Luciano Fusina
email: fusina@ms.fci.unibo.it

Elisabetta Canè

Riccardo Tarroni

Klaus Burczyk

Elisabetta Cané

Prof Luciano Fusina

+ ν5 high resolution infrared bands of FClO3

Keywords: perchloryl fluoride, high resolution infrared spectra, combination bands, rovibrational analysis, anharmonic resonance perchloryl fluoride, high resolution infrared spectra, combination bands, rovibrational analysis, anharmonic resonance, anharmonicity constants

published or not. The documents may come     The ν2 + ν4 and 2ν2 + ν5 high resolution infrared bands of FClO3

Introduction

Among the prolate symmetric rotors of C 3v symmetry, perchloryl fluoride has been the subject of extensive spectroscopic investigations during the last 20 years. The great majority of these studies have been performed using mono isotopic samples, F 35 Cl 16 O 3 , F 37 Cl 16 O 3 , F 35 Cl 18 O 3 , and F 37 Cl 18 O 3 , to reduce the spectral congestion due to the isotopic effect of the chlorine atom. The fundamental bands of its six vibration normal modes, ν 1 , ν 2 , and ν 3 , of A 1 symmetry, ν 4 , ν 5 , and ν 6 , of E symmetry, have been ro-vibrationally analysed from high resolution infrared spectra [1][2][3][4][5][6][7][8][9]. Furthermore, the analysis was extended to first overtone and combination bands for all isotopic species and several anharmonicity constants have been derived [10][11][12][13][14]. The ground state of this quasi-spherical rotor has been fully characterized too [5] and the equilibrium molecular structure has been derived [9]. The conspicuous amount of available experimental data prompted the ab initio calculations of the anharmonic force field of perchloryl fluoride at high level of theory [14]. In most cases, theoretical values of the molecular parameters, fundamental frequencies and anharmonicity constants, compare well with the experimental counterparts.

24

x is one of the anharmonicity constants whose experimental values are still lacking.

The theoretical values, -0.05, 0.07, 0.50, and 1.17 cm -1 ( the positive signs of the last three constants were reported incorrectly in [14]), for F 35 Cl 16 O 3 , F 37 Cl 16 O 3 , F 35 Cl 18 O 3 , and F 37 Cl 18 O 3 , respectively, are too small and the positive signs are opposite in comparison with those of the other ij x constants, which are in the range from -8 to -0.3 cm -1 . A possible explanation of this anomaly could be the neglect of the anharmonic interaction affecting the v 2 = v 4 = 1 state in the calculation. In fact, the cubic term of the anharmonic potential, 245 2 4 5 q q q φ , is responsible of the interaction between the v 4 = 1 and the v 2 = v 5 = 1 states [3,8,9] and between higher excited levels of the same symmetry, characterized by at least one and ν 2 + ν 4 + ν 5 bands of F 35 Cl 16 O 3 [11].

To clarify this point, further investigations are requested. The 24 x constants can be obtained from the ro-vibration analyses of the ν 2 + ν 4 and 2ν 2 + ν 5 bands, including the interaction between the excited states in the model Hamiltonian. In addition, the ab initio values of the 24 x constants should be recalculated taking into account the anharmonic resonance contribution.

This work presents the analysis of the ν 2 + ν 4 and 2ν 2 + ν 5 bands from the spectra recorded using mono isotopic pure samples of the four symmetric isotopologues of perchloryl fluoride. The combination bands are of perpendicular type and are located from 1920 to 2060 cm -1 . The results of the ro-vibration analysis of these combination bands will be outlined in the following sections.

In addition, the results of the calculation of the ab initio values of 24 25 45 45 , , and x x x g will be presented. For this calculation we have removed the resonant term from the analytic perturbative formulae. Finally, the experimental 24 x constants will be obtained combining the deperturbed ν 2 + ν 4 band origins with those of the ν 2 and ν 4 fundamentals [1][2][3][4]8,9] and compared to the corresponding ab initio values.

