D 3 " r2, `8rˆR ´, D d,f " td ą 0, f ą 0u, D x,t " " x ě 0, t ď 0| x 2 ´α β t 2 `β 2 ´α ą 1 * and Hpx, t, d, f q " 1 4 ´ta 2f `a `2 ´x? d ´b¯2 `b2 ´d2 8 x 2 ´α β `pa ´2q 2 ´4f 2 8 t 2 `β 2 ´α `d 2 p1 `f q `ab 4 .
Proof. As pp a T , p b T q " gp a X T {T , S T , Σ T , L T q where g is the function defined on tpx, y, z, tq P R 4 |yz ´1 ‰ 0u by gpx, y, z, tq " ˆy p2z ´t2 1 tď0 `t1 tą0 q ´x2 yz ´1 , t 2 1 tď0 ´t1 tą0 `px 2 ´2qz yz ´1

˙, the contraction principle and Lemma 6.1 give us that the couple pp a T , p b T q satisfies an LDP with good rate function I a,b given by (6.6) I a,b pα, βq " inf tΛ ˚px, y, z, tq|gpx, y, z, tq " pα, βqu .

We can reduce the domain over which we look for the infimum using (6.1). The condition gpx, y, z, tq " pα, βq gives us a link between x and y and one between t and z:

(6.7) βy " x 2 ´α and p2 ´αq z " t 2 `β.

We first notice that if α ď 0 and β ă 0 then y is negative and for all x, z, t, Λ ˚px, y, z, tq " `8 such as I a,b pα, βq. Similarly, if β ě 0 and α ą 2, z is negative and Λ ˚px, y, z, tq " `8 for all x, y, t, then I a,b pα, βq " `8. If β " 0 and α ă 0, the first condition in (6.7) leads to x 2 negative and if α " 2 and β ą 0 the second condition gives t 2 negative. So, in both cases, we get I a,b pα, βq " `8. We now focus on the values of α and β for which I a,b is not clearly infinite. We first consider the two remaining limit cases : p0, 0q and p2, 0q. If α " β " 0 then the first condition of (6.7) gives x 2 " α " 0 so that (6.8) I a,b p0, 0q " K a,b p0, 0q.

Similarly, if α " 2 and β " 0, the second condition implies that t 2 " p2 ´αqz " 0 and consequently (6.9)

I a,b p2, 0q " J a,b p2, 0q.
For all remaining values of pα, βq, we define the function (6.10)

Hpx, t, d, f q " 1 4 ´ta 2f `a `2 ´x? d ´b¯2 `b2 ´d2 8 x 2 ´α β `pa ´2q 2 ´4f 2 8 t 2 `β 2 ´α `d 2 p1 `f q `ab 4
and obtain the announced result:

(6.11) I a,b pα, βq " inf Dx,t sup D d,f
Hpx, t, d, f q, where D x,t "

! x ě 0, t ď 0| x 2 ´α β t 2 `β 2´α ą 1
) and D d,f " td ą 0, f ą 0u.

We were not able to compute I a,b directly. It is the aim of the next subsection. Hpx, 0, d, f q.

Thus it easily follows that (6.13) I a,b pα, βq ď min pJ a,b pα, βq, K a,b pα, βqq .

So, we just have to show the other inequality. We name p θ T " ´p a T , p b T ¯and θ T " ´r a T , r b T ¯1X T ě1 `´q a T , q b T ¯1X T ă1 .

Lemma 6.3. The estimators θ T and p θ T are exponentially equivalent, which means that for all ε ą 0, lim

T Ñ`8 1 T log P ´ p θ T ´θT ą ε ¯" ´8.
In particular, as the sequence p p θ T q satisfies an LDP with good rate function I a,b , then the same LDP holds true for pθ T q.

Proof. From the definition of each estimator, we get that

p a T ´pr a T 1 X T ă1 `q a T 1 X T ě1 q " S T L T 1 X T ě1 ´XT T 1 X T ă1 V T and p b T ´´r b T 1 X T ă1 `q b T 1 X T ě1 ¯" X T T Σ T 1 X T ă1 ´LT 1 X T ě1 V T .
Thus, for all ε ą 0,

P ´ p θ T ´θT ą ε ¯ď P ε T `Qε T `pε T `qε T where P ε T " P ˆˇˇˇS T L T 1 X T ě1 V T ˇˇˇě ε 2 ? 2 ˙, Q ε T " P ˜ˇˇˇˇX T T 1 X T ă1 V T ˇˇˇˇě ε 2 ? 2 ¸, p ε T " P ˜ˇˇˇˇX T T Σ T 1 X T ă1 V T ˇˇˇˇě ε 2 ? 2 ¸and q ε T " P ˆˇˇˇL T 1 X T ě1 V T ˇˇˇě ε 2 ? 2 ˙.
For all η ą 0, we have the following upper bounds:

P ε T ď P ˆ|S T | ě ε 2η ? 2
˙`P ˆ|L T 1 X T ě1 | V T ě η ď P ˆST ě ε 2η ? 2 ˙`P `LT 1 X T ě1 ě η 2 ˘`P pV T ď ηq , (6.14) (6.15) p ε T ď P ˆΣT ě ε 2η ? 2
˙`P ˆXT T 1 X T ă1 ě η 2 ˙`P pV T ď ηq , (6.16) q ε T ď P ˆLT 1 X T ě1 ě εη 2 ? 2
˙`P pV T ď ηq , and (6.17)

Q ε T ď P ˆXT T 1 X T ă1 ě εη 2 ? 2 ˙`P pV T ď ηq .
First of all, using Theorem 3.1, we show that for all c ą ´a b , (6.18) lim T Ñ`8 We now consider the parts involving L T . For all c ą 0 and λ ą 0:

P pL T 1 X T ě1 ě cq " P plog X T 1 X T ě1 ě cT q ď E " e λ log X T ‰ e ´λcT . Hence 1 T log P pL T 1 X T ě1 ě cq ď ´λc `1 T log `E " X λ T ‰˘.
Asymptotic properties of the moments of the process X T as T tends to infinity can be found in Proposition 3 of [START_REF] Ben Alaya | Asymptotic Behavior of The Maximum Likelihood Estimator For Ergodic and Nonergodic Square-Root Diffusions[END_REF], and give that the second term tends to zero for T going to infinity. Thus, for any λ ą 0 and c ą 0, we have the following upper bound lim sup

T Ñ`8 1 T log P pL T 1 X T ě1 ě cq ď ´λc.
Consequently, letting λ go to infinity, we obtain that for all c ą 0, (6.21) lim sup

T Ñ`8 1 T log P pL T 1 X T ě1 ě cq " ´8.
Finally, we consider the terms involving X T T . For all c ą 0 and λ ą 0:

P ˆXT T 1 X T ă1 ě c ˙ď E " e λX T 1 X T ă1 ‰ e ´λcT ď e λ´λcT . Hence 1 T log P ˆXT T 1 X T ă1 ě c ˙ď ´λc `λ T .
Thus, for any λ ą 0 and c ą 0, we have the following upper bound lim sup

T Ñ`8 1 T log P ˆXT T 1 X T ă1 ě c ˙ď ´λc.
Consequently, letting λ go to infinity, we obtain that for all c ą 0, (6.22) lim sup

T Ñ`8 1 T log P ˆXT T 1 X T ă1 ě c ˙" ´8.
Consequently, combining the limits (6.18) to (6.22) , we are able to compute the asymptotic behaviour of the bounds (6.14) to (6.17) and we show that for all ε ą 0 and all η ą 0 small enough, lim sup

T Ñ`8 1 T log P ´ p θ T ´θT ą ε ¯ď ´Mε,η , where M ε,η " min " I ˆε 2η ? 2 ˙, J ˆε 2η ? 2 ˙, K pηq * .
Each term in this minimum tends to infinity as η goes to zero, so that M ε,η itself tends to infinity. This gives the announced result.

Proof of Theorem 2.3. We have already shown in Lemma 6.2 that I a,b p2, 0q " J a,b p2, 0q and I a,b p0, 0q " K a,b p0, 0q and that, except at this two points, I a,b is infinite over R ´ˆR ´and over r2, `8r ˆR`. We also know by (6.13) that I a,b pα, βq ď min pJ a,b pα, βq; K a,b pα, βqq , so we still need to establish the other inequality over the remaining domain. In the sequel, we show that, for all compact subsets C Ă R 2 , (6.23) lim sup

T Ñ`8 1 T log P `θT P C ˘ď ´inf pα,βqPC mintJ a,b pα, βq, K a,b pα, βqu.
It is sufficient to consider compact subsets of R 2 instead of closed ones, as we already know that the sequence θ T satisfies an LDP with good rate function I a,b and R 2 is locally compact so that the family `P `θT P ' ˘˘T is exponentially tight (see Lemma 1.2.18 and Exercise 1.2.19 of [START_REF] Dembo | Large deviations techniques and applications[END_REF]). And this will prove the announced result as, by Lemma 6.3, the sequences θ T and p θ T share the same LDP. First of all, we notice that θ T " gp a X T {T , S T , Σ T , L T q where g is the function defined over tpx, y, z, tq P R 4 |yz ´1 ‰ 0u by (6.24) gpx, y, z, tq " ˆy p2z ´t2

1 tă0 q ´x2 1 tě0 yz ´1 , t 2 1 tă0 `px 2 1 tě0 ´2qz yz ´1 ˙,
and the quadruplet p a X T {T , S T , Σ T , L T q satisfies an LDP with good rate function Λ ˚given by Lemma 6.1. As the function g given by (6.24) is not continuous, we cannot apply directly the contraction principle. However, (6.25) lim sup

T Ñ`8 1 T log P `θT P C ˘ď ´inf g ´1pCq Λ ˚.
We need to describe the subset g ´1 pCq. A quadruplet px, y, z, tq of R 4 belongs to g ´1 pCq if and only if there exists a sequence px n , y n , z n , t n q n and a sequence pα n , β n q n P C such that, as n tends to infinity, (6.26) px n , y n , z n , t n q Ñ px, y, z, tq and for all n (6.27) g px n , y n , z n , t n q " pα n , β n q

As C is a compact subset, up to a subsequence, there exists pα, βq P C such that pα n , β n q converges to pα, βq as n goes to infinity. Moreover, (6.27) is equivalent to the following conditions for all n:

β n y n " x 2 n 1 tně0 ´αn and p2 ´αn q z n " t 2 n 1 tnă0 `βn .

Up to a subsequence again, both indicator functions converge toward 1 or 0. Thus, letting n go to infinity, we obtain conditions on px, y, z, tq which lead to

g ´1 pCq " ď pα,βqPC D ὰ,β Y D ά,β
where D ὰ,β " px, y, z, tq P R 3 ˆR`| βy " x 2 ´α and p2 ´αq z " β ( and D ά,β " px, y, z, tq P R 3 ˆR´| βy " ´α and p2 ´αq z " t 2 `β( . Thus, (6.25) becomes (6.28) lim sup

T Ñ`8 1 T log P `θT P C ˘ď ´inf pα,βqPC min # inf D ὰ,β Λ ˚; inf D ά,β Λ ˚+ .
For all α ‰ 2 and β ‰ 0, we easily rewrite D ὰ,β "

!´x , x 2 ´α β , β 2´α , t ¯, x P R, t ě 0 ) and D ά,β " !´x , ´α β , t 2 `β 2´α , t ¯, x P R, t ď 0
) . As the rate function Λ ˚given by Lemma 6.1 is infinite for t ą 0, the infimum over D ὰ,β reduces to the infimum over We apply the Gärtner-Ellis theorem (see [START_REF] Dembo | Large deviations techniques and applications[END_REF]). It is easy to deduce from Theorem 4.1 that the pointwise limit r Λ of the normalized cumulant generating function r Λ T of the couple pS T , Σ T q is given by (A. ) .

!´x , x 2 ´α β , β 2´α , 0 ¯, x P R ) which is equal to J a,
First we note that (A.3) for x ď 0 or y ď 0, Ipx, yq " `8, because r Λpµ, νq tends to ´8 as ν or µ tends to ´8. Only the case x ą 0 and y ą 0 remains to be studied. We look for the critical points. If xy ´1 ‰ 0, we obtain

(A.4) ˆd0 f 0 ˙" 1 xy ´1 ˆ2y 1 ˙.
But d 0 and f 0 must both be positive, so the solution is in the domain if and only if xy ´1 ą 0. It is easy to check that this critical point corresponds to a maximum of r Λ. Using the fact that µ " To conclude, we need to examine the case x ą 0, y ą 0 and xy ´1 ă 0. We already know that for ν and µ tending to ´8, ´r Λpµ, νq tends to `8. However, as x and y are non negative, we cannot conclude directly. But it is possible to find a direction in which r Λ dominate the expression. Note that, for ´ν and ´µ large enough, xµ `yν ´r Λpµ, νq " xµ `yν `?µν.

Let k ą 0 and ν " kµ. We just have to find a k ą 0 that satisfies k 2 py 2 ´1q `2xyk `x2 ă 0.

If y ă 1 then all k ą ´x y´1 fit. Else, if y ě 1, necessarily x ă 1 and we use the same argument with µ " kν this time. We have found directions for which xµ `yν ´r Λpµ, νq tends to `8, so that the supremum itself is equal to `8. And, (A.6) for x ą 0, y ą 0 such that xy ´1 ă 0, Ipx, yq " `8. dy.

We consider separately the integrals over r0, Besides, using the lower bound of (B.2), we clearly have, for all γ ą 0 and λ P R,

(B.6) H T ě β f T e ´αT Γpf `1q2 f ż 1 0 e λ ? T ? y´γ ? ´T log y y 2f `a´2 4
dy.

To obtain the announced lower bound, we need to consider separately the integral over r0, 1 T s and over r 1 T , 1s. On the one hand, the integral over r

K T ď pβ T q f 2 f Γp1 `f q ż `8 1 e β T ?
y`λ ?