In the following the studied isotopologues will be designated by a short notation, i.e. (35,[START_REF] Mills | Theoretical Chemistry[END_REF] is used to indicate F 35 Cl 16 O 3 .

Experimental

The synthesis of the mono isotopic samples was reported in [5]. In the same paper the recording conditions and the calibration details of the Fourier transform infrared spectra are listed in Table 1,columns X and V,for (35,[START_REF] Mills | Theoretical Chemistry[END_REF] and (37,[START_REF] Mills | Theoretical Chemistry[END_REF], respectively, and in and L, for (35,18) and (37,[START_REF] Papousek | Molecular Vibrational-Rotational Spectra[END_REF], respectively. All spectra were recorded using a Bruker 120 HR interferometer, at Wuppertal. The resolution of (35,[START_REF] Mills | Theoretical Chemistry[END_REF] and (37,[START_REF] Mills | Theoretical Chemistry[END_REF] spectra is quoted 4.4×10 -3 cm -1 , the data precision is estimated to be 3.0×10 -4 cm -1 , while the resolution of (35,[START_REF] Papousek | Molecular Vibrational-Rotational Spectra[END_REF] and (37,[START_REF] Papousek | Molecular Vibrational-Rotational Spectra[END_REF] spectra is 4.3×10 -3 cm -1 and the data precision 2.0×10 -4 cm -1 .

Spectra description

The ν 2 + ν 4 bands have medium strong intensities at the chosen recording conditions and are clearly apparent in each spectrum, being centred at about 2032, 2007, 1984 and 1958 cm -1 for (35,16), (37,[START_REF] Mills | Theoretical Chemistry[END_REF], (35,[START_REF] Papousek | Molecular Vibrational-Rotational Spectra[END_REF] and (37,[START_REF] Papousek | Molecular Vibrational-Rotational Spectra[END_REF], respectively. On the contrary, the 2ν 2 + ν 5 bands are very weak and their presence could be inferred from the anomalous features observed in the ∆K = -1 side of the ν 2 + ν 4 bands, where the 2ν 2 + ν 5 band centres are located, see Figure 1. The ν 2 + ν 4 and 2ν 2 + ν 5 band centres differ by about 16, 15, 11, and 9 cm -1 in (35,16), (37,[START_REF] Mills | Theoretical Chemistry[END_REF], (35,[START_REF] Papousek | Molecular Vibrational-Rotational Spectra[END_REF] and (37,[START_REF] Papousek | Molecular Vibrational-Rotational Spectra[END_REF], respectively, with the 2ν 2 + ν 5 bands located at lower wavenumbers.

[Insert figure 1]

The general appearance of ν 2 + ν 4 is similar for the four isotopologues, so the following description refers to all the analysed spectra and the observed minor differences will be pointed out. The bands extend for about 50 cm -1 and their shapes are characterized by the p Q K and r Q K sub-branches which extend to higher and lower wavenumbers from the r Q 0 position. Only the sub-branches with " K = 3, 6, 9,… are observed since the allowed levels are those with k l -= 3p, where p is an integer, because the three identical oxygen atoms have nuclear spin quantum number I = 0. Successive p Q K and r Q K sub-branches are almost equidistant, with a separation of ca. 0.45 cm -1 , with the J structure degrading to lower wavenumbers. The ∆K = -1 side of the band is mainly constituted by patterns comprising p P K (J), p P K+3 (J+1), p P K+6 (J+2)… transitions, which extend from low to high wavenumbers, in (35,[START_REF] Mills | Theoretical Chemistry[END_REF], (37,[START_REF] Mills | Theoretical Chemistry[END_REF] and (35,[START_REF] Papousek | Molecular Vibrational-Rotational Spectra[END_REF], see Figure 2. In (37,[START_REF] Papousek | Molecular Vibrational-Rotational Spectra[END_REF] the transitions in a pattern reverse the direction extending from high to low wavenumbers for " 9 K ≤ , see Figure 3.