T y´α T y y γ`2 f `a´2 4 dy and (B.9)

K T ě pβ T q f 2 f Γp1 `f q ż `8 1 e λ ? T y´α T y y γ`2 f `a´2 4 dy.
To go further, we need to consider the sign of the exponent γ `2f`a´2 , we obtain the following asymptotic behaviour for the upper bound of K T ,

K T ď pβ T q f 2 f Γp1 `f q ż `8 1 e p β T `λ?
T q ? y´α T y dy

" 2pβ T q f 2 f Γp1`f q e pλ ? T `βT q 2 4α T ż `8 1´λ ? T `βT 2α T e ´αT u 2 ˆu `λ? T `βT 2α T ˙du ď 2pβ T q f 2 f Γp1`f q e pλ ? T `βT q 2 4α T ´A1 `βT 2α T A 2
where A 1 and A 2 are given by: (B. Thus:

(B.12) K T ď 2pβ T q f 2 f Γp1`f q e pλ ? T `βT q 2 4α T 1 2α T e ´αT ˆ1´λ ? T `βT 2α T ˙2 ˜1 `βT 2α T 1 ´λ? T `βT 2α T " pβ T q f 2 f Γp1`f qα T ˜1 `βT 2α T 1 ´λ? T `βT 2α T ¸exp `λ? T `βT ´αT ˘. ' If γ `2f`a´2 4
ą 0: with formula 3.462(1) in [START_REF] Gradshteyn | Table of integrals, series, and products[END_REF], we get .

K T ď pβ T q f 2 f Γp1`f q ˆ2 p2α T q ´pγ`
Thus K T " Opβ f T q as T goes to infinity. If λ " 0, we use formula 9.246(2) in [START_REF] Gradshteyn | Table of integrals, series, and products[END_REF].

D ´2pγ`gq ˆ´β T ? 2α T ˙" ? 2π Γ p2γ `2gq ˆβT ? 2α T ˙2γ`2g´1 e pβ T q 2 8α T .
This leads to the same conclusion: as T goes to infinity,

(B.13) K T " O ´pβ T q f ¯.
Otherwise, for the proof of Lemma 4.1, we also need a lower bound for K T when γ ě 0. We note that over r1, `8r, y γ`g´1 ě 1, so that

(B.14) K T ě 2 ´f pβ T q f Γp1 `f q ż `8 1 e λ ? T
? y´α T y dy.

With the change of variable given by u " ? y ´λ? T 2α T , it becomes ´b pβ T q 1`f α T Γp1 `f q e pλ ?

(B.15) K T ě 2 1´f pβ T q f Γp1`f q e λ 2 T 4α T ż `8 1´λ ? T 2α T e ´αT u 2 pu `λ? T 2α T q du ě 2 1´f pβ T q f Γp1`f q e λ 2 T
T `βT q 2 4α T

.

For the lower bound, with the change of variable given by u " ? y ´λ? T 2α T , we obtain With the change of variable given by u " ? y ´λ? T 2α T , it becomes

K T ě 2 ´f pβ T q f Γp1`f q ż `8
K T ě 2 1´f pβ T q f Γp1`f q e λ 2 T 4α T ş `8 1´λ ? T 2α T e ´αT u 2 pu
`λ? T 2α T q du ě 2 1´f pβ T q f Γp1`f q e λ 2 T 4α T ˆş`8 Moreover, we establish the upper bound with formula 3.462(1) of [START_REF] Gradshteyn | Table of integrals, series, and products[END_REF] as in the previous proof.

K T ď pβ T q f
2 f Γp1`f q ˆ2 p2α T q ´pγ`gq Γ p2γ `2gq e T y´α T y y γ`g´1 dy ď 2 1´f pβ T q f p2α T q ´pγ`gq Γp2γ`2gq Γp1`f q e pλ ? .

T
This easily leads to the announced result.

because E pX T q converges almost surely to E pX 8 q " ´a b (see [START_REF] Overbeck | Estimation for continuous branching processes[END_REF] Lemma 3) and, as the parameter a is supposed greater than 2, we obtain from Proposition 3 in [START_REF] Ben Alaya | Asymptotic Behavior of The Maximum Likelihood Estimator For Ergodic and Nonergodic Square-Root Diffusions[END_REF] that

E `X´1 T ˘Ñ ´b 2 Γpa{2 ´1q Γpa{2q " E `X´1 8 ˘.
This gives the announced convergence in probability to zero. Thus, with Slutsky's lemma, the simplified estimators pr a T , r b T q satisfy the same asymptotic normality result than the MLE. Similarly, for the second couple of simplified estimators, we have (E.2) ? T ˆq a T ´a q b T ´b˙" ? T ˆp a T ´a p b T ´b˙`?

T ˜XT T V T ´XT Σ T T V T ¸.
As Σ T converges almost surely to ´b a´2 and V T to 2 a´2 , we only have to show that X T { ? T converges to zero in probability. For all ǫ ą 0, P ˆˇˇˇX T ? T ˇˇˇě ε ˙" P ´XT ě ? T ε ¯ď E pX T q

´?T ε T Ñ`8

Ý ÝÝÝ Ñ 0 with the same argument than before. Thus pq a T , q b T q also satisfies the same asymptotic normality result.

Figure 1 .

 1 Figure 1. Rate function for the couple of simplified estimators pr a T , r b T q

Figure 2 .

 2 Figure 2. Rate function for the couple of simplified estimators pq a T , q b T q.

Theorem 2 . 3 .Corollary 2 . 5 .Corollary 2 . 6 .Figure 3 .

 2325263 Figure 3. Rate functions for dimensional and drift parameters.

)

  , which is equal to K a,b p2, βq. Thus, as J a,b p2, βq is equal to infinity, we can conclude that I a,b p2, βq " min tJ a,b p2, βq ; K a,b p2, βqu .

  Appendix A: proof of Lemma 3.1

4 .' 4 ď 1

 441 If γ `2f`a´2 4 ď 0: using the fact that y γ`2 f `a´2 and with the change of variable u " ? y ´λ? T `βT 2α T

?ee('?

  y´α T y y γ`g´1 dy " 2 1´f pβ T q f Γp1`f q e ´αT u 2 p2uq 2pγ`g´1q du ȩ2 1´f pβ T q fΓp1`f q e ´αT u 2 du.By dominated convergence, the last integral converges as T tends to infinity to the positive constant 2 a π d´b . For T large enough, this leads to If γ `g ´1 ą 0: over r1, `8r, we notice that y γ`g´1 ě 1. Thus (B.22) K T ě 2 ´f pβ T q f Γp1 `y´α T y dy.

e

  ´αT u 2 u du `λ? T 2α T ş `8 1´λ ? T 2α T e ´αT u 2 du Ḃy dominated convergence, the first integral in the last expression tends to zero as T goes to infinity and the second integral converges to the positive constant 2 a π d´b .It gives, for T large enough,

  Lemma 3.1. The couple pS T , Σ T q satisfies an LDP with good rate function

	Ipx, yq "	$ & %	y 2pxy ´1q `8	`b2 8	x	`pa ´2q 2 8	y	`ab 4	if x ą 0, y ą 0 and xy ´1 ą 0 otherwise.