[Insert figures 2 and 3 near here]

The structure of the R branches can be described as a succession of r R K (J), r R K+3 (J+1), r R K+6 (J+2)… transitions extending to low wavenumbers for increasing J and K values but reversing their direction for " 21 K ≥ in (35,[START_REF] Mills | Theoretical Chemistry[END_REF], " 15 K ≥ in (37,16), and " 9 K ≥ in (35,[START_REF] Papousek | Molecular Vibrational-Rotational Spectra[END_REF], as it can be observed in Figures 456.

[Insert figures 4,5,6 near here] In (37,[START_REF] Papousek | Molecular Vibrational-Rotational Spectra[END_REF] the r R K (J) transitions constitute groups characterized by the same J value and increasing K from high to low wavenumbers and reversing their direction for " 12 K ≥ , see Figure 7.

[Insert figure 7 near here] The p R and r P branches are generally weak ( see Figure 2 ) and have been mainly identified from the assigned r R and p P transitions with the help of ground state combination differences (GSCD) obtained from the precise ground state molecular constants [5]. In total, 3334, 2334, 2521, and 2441 transitions have been assigned to ν 2 + ν 4 of (35,[START_REF] Mills | Theoretical Chemistry[END_REF], (37,[START_REF] Mills | Theoretical Chemistry[END_REF], (35,[START_REF] Papousek | Molecular Vibrational-Rotational Spectra[END_REF] and (37,[START_REF] Papousek | Molecular Vibrational-Rotational Spectra[END_REF], respectively. The presence of the 2ν 2 + ν 5 bands in the investigated spectral range is highlighted by the enhancement of the line densities, due to the Q branch structure close to the band centres, at about 2017 cm -1 in (35,16), 1993 cm -1 in (37, [START_REF] Mills | Theoretical Chemistry[END_REF], 1974 cm -1 in (35,18) and 1952 cm -1 in (37,18), see Figure 1. The transitions of 2ν 2 + ν 5 belonging to the and v 2 = 2, v 5 = 1 levels, were identified after the completion of the assignments of ν 2 + ν 4 .

Two r R K (J) sub-branches of 2ν 2 + ν 5 for (37,[START_REF] Papousek | Molecular Vibrational-Rotational Spectra[END_REF] are shown in Figure 3; their intensities are comparable to those of the p P K (J) transitions of ν 2 + ν 4 observed in the same region. This is peculiar of (37,[START_REF] Papousek | Molecular Vibrational-Rotational Spectra[END_REF] since the anharmonic resonance is effective on levels with values of the K quantum number lower than those in (35,[START_REF] Papousek | Molecular Vibrational-Rotational Spectra[END_REF], (37,[START_REF] Mills | Theoretical Chemistry[END_REF] and (35,[START_REF] Mills | Theoretical Chemistry[END_REF]. A similar behaviour was evidenced in the analyses of ν 4 / ν 2 + ν 5 [3,8,9]. The number of the transitions assigned to 2ν 2 + ν 5 are 64, 34, 236, and 757, for (35,16), (37,[START_REF] Mills | Theoretical Chemistry[END_REF], (35,[START_REF] Papousek | Molecular Vibrational-Rotational Spectra[END_REF] and (37,[START_REF] Papousek | Molecular Vibrational-Rotational Spectra[END_REF], respectively.

Analysis

Band analyses

We started the analysis from the ν 2 + ν 4 band of (35,[START_REF] Mills | Theoretical Chemistry[END_REF], which is characterised by well distinguishable features. First, about one hundred strong lines constituting the regular patterns of the p P branches were tentatively assigned and the corresponding p Q transitions were identified by means of GSCD. Subsequently, we concentrated on the ∆K = +1 side of the band, at a first glance more irregular in its general appearance. After having plausibly assigned about fifty r R transitions, the overall data set was fitted as outlined in the following.