  Theorem 3.2. The sequence pV T q verifies an LDP with good rate function

	Kpxq "	$ ' & ' %	´b 4 `8 d	px `1q ˆpa ´2q 2	`4 x	˙`ab 4	if x ą 0 if x ď 0.
	Proof. It follows immediately from Lemma 3.1 together with the contraction princi-
	ple [5]. It only remains to explicitly evaluate the rate function K given, for all real
	z, by
			Kpzq "	inf tpx,yq|z"xy´1u

  4.1. Let Λ T pλ, µ, ν, γq be the normalized cumulant generating function of the quadruplet ´aX T {T , S T , Σ T , L T ¯. Denote by Λ the pointwise limit of Λ T as T tends to `8. For all λ, γ P R, µ ă b 2 8 and ν ă pa´2q 2

	$ ' ' ' ' ' ' ' ' ' ' &	´d 2	p1 `f q	´ab 4	`λ2 d ´b	, if λ ą 0 and γ ě 0 8 or if γ ă 0, λ ą 0 and γ 2 λ 2 ă 2f `a`2 d´b ,
	Λpλ, µ, ν, γq " where d " a b 2 ´8µ and f " ´d 2 p1 `f q ´ab 4 ' ' ' ' ' ' ' ' ' ' % ´d 2 p1 `f q ´ab 4 b 1 2 pa ´2q 2 ´8ν. `γ2 2f `a `2 if λ ď 0 and γ ă 0 or if γ ă 0, λ ą 0 and γ 2 λ 2 ě 2f `a`2 d´b , if λ ď 0 and γ ě 0,
	Remark 4.1. The function Λ is steep. Indeed, Λ is differentiable over its domain
	D Λ " R 4 ˆr´8, b 2 8 rˆr´8, pa´2q 2 8	rˆR 4 and its gradient is given by
	(4.2)		∇Λ "

  1 T log J T . We split J T into two terms: J T " H T `KT where

	(4.8)	H T "	ż 1 0	e λ ?	T y´γ	? ´T log y´α T y y	a´2 4 I f pβ T	? yq dy
	and (4.9)	K T "	ż 8 1	e λ ?	T y´α T y y γ`a ´2 4 I f pβ T	? yq dy
	with (4.10)	α T "	d cothpdT {2q 4	´b	and β T "	d 2 sinhpdT {2q ? x	.

  Second case: β ą 0. Still for the same reasons, if α ą 2, J a,b pα, βq " `8. We now suppose α ă 2. As β ą 0 the condition βy `α ě 0 becomes y ě ´α β which is smaller than 2´α β . Consequently, J a,b pα, βq " inf

	ˆ2 ' 2´α ´α β ´4 b ´β ˙" pa ´2q 2 β 8p2 ´αq β ăy gpyq. ˆ1	`p2 ´αqb βpa ´2q	˙2 `2β ´b.
	The derivative is equal to zero for y " 2´α β ´4 b´β which is always greater than 2´α β so inside the domain. We get J a,b pα, βq " g ˆ2 ´α β ´4 b ´β ˙" pa ´2q 2 β 8p2 ´αq ˆ1 `p2 ´αqb βpa ´2q ˙2 `2β ´b.

2´α β ´4 b´β ą 2´α β is always satisfied and 2´α β ´4 b´β ď ´α β if and only if β ě b 3 . Consequently, if b ă β ď b 3 , the derivative does not vanish on the domain and we find the same value of J a,b as above, while if b 3 ă β ă 0, we get J a,b pα, βq " g 5.2. Proofs of Corollaries 2.2 and 2.1. Using the contraction principle again, we deduce from theorem 2.1 LDPs for both estimators. We begin with r b T because the calculations are really straightforward. Proof of Corollary 2.2. From the contraction principle, we know that

  For I 2 , with the calculations of the second case, we already know that the derivative equals zero for β 0 verifying This is well defined if and only if pa ´2q 2 `16 p2 ´αq ą 0. And, β 0 is in the domain if and only if β 0 ă 0 and β 2 0 ď b 2 9 . All those conditions are fulfilled if and only if 9p2 ´αq 2 ď pa ´2q 2 `16p2 ´αq. As α ą 2, we obtain the condition

				β 2 0 "	p2 ´αq 2 b 2 pa ´2q 2 `16 p2 ´αq	.
		α ă ℓ a :"	10 9 `1 9	a	64 `9pa ´2q 2
	and β 0 "	´p2 ´αq b a pa ´2q 2 `16 p2 ´αq	. Thus, for 2 ă α ă ℓ a
		2 β 8p2 ´αq	ˆ1	βpa ´2q `p2 ´αqb	˙2	´β 4	ˆ1	´b β	˙2+
	and (5.6)	I 2 " inf b 3 ďβă0	#	pa ´2q 2 β 8p2 ´αq	ˆ1	`p2 ´αqb βpa ´2q	˙2 `2β	´b+	.

  5.2. The sequence tpS T , Σ T , L T qu satisfies an LDP with good rate function The proof is postponed to Appendix D.By the contraction principle applied to the rate function r I of Lemma 5.2, we obtain that ´q a T , q b T ¯satisfies an LDP with good rate function K a,b given by

	r Ipy, z, tq "	$ ' ' & ' ' %	ab 4 `8 `b2 8	y	`pa ´2q 2 8	z	`a 4	t 2 `4zpyt 2 `1q `t4 y 8pyz ´1q	if t ď 0, y ą 0, z ą 0 and yz ´1 ą 0, otherwise.
	Proof. D α,β	"	ab 4	`b2 8	y	`pa ´2q 2 8	z	`a 4	t 2 `4zpyt 2 `1q `t4 y 8pyz ´1q

K a,b pα, βq " inf ! r Ipy, z, tq|hpy, z, tq " pα, βq, py, z, tq P R 3 , yz ´1 ‰ 0 ) which reduces to K a,b pα, βq " inf

  1 y . Assuming that all conditions are fulfilled, we derive from Lemma 5.2 that where C α and A α,β do not depend on z, and are defined by First case: α ă 0. We have already shown that K a,b pα, βq " `8 if β ă 0. For β ą 0, 0 ă β 2´α ď β ´α so D z " z ą ´β α ( . We look for critical points of A α,β `Cα z ´2βz

	(5.11)	C α "	1 8	pa ´αq 2 `2 ´α and A α,β "	´α β	b 2 8	`ab 4	´aβ 4	`αβ 8	.
	Thus,
	(5.12) where D z " z ą 0|z ą ´β α and p2 ´αqz ´β ě 0 K a,b pα, βq " inf Dz " A α,β `Cα z ( . ´2βz αz `β *
	'
		r I	ˆ´α β	, z,	a	p2 ´αqz	´β˙"	A α,β `Cα z	´2βz αz `β

  then C α is null or negative so the derivative cannot vanish and K a,b is given by (5.16). then C α is positive but as α ą α a the critical point z 0 is greater than β 2´α so outside the domain D z . The infimum is reached at the boundary β

	5.16)	K a,b pα, βq "	a 4	pb ´βq	´α 8β	`b2 ´β2 ˘´β pa ´αq 2 8 pα ´2q	.
	If α P ra `4 as, If α ą a`4`2 ´2? a; a `4 `2? ? a,

2´α

of the domain and is still given by (5.16).

5.4. Proofs of Corollaries 2.3 and 2.4.

Proof of Corollary 2.4. The result is a direct application of the contraction principle to the rate function K a,b of Theorem 2.2. We did not obtain an explicit expression of the infimum.

Proof of Corollary 2.3. With the contraction principle again, we know that the pq a T q satisfies an LDP with good rate function K a pαq " inf βPR K a,b pα, βq.

  6.2. Evaluating the rate function I a,b . We want to show that (6.12) I a,b pα, βq " min pJ a,b pα, βq, K a,b pα, βqq where J a,b and K a,b are the rate functions for the two couples of simplified estimators (see Theorems 2.1 and 2.2) and I a,b is given by Lemma 6.2 We notice that K a,b pα, βq " inf

	tď0	sup D d,f	Hp0, t, d, f q and J a,b pα, βq " inf xě0	sup D d,f

  b pα, βq. Over D ά,β , we know by Lemma 6.1 that We now need to investigate the cases α " 2 and β " 0 before concluding. For β " 0 and α R s0, 2r, we already know the value of I a,b . If α P s0, 2r, D ὰ,0 " px, y, z, tq P R 3 ˆR`| x 2 " α and z " 0 ( and D ά,0 " H.And with the argument than before we obtain that for all α P s0, 2r,I a,b pα, 0q " J a,b pα, 0q " min tJ a,b pα, 0q ; K a,b pα, 0quas K a,b pα, 0q is equal to infinity. Now, for α " 2, we have already computed I a,b for all β ě 0. For β ă 0, D 2,β " H and D 2,β " px, y, z, tq P R 3 ˆR´| βy " ´2 and t 2 " ´β( .

	Λ ˚ˆx,	´α β	,	t 2 `β 2 ´α , t ˙" sup D d,f	" 1 4	´x? d ´b	´ta	ϕpf q ¯2 `ψd,f ptq
	Λ ˚ˆx, the infimum over D ά,β is greater than the infimum over ´α β , t 2 `β 2 ´α , t ˙ě Λ ˚ˆ0, ´α β , t 2 `β 2 ´α , t ȧnd !´0 , ´α We obtain that D 2,β " "ˆx , ´2 β , z, t ˙, x P R, z P R, t ď 0 *
	and with the same argument than before, the infimum over this subset is greater !´0
	than the infimum over	, ´2

*

where ψ d,f ptq does not depend on x. Thus, β , t 2 `β 2´α , t ¯, t ď 0 ) which is equal to K a,b pα, βq. β , z, t ¯, z P R, t ď 0

  1 T s and over r 1 T , 1s. On the one hand, ż 1By dominated convergence as γ ă 0, the first integral tends to zero when T goes to infinity and the second one tends to the positive constant

														b	π g . This leads to the
	following bounds, for T large enough: ż 1 T 0 e ´γ? ´T log y y g´1 dy ď 2	c π g	|γ| ? g	T	exp	4g ˆγ2 T	ż
					1 T 0	e ´γ? ´T log y y g´1 dy ě	1 2	c π g	|γ| ? g	T	exp	4g ˆγ2 T	˙.
	Combined with the result over r 1 T , 1s, it gives the announced result.
	B.2. Proof of Lemma 4.3. The upper bound easily follows from (B.2). Actually,
	for all γ ą 0 and λ P R, we obtain (B.5) H T ď β f T Γpf `1q2 f ż 1 0 ď β f T Γpf `1q2 f e |λ|	e pλ ? ? T `βT . T `βT q	? y´γ	? ´T log y y	2f `a´2 4	dy
	1 T On the other hand, with the same argument, we show that e β T ? y`λ ? T y´γ ? ´T log y´α T y y 2f `a´2 4 ż 1 e β T ? y`λ ? T y´γ dy ď 1 T ´eβ T ? y`λ ? ? ´T log y dy T y´γ ? ´T log y ď 1 ˆsup r 1 T ,1s e λ ? T `βT ´γ? ´T log 1 T ? T ? log T ¯. " O ´epλ´γq	ď
					ż 1 1 T	e λ ?	T y´γ	?	´T log y´α T y y	2f `a´2 4	dy " O ´epλ´γq ?	T	?	log T	¯.
	Over r0, 1 T s, e λ ? ż 1 T 0 e λ ? T y´γ T y , e βt ? y and e ´αT y are bounded. So we have ? ´T log y´α T y y 2f `a´2 4 0 dy ě e ´αT {T ´|λ| ż 1 T e ´γ? ´T log y y	2f `a´2 4	dy
	and
	ż 1 T 0	e β T	? y`λ ?	T y´γ	? ´T log y´α T y y	2f `a´2 4	dy ď e |λ|`α T {T `βT {	?	T	0 ż 1 T	e ´γ? ´T log y y	2f `a´2 2	dy.
	Using the change of variable given by z " obtain:	?	´log y	`γ? T 2g , where g " 2f `a`2 4	, we
	ż 1 T 0	e ´γ? ´T log y y g´1 dy " 2e " 2e	γ 2 T 4g γ 2 T 4g	ż `8 ? log T ˜ż `8 `γ? T 2g ? log T `γ? e ´g z 2 T 2g e ´g z 2 z dz ˆz ´γ? T 2g ´γ? ˙dz T 2g ż `8 ? log T	2g `γ? T	e ´g z 2 dz	¸.

  gq Γ p2γ `2gq e

	1	e β T	? y`λ ?	pλ T y´α T y y γ`g´1 dy ? T `βT q 2 8α T ď D ´2pγ`gq ´´β T ´λ? ż T ? 2α T
	0 2pβ T q f p2α T q ´pγ`gq Γp2γ`2gq 2 f Γp1`f q	e	pλ	? T `βT q 2 8α T	D ´2pγ`gq	´´β T ?	´λ? 2α T T	¯,
	where D ´2pγ`gq is the parabolic cylinder function defined by 9.250 in [8]. But, if λ ă 0, for T large enough, ´βT ´λ? T ą 0 so formula 9.246(1) of [8] gives D ´2pγ`gq ˆ´β T ´λ? T ? 2α T ˙" ˆ´β T ´λ? T ? 2α T ˙´2pγ`gq e ´pλ ? T `βT q 2 8α T

  All those results are still true for λ " 0. B.4. Proof of Lemma 4.5. As previously, we use (B.2) to find lower and upper bounds for K T as T goes to infinity, for all λ ą 0. We consider two cases depending on the sign of the exponent γ `g ´1. ' If γ `g ´1 ď 0: As in the proof of Lemma 4.4, we have the following upper bound

	(B.19) gence, and A 2 " A 1 " ż `8 1´λ ? T `βT K T ď 2α T e ´αT u 2 u du tends to zero for all λ ą 0 by dominated conver-2 pβ T q f 2 f Γp1 `f q e pλ ? ˆA1 ẇhere T `βT q 2 4α T `βT A 2 2α T ż `8 1´λ ? e ´αT u 2 du tends to the positive constant 2 a π d´b . Thus, T `βT 2α T for T large enough,
	(B.20)	K T ď 2 1´f	c	d	π
		4α T	˜ż	`8 1´λ ? T 2α T	e ´αT u 2 u du	`λ? T 2α T	2α T 1´λ T ? ż `8	e ´αT u 2 du	Ḩowever,
	(B.16)	the first integral is easily computable: ż `8 1´λ ? T 2α T e ´αT u 2 u du " 1 2α T exp ˜´α T ˆ1	2α T ´λ? T	˙2a
	nd for the second one, we clearly have the following upper bound, (B.17) ş `8 1´λ ? T 2α T e ´αT u 2 du " ş `8 1´λ ? T 2α T e ´αT u 2 u ˆ1 u du ď 1 1´λ ? T 2α T ş `8 1´λ ? T 2α T e ´αT u 2 u du ď 1 1´λ ? T 2α T 1 ´λ? T 2α T 2α T exp ˆ´α T ´1	¯2˙.
			e ´αT 2α T	˜1	`λ? T 1 2α T ´λ? T 2α T	¸exp	T ´λ?	¯.