The ro-vibration analyses of ν 4 / ν 2 + ν 5 [3] and 2ν 4 / ν 2 + ν 4 + ν 5 [11] dyads for (35,[START_REF] Mills | Theoretical Chemistry[END_REF] evidenced the existence of perturbations due to the anharmonic resonance between the excited states. We expected that the same interaction could affect the rotational structure of the v 2 = v 4 = 1 and v 2 = 2, v 5 = 1 states. In consequence, the ν 2 + ν 4 assigned transitions were fitted taking into account the v 2 = 2, v 5 = 1 levels and the anharmonic interaction perturbing them. The initial values of the rotational and centrifugal distortion constants of the excited states were obtained from the corresponding ground state parameters [5] and their vibration (v 1, v 1, 1) ( ) ( )

ν ν = = = ± = + + G l x (1) 2 5 5 2 0 2 5 0 22 25 (v 2, v 1, 1) ( ) ( ) 2
ν ν ν = = = ± = + + + + G l x x ( 2
)
where 0 ( ) i ν is the origin of the indicated fundamentals or combination bands [2,3] and ij

x are the anharmonicity constants [3,10,14]. Equations ( 1) and ( 2) have been obtained from the classical formula of vibrational term value referred to the ground state [START_REF] Herzberg | Molecular Spectra and Molecular Structure. II Infrared and Raman Spectra of Polyatomic Molecules[END_REF].

The term values of the excited ro-vibration levels were obtained as

E v (J, k, l )/hc = 0 v E + B v [J(J + 1) -k 2 ] + A v k 2 -2(Aζ) v k l -D vJ [J(J + 1)] 2 -D vJK [J(J + 1)]k 2 -D vK k 4 + η vJ [J(J + 1)]k l + η vK k 3 l ( 3
)
where v = (v 2 = v 4 = 1) or v = (v 2 = 2, v 5 = 1), the signed quantum number k = ±K, and l = ± 1.

The off diagonal matrix element resulting from the anharmonic resonance term in the Hamiltonian model is: The ground state term values used to calculate the transition wavenumbers were derived from the corresponding parameters which include the coefficient of the k ∆ = ± 3 interaction term 0 ε [5]. They are reported in Table 1 of this paper for completeness. The ground state eigenvalues were obtained from the appropriate energy matrices. The diagonal elements were calculated from

2 4 4 5 5 30 32 2 4 4 5 5 v 2, v 0 0, v 1 1, , ( ) / v 1, v 1 1, v 0 0, , = = = = = ± + = = = ± = = JK l l J k H H hc l l J k = 2 245 245 245 ( 1) K J W W k W J J + + + (4)
E 0 (J, k)/hc = B 0 [J(J + 1) -k 2 ] + A 0 k 2 -D 0J [J(J + 1)] 2 -D 0JK [J(J + 1)]k 2 -D 0K k 4 (5)
and the off-diagonal ones from <v, l , J, kH/hc v, l , J, k

±3 > = ε v (2k ± 3 ) [J(J+1) -k(k ±1)] 1/2 [J(J+1) -(k ±1) (k ±2)] 1/2 [J(J+1) -(k ±2) (k ±3)] 1/2 (6)
with v = 0 and l = 0.

Preliminary fits of the transitions assigned to ν 2 + ν 4 were encouraging: the attributed J and K values were found to be correct and the starting value of the deperturbed ν 2 + ν 4 origin was notably improved. Therefore, the assignments of the ν 2 + ν 4 transitions were greatly extended to higher J and K values, using as a guide their predicted positions and the GSCD. Thanks to these assignments the rotational and the centrifugal distortion constants were refined, improving also the overall quality of the fit. However, transitions with " 12 K ≤ and " J > 40 evidenced that two ro-vibrational perturbations, with ∆l = 0, ∆k = ± 3, and ∆l = ∆k = ± 2 selection rules, are active within the v 2 = v 4 = 1 state. The corresponding offdiagonal matrix elements were included in the calculations. They are, respectively, equation