Using the fact that λ ă 0 and combining (B.15), (B.16) and (B.17), we show the announced result:

(B.18) K T ě 2 1´f pβ T q f Γp1 `f q

Appendix B: Proofs of Lemmas 4.2 to 4.5

The four following proofs rely on lower and upper bounds for the modified Bessel function of the first kind, given by formula (6.25) of [START_REF] Luke | Inequalities for generalized hypergeometric functions[END_REF]. More precisely, for all z ą 0 and ν ą ´1 2 , we have:

Appendix C: proof of Lemma 5.1

We call L the pointwise limit of the cumulant generating function of the triplet pX T {T, S T , Σ T q. We notice that Lpλ, µ, νq " Λpλ, µ, ν, 0q where Λ is given by Lemma 4.1. We easily deduce from Remark 4.1 that the function L is steep. We apply the Gärtner-Ellis theorem: the rate function is given by the Fenchel-Legendre transform of L on its effective domain. txλ `yµ `zν ´Lpλ, µ, νqu .

For y and z we recognize the exact same term than in Theorem 3.2. So with the same argument, we show that (C.2) for y ď 0, z ď 0 or yz ´1 ď 0, Ipx, y, zq " `8.

Besides, if x ă 0 then for λ tending to ´8, λx Ñ `8 and Ipx, y, zq " `8 because Λ does not depend on λ for λ ă 0. Moreover, for x ą 0, the term on λ is always negative for λ negative and sometimes positive if λ is positive. So the supremum is necessarily reached for λ ą 0. We finally have to calculate:

txλ `yµ `zν ´Λpλ, µ, νqu with x ě 0, y ą 0, z ą 0 and yz ´1 ą 0. We do exactly as in Appendix A, we are looking for critical points and the calculations are very similar. We find:

pyz´1q 2 ¯and ν 0 " 1 8 ´pa ´2q 2 ´px 2 `2q 2 pyz´1q 2 ¯, which leads easily to I.

Appendix D: Proof of Lemma 5.2

The pointwise limit of the cumulant generating function of the considered triplet is easily given by Λp0, µ, ν, γq where Λ is defined in Lemma 4.1: With the same argument than for the other couple of simplified estimators, we show that r Ipy, z, tq " `8 for y ă 0, z ă 0 or yz ´1 ă 0. We also notice that for t ą 0 the expression inside the supremum tends to infinity as γ goes to infinity. So r Ipy, z, tq " `8 for t ą 0. Besides, for t ď 0, the part involving γ is always negative for γ ě 0 and sometimes positive for γ ă 0. It implies that the supremum is necesseraly reached for γ ă 0. Replacing µ and ν by their expression on d and f , we obtain, for y ą 0, z ą 0, yz ´1 ą 0 and t ď 0, r Ipy, z, tq " sup

We investigate critical points. We obtain

Replacing it into the expression of r I, we easily get the announced result.

Appendix E: Proofs of asymptotic normality results for the two couples of simplified estimators

The key to obtain those results is Slutsky's lemma. Indeed, we have

and we show that the right-hand side of (E.1) converges to zero in probability. Namely, it is well known (see for instance Lemma 3 of [START_REF] Overbeck | Estimation for continuous branching processes[END_REF]) that S T converges almost surely to ´a b and V T to 2 a´2 . And for all ε ą 0, we have Ý ÝÝÝ Ñ 0


Barbara Gallina
email: barbara.gallina@mdh.se

Mattias Nyberg
email: mattias.nyberg@scania.com

Reconciling the ISO 26262-compliant and the agile documentation management in the Swedish context

Keywords: Safety-critical systems, ISO 26262, Swedish culture, documentation management, safety cases

. Thus, agile and ISO 26262-compliant documentation management styles seem to form an odd couple. When this couple is located in the Swedish cultural context, reconciliation and negotiation within it represent a true challenge. In this paper, based on the state of practice in industrial settings, we report about our findings and propose our envisioned solution to face this challenge and get a balance. Finally, conclusion and perspectives for future work are also drawn.

I. INTRODUCTION

Safety standards (e.g. ISO 26262 [START_REF]Road vehicles Functional safety. International Standard[END_REF]) define safety lifecycles to be adopted during the development of safety-critical systems. Typically, these life-cycles describe or prescribe: how to break down the work in order to develop the desired systems (i.e., breaking down the work into atomic units of work, e.g., steps or into composite units of work, e.g., phases, activities, etc.); the roles that should be involved, the work products that should be taken in input or should be produced in output, the methods to be adopted to perform the work.

Besides, safety life-cycles targeting the development, the standards also proposes/imposes guidelines for managing the hundreds of hundreds of documents. Managing the documentation represents a relevant activity especially in safety-critical systems engineering. The documentation process is usually tightly coupled with the development life-cycle. Life-cycle's work products represent immediate as well as direct evidence to be used during the safety assessment process to support the claims about system's safety [START_REF] Nair | Safety evidence traceability: Problem analysis and model[END_REF]. More specifically, the left-hand side work products of the V-model (e.g., requirement specification) represent immediate evidence; while the right-hand side work products of the V-model (e.g., verification results) represent direct evidence. Improper documentation/evidence management may indirectly result in certification risk [START_REF] Nair | Safety evidence traceability: Problem analysis and model[END_REF]. In the context of ISO 26262, for instance, the goal of the documentation process is to make documentation available: 1) during each phase of the entire safety lifecycle for the effective completion of the phases and verification activities; 2) for the management of functional safety, and 3) as an input to the functional safety assessment.

It should be noted that ISO 26262, which was released in 2011, can still be considered a new entry in the automo-tive domain. At the time being ISO 26262 addresses functional safety of electrical/electronic (E/E) and programmable electronic safety-related systems within road vehicles with a maximum gross weight of 3.5 tons. Heavy trucks are not yet contemplated by the standard. A new version of the standard is expected to be issued by 2018. Manufacturers of heavy trucks, however, are keeping a constant eye on the ongoing revision of the standard and they are strategically planning its timely adoption by 2018 [START_REF] Dardar | Industrial experiences of building a safety case in compliance with iso 26262[END_REF], [START_REF] Gallina | VROOM & cC: a Method to Build Safety Cases for ISO 26262-compliant Product Lines[END_REF]. Gap analysis as well as alignment investigations are typically performed to identify necessary adaptations/changes to be introduced in order to reach a state of practice that can be considered compliant with the state of the art, represented by the standard. In some contexts, the adoption of ISO 26262 may result much more challenging than in others due to cultural background as well as agilityoriented mentality. In this paper, we focus our attention on one of such challenging context: the Swedish (due to its cultural specificity) and agility-oriented context. After having analyzed such context, we provide our solution proposal aimed at easing a smoothly introduction of ISO 26262 within such context.

The rest of the paper is organized as follows. In Section II, we provide essential background information. In Section III, we identify a true challenge related to documentation management. In Section IV we present our proposal aimed at reconciling the Swedish odd couple. Finally, in Section V, we present some concluding remarks and future work.

II. BACKGROUND

In this section, we present the background information related to the problem space. In particular, in Section II-A, we recall ISO 26262 requirements in general and more specifically we focus on requirements related to documentation management. In Section II-B, we recall agile principles. In Section II-C, we present the current findings concerning the Swedish culture. Finally, in Section II-D, we introduce the Swedish and agile-oriented context at Scania.

A. ISO 26262-compliant documentation management

In this section, to make the paper self-contained, we recall essential information related to ISO 26262-compliant development life-cycle and documentation management. The intention is to let the reader realize the complexity of documentation management in the context of automotive safety-critical systems engineering due to the huge amount of work-products that are expected to be provided. The intention is also to let the reader realize the necessity of roles and responsibility in the context of automotive safety-critical systems engineering.

ISO 26262 [START_REF]Road vehicles Functional safety. International Standard[END_REF] is a functional safety standard that targets the automotive domain. This standard introduces safety integrity levels, called ASILs, which are specific for the automotive domain and requires the adoption of specific processes for developing the system, managing the documentation, qualifying tools, etc.. Thus, to develop an item in compliance with the standard, the development process shall be adopted and all the work-products shall be provided, including a safety case, which is supposed to progressively compile the work products that are generated during the processes. Similarly, the requirements concerning documentation management should be fulfilled. According to part 10.2 of ISO 26262, the documentation can take various forms and structures and tools can be used to generate documents automatically. In part 10 of the standard it stated that "the documentation process shall be planned in order to make documentation available. The documents should be: a) precise and concise, b) structured in a clear manner, c) easy to understand by the intended users, and d) maintainable". In part 10.4.4, it is stated that the structure of the entire documentation should consider inhouse procedures and working practices. It shall be organized to facilitate the search for relevant information. The standard, however, is not fully rigid. Flexibility is allowed if properly introduced, i.e., via the application of tailoring rules [START_REF]Road vehicles Functional safety. International Standard[END_REF]. Moreover, since the standard is a human creation and thus only perfectible but not perfect, it is likely that additional shortcomings, beyond those that have already been spotted, are present. Moreover a general consensus on its interpretation is not available yet [START_REF]5th International Conference on ISO 26262[END_REF], [START_REF] Ellims | Agonising over asils: Controllability and the in-wheel motor[END_REF]. In part 6.4.2.1, it is also stated that a "project manager shall be appointed at the initiation of the item development". Once appointed, he/she shall ensure that the safety manager is appointed, who shall be responsible for the planning and coordination of the functional safety activities in the development phases of the safety process.

B. Agile documentation management

The agile manifesto is constituted of twelve principles [START_REF]Agile Manifesto[END_REF]. These principles can be summarized by the following four slogans: 1) individuals and interactions over processes and tools; 2) working software over comprehensive documentation; 3) customer collaboration over contract negotiation; 4) responding to change over following a plan. From the above self-explanatory principles, we can infer that the agile way of working is radically informal and flexible.

C. Swedish (business/management) culture

The Swedish culture is characterized by some fundamental principles, which can be identified by a handful of interrelated keywords: consensus-based informal decision making, flat hierarchy, trust, team-focused leadership, fika. In what follows, we explain these keywords since we believe that to the uninitiated they might be unintelligible or misleading. Our explanation builds on top of research studies [START_REF] Salminem-Karlsson | Swedish and indian teams: Consensus culture meets hierarchy culture in offshoring[END_REF], [START_REF] Isaksson | Chinese views on swedish management-consensus, conflict-handling and the role of the team[END_REF], which were aimed at identifying and describing such principles. Consensus-based informal decision making -Each teammember is expected to express his/her opinion on what product to make and how to make it. Each team-member is typically considered as a critical resource for market know how. Topdown imposition of decisions are typically considered dysfunctional for working well-being. Flat hierarchy -A semidemocracy characterize the working places. Distributed control replaces centralized and vertical control. Trust -Juniors are expected to feel responsible and act. Seniors are expected to abstain themselves and coach and delegate. Trust replaces command and control. Team-focused leadership -Formal leaders are expected to empower the team. Their role is not to be dominant. They should inspire and set examples. They should never override others' responsibility. Lagom -this Swedish term can be translated in English as moderation. "Lagom"may imply that the employees take care of each other' s well-being and resist managerial pressures by conforming to this cultural norm [START_REF] Wieland | Struggling to manage work as a part of everyday life: Complicating control, rethinking resistance, and contextualizing work/life studies[END_REF].

These main keywords are then made cohesive via another one: fika. Fika is a Swedish term that denotes a recreation time-slot that Swedish employees introduce twice a day. A fika is supposed to be spent collectively in order to strengthen the team. During a fika, employees share time and space, practice the art of behaving lagomly, build trust among each other via conviviality, and, as a consequence, enable the creation of flat hierarchy, where team-focused leadership becomes natural. This amalgam of keyword permeates the Swedish society and functions in homogeneous contexts, where people share the cultural background. As investigated in the previously cited studies, when Swedes have to co-work with non-Swedes, some adaptations and negotiations are needed.

D. Safety culture and state of practice at Scania

Scania is a major Swedish automotive industry manufacturer of commercial vehicles -specifically heavy trucks and buses. Scania develops, manufactures and sells trucks with a gross vehicle weight of more than 16 tonnes intended for long-distance haulage, regional, and local distribution of goods, as well as construction haulage. Scania's bus range is concentrated on bus chassis, intended for use in tourist coaches, as well as urban and intercity traffic. Both trucks and buses are classified in typed series and characterized by various variations. From a management point of view, Scania fully incorporates the Swedish management style introduced in Section II-C. On one page of Scania's website [START_REF]Scania-strategy[END_REF], for instance, we can read: "Respect for the individual means recognizing and utilising each employees knowledge, experience and ambition in order to continuously improve and develop working methods. Inspiration and new ideas are born out of day-to-day operations. This helps ensure higher quality, efficiency and job satisfaction." At Scania, imposed processes, assigned jobs, and status rather than merit are not welcome. Roles are weakly defined and destructive interference is carefully avoided. Concerning quality, Scania states "Elimination of all forms of waste is the way Scania can ensure that all deliveries meet the expectations of demanding customers. Deviations from targets and standards are used as a valuable source of continuous improvement in Scanias processes." [START_REF]Scania-strategy[END_REF]. This statement is clearly in line with the agile manifesto.

III. A TRUE CHALLENGE

The background has set the stage and introduced the main characters and their context. From the background, we have learnt that documentation management is perceived in an almost opposite way from the agile-community and the safety-standard community. In the context of automotive safety-critical software engineering, these two communities necessarily come together. However, given their opposite way of working, these two communities represent an odd couple. When this couple lives in Sweden, reconciliation of and negotiation within these two communities represent a true challenge. As we can retrieve from the background, the Swedish culture and the agile manifesto exhibit some similarities. For instance, the agile slogan "Individuals and interactions over processes and tools" is related to the Swedish trust and flat hierarchy. Introduction of specific roles, conflicts with the coaching and team-based leadership style.

IV. SOFT AND HARD PIECES OF SOLUTION

In this section, we present our solution which comprises soft and hard pieces. Both pieces borrow from OSLC (Open Services for Lifecycle Collaboration). Thus, to let the reader follow the discussion, first of all, in Section IV-A,we recall basic information on OSLC. Then, in Section IV-B, we present the soft piece of our solution, while the hard piece is presented in Section IV-C. These pieces of solution are being developed in the framework of two ongoing projects: Espresso and Gen&ReuseSafetyCases. Espresso [START_REF]-04446-ESPRESSO[END_REF] is aimed at building a Scania-specific model-based solution aimed at making documentation management more efficient and more aligned to ISO 26262. To reach its goal, Espresso will benefit from the cooperation with Gen&ReuseSafetyCases [START_REF]Gen&ReuseSafetyCases-SSF[END_REF].