(6) with v = (v 2 = v 4 = 1), 4 1 = ± l and < v 2 = v 4 = 1, 4 l = +1, J, k +2H/hc v 2 = v 4 = 1, 4 l = -1, J, k > = -½ t q [J(J+1) -k(k ±1)] 1/2 [J(J+1) -k(k m 1)] 1/2 (7)
As a consequence, a better reproduction of the wavenumbers for transitions with " 12 K ≤ and " J > 40 was obtained. The assignments of ν 2 + ν 4 were completed and in total 3334 transitions were collected, with " and " up to 81 and 63 J K .

In the course of the analysis of ν 2 + ν 4 the lines of the 2ν 2 + ν 5 transitions were also searched for, in particular those of the ∆K= +1 sub-branches, whose intensities are expected to increase due to the anharmonic resonance. In fact, this should be effective on the kl > 0 levels due to the opposite values of Aζ = = , 0.0936 and -0.0685 cm -1 , respectively [3,11]. However, the results of the search were very scarce, notwithstanding the line positions were predicted with sufficient reliability. Only 64 lines were assigned to 2ν 2 + ν 5 of (35,16), with kl ≥ 42, the value corresponding to the avoided crossing of the interacting levels as evidenced in the eigenvectors of (35,[START_REF] Mills | Theoretical Chemistry[END_REF].

In the least-squares fits a unitary statistical weight has been attributed to each transition that corresponds to an isolated line, while the weight has been reduced to doublets has not been observed. In the last cycle of the refinements all the experimental data that differed from the corresponding calculated values by more than 0.003 cm -1 , about ten times the estimated precision of the measurements, have been excluded from the data set. The spectroscopic parameters, the numbers of assigned transitions and of rejected lines, the values of " max J and " max K , and the standard deviations of the fit are collected in Table 2. For the 2ν 2 + ν 5 band only the deperturbed origin, B and Aζ were determined, all the other parameters were constrained either to the values calculated as described above or to zero.

The ν 2 + ν 4 / 2ν 2 + ν 5 band analyses for (37,[START_REF] Mills | Theoretical Chemistry[END_REF], (35,[START_REF] Papousek | Molecular Vibrational-Rotational Spectra[END_REF] and (37,[START_REF] Papousek | Molecular Vibrational-Rotational Spectra[END_REF] were performed according to the same procedure adopted for (35,[START_REF] Mills | Theoretical Chemistry[END_REF]. In (37,[START_REF] Mills | Theoretical Chemistry[END_REF] the bands are weaker than in (35,[START_REF] Mills | Theoretical Chemistry[END_REF] so the transitions assigned to ν 2 + ν 4 and 2ν 2 + ν 5 are less numerous, equal to 2334 and 34, respectively. The avoided crossing of the interacting levels is predicted for kl ≥ 36 but, due to the weakness of the band, transitions of the ∆K = +1 branches with " 36 K ≥ were not detected. This explains the very small number of transitions assigned to the 2ν 2 + ν 5 band.