Gen&ReuseSafetyCases is aimed at proposing Scania-specific solutions for enabling model-based self assessment via safety case generation in compliance with ISO 26262.

A. OSLC

OSLC is a standard that targets tools used during a product's life cycle and enables their integration and interoperability. OSLC 3.0 [START_REF]Open Services for Lifecycle[END_REF] is the current version of OSLC. Tools for requirements engineering, design, implementation, etc. are expected to interoperate in a traceable manner i.e. traceability between the respective work products can be easily retrieved and shown. To enable interoperability, different specifications, called domains, need to be provided. Requirements Management domain (RM) represents a sample of these domains. OSLC builds on top of Linked Data [START_REF]Linked Data[END_REF], Resource Description Framework (RDF) [START_REF]RDF Primer[END_REF], and HTTP protocol. Each work product is described as an HTTP resource, identified via a Uniform Resource Identifier (URI). Work products are manipulated via HTTP methods (i.e., GET, POST, etc.). To interoperate via a work product, a tool that acts as a provider has to associate an URI to the work product and post it; a tool acting as consumer can get the work product via the URI.

B. Soft piece of solution

The soft piece of solution focuses on cultural and management aspects and consists of a negotiation and reconciliation strategy compatible with the Swedish environment. Since agility-oriented (documentation) management can hardly coexist with the ISO 26262-compliant (documentation) management, trade-offs are needed. Our soft piece of solution is aimed at triggering the employee's sense of responsibility and consequent participation in establishing trade-offs to be implemented in the hard solution.

As we have recalled in Section II-A, ISO 26262 is under revision and a consensus concerning its interpretation is still to be achieved. We believe that these undefined circumstances are favorable and may constitute a fertile soil to align employees towards a common strategy: achieving a Scania interpretation of ISO 26262, jointly with safety assessors, in order to comply with the standard in an agile manner, i.e. complementary extension of the current state of practice. The idea is to build an ISO 26262-compliant, semantic, and distributed-butinterconnected Scania management. Figure 1, adapted from the original OSLC figure [START_REF]OSLC Community -open-services[END_REF], gives an intuition of the ISO 26262compliant Open-minded Teams for Lifecycle Collaboration. In line with the Swedish management style, teams will keep their autonomy. However, interoperability will be enabled in order to identify and address cross-team needs. According to Note 1 in part 6.4.3.1, the safety manager can delegate tasks to persons that possess the required skills, competences and qualifications. Already in place team leaders will be trained with respect to ISO 26262. Teams, working at different abstraction levels will take the responsibility to choose appropriate ways to comply with the standard according to their current way of working. Their responsibility entails the selection of process elements with the adequate ASIL-related stringency (i.e., activities and associated methods and tools (if any) as well as work products). To maximize agility and stick to the current Scania management, lagomly the just enough number of work products will be provided. ISO 26262-compliant tailoring will be applied under the supervision of safety assessors in order to identify allowed minima. Tool-supported techniques will be privileged with respect to manual techniques in order to increase automation and enable semi-automatic generation.

C. Hard piece of solution

Once cultural and managerial aspects are reconciled via establishment of trade-offs, the hard piece of solution may focus on the infrastructure for documentation management. More specifically, the solution consists of a Scania-specific modelbased development aimed at integrating current and future in-house/off-the-shelf/outsourced ISO 26262-compliant tools within an interoperable tool-chain satisfying Scania purposes and requirements, including scalability. To make the tool-chain interoperable, we intend to adopt OSLC and create a family of interoperable tools that support all phases of the product lifecycle, via the exploitation of the OSLC-related family of core web service specifications plus Scania-specific OSLCcompliant domains. We opt for OSLC since, as stated by Biro [START_REF] Biro | Open services for software process compliance engineering[END_REF], "it has the potential to have a determining impact on the future of software process compliance" and we believe that this potential is for any process compliance, including documentation management related to the item development. We also believe that in terms of the so called 'Hype-cycle", OSLC will reach the plateau of productivity phase. The intermediateterm goal is to explore real feasibility concerning large-scale tool integration. The long-term goal intention is to make this tool-chain available to enable cooperation throughout the supplier-manufacturer chain.

Within ESPRESSO, the current Scania documentation (mainly in .docx and .excel-based format) should be converted into machine-representable or even better machineunderstandable Linked Data [START_REF]Linked Data[END_REF] (more precisely, RDF-based data formats [START_REF]RDF Primer[END_REF]) to be consumed/provided within the OSLCbased documentation/development management tool-chain. At the time being, only a portion of the tool-chain is implemented. A tool called Requirements Specifier (RS), for instance, permits requirements engineers to specify requirements at each level of the ISO 26262-compliant hierarchy. Thus, Safety Goals (SGs), Functional Safety Requirements (FSRs), Technical Safety Requirements (TSRs), Hardware and software requirements (HSRs and SSRs) are supported. RS implements the notion of contract structure proposed by J. Westman et al [START_REF] Westman | Extending contract theory with safety integrity levels[END_REF]. RS is expected to provide URI-tagged resources based on ISO 26262-compliant and Scania-specific RM domain. Assumed that safety goals are formulated to avoid unreasonable risk, this traceable and hierarchical requirements structure can be used for arguing about absence of such risk [START_REF] Dardar | Industrial experiences of building a safety case in compliance with iso 26262[END_REF]. Within Gen&ReuseSafetyCases, to generate such arguments, we build upon model-based safety certification ideas developed by I. Sljivo et al [START_REF] Sljivo | Generation of safety case argument-fragments from safety contracts[END_REF] and by B. Gallina [START_REF] Gallina | A model-driven safety certification method for process compliance[END_REF] and we develop a tool for safety case creation. In principle the safety case creator should generate the safety case representation by consuming interrelated work products, which constitute immediate, direct, indirect evidence as well as (a)tomic goals. To argue about process compliance, ISO 26262-compliant process information can be inferred via the interconnected work products. Thus a process is not imposed from the top but is obtained bottom-up semi-automatically. By doing so, we can avoid the introduction of additional hierarchical human roles and thus flat hierarchy is preserved. The unique safety manager will be replaced by a safety case generator, which will consume and compile the work performed by the different teams according to thoroughly specified generative rules. The tool-chain is expected to guarantee the availability/searchability/maintainability of the documents as well as enhanced precision and structure as required by ISO 26262, part 10. Finally, semi-automation is expected to save time and thus reduce waste, in line with the agile practices within Scania.

V. CONCLUSION AND FUTURE WORK

In this paper, we have presented a true challenge: ISO 26262-compliant documentation management in agilityoriented Swedish context. After having introduced a broader perspective on the challenge, we have reported our finding within Scania. Then, we have proposed our OSLC-based soft and hard solution to contribute in facing such challenge and our research agenda towards the concretization of our solution.

Fig. 1 .

 1 Fig. 1. Open-minded Teams for Lifecycle Collaboration.

Acknowledgments: This work has been partially financially supported by the Swedish Foundation for Strategic Research via the SSF Gen&ReuseSafetyCases project [13] project and the Vinnova Espresso [12] project. We also thank C. Mattila.