The results of the fit are listed together with those of (35,[START_REF] Mills | Theoretical Chemistry[END_REF] in Table 2. The analysis of the spectra for the heavier isotopologues was difficult in the early stages, because the assignment of the first nucleus of r R transitions was complicated by the irregular appearance of the lines, particularly in (37,[START_REF] Papousek | Molecular Vibrational-Rotational Spectra[END_REF]. The avoided crossings of the interacting levels are observed for kl ≥ 27 and 12 in (35,18) and (37,18), respectively. Therefore, in addition to the transitions assigned to ν 2 + ν 4 of (35,18) and (37,18), 2521 and 2441, a large number of transitions were identified in the 2ν 2 + ν 5 bands of (35,18) and (37,18), 236 and 757, respectively. Thank to the more conspicuous data sets , in addition to the band origin, B and Aζ, also A, D J , D JK and η K of the v 2 = 2, v 5 =1 states resulted statistically determined. The sets of parameters for (35,[START_REF] Papousek | Molecular Vibrational-Rotational Spectra[END_REF] and (37,[START_REF] Papousek | Molecular Vibrational-Rotational Spectra[END_REF] are collected in Table 3. As it can be seen in Tables 2 and3, less than 4% of the total number of assigned transitions were discarded in each fit according to the rejection criterion. They are randomly distributed in the various sub-branches and correspond to blended or very weak lines. Four tables with the full list of assigned transition wavenumbers, quantum numbers and residuals from the fits associated with ν 2 + ν 4 and 2ν 2 + ν 5 bands of the four FClO 3 isotopologues are deposited with the British Library under the Supplementary Publications scheme. The tables are also available from the authors.

Computed anharmonicity constants

In Ref. [14] we have presented a high-level full-quartic ab initio force field for FClO 3 . From the quartic force field, the main spectroscopic constants have been derived for all known isotopologues, following standard second-order perturbation approach [START_REF] Papousek | Molecular Vibrational-Rotational Spectra[END_REF]. No

Fermi resonance was taken into account at that stage. However, a few of the computed anharmonicity constants presented in [14], i.e. x 24 , x 25 , x 45 , and g 45 , showed irregular trends among the various isotopologues, calling for a re-examination of the problem. In the light of the findings of the present paper, such anomalous behaviour can be attributed to the improper handling of the Fermi resonance between the v 4 = 1 and v 2 = v 5 = 1 vibrational levels. In fact, it is known [START_REF] Califano | Vibrational States[END_REF] that in case of Fermi resonance, the resonant term, i.e. the term containing the 245 φ force constant, must be removed from the analytic perturbative formulae for the anharmonicity constants. The aforementioned constants have then been recomputed by means of the SPECTRO software [START_REF] Gaw | Advances in Molecular Vibrations and Collision Dynamics[END_REF]. The results are reported in Table 4 and compared to the previous ones and to experimental values determined in the present study or available in the literature. The values of the spectroscopic parameters listed in Tables 2 and3 compare well with the corresponding ones in the ground state, in magnitude, sign and uncertainty. The corresponding parameters of (35,16) and (37,16) are in very good agreement and their uncertainties are comparable for both the ν 2 + ν 4 and 2ν 2 + ν 5 bands. Similar results are observed for (35,18) and (37,18). Some of the parameters of the v 2 = 2, v 5 = 1 state, A , J D , JK D and K D for (35,[START_REF] Mills | Theoretical Chemistry[END_REF] and (37,[START_REF] Mills | Theoretical Chemistry[END_REF] and K D for (35,[START_REF] Papousek | Molecular Vibrational-Rotational Spectra[END_REF] and (37,[START_REF] Papousek | Molecular Vibrational-Rotational Spectra[END_REF] were kept fixed to their initial values since, when refined, they resulted statistically undetermined and ineffective in improving the standard deviation of the fits. On the contrary, the K D constants of v 2 = v 4 = 1 for (35,18) and (37,18) were fixed to the values in Table 3 since they resulted 100% correlated with the A constants in the refinement procedures. Those refined values, truncated to the number of statistically significant figures, differ from the initial ones and contribute to improve the standard deviations of the fits. For all isotopologues the [3,8,9], respectively, both in sign and magnitude.

Results and discussion

The anharmonic resonance coefficients 245 W for (35,[START_REF] Papousek | Molecular Vibrational-Rotational Spectra[END_REF] and (37,18) have been determined

with very good precision, while for (35,16) and (37,16) they resulted 100% correlated with the ν 2 + ν 4 and 2ν 2 + ν 5 deperturbed band origins. In order to prevent divergence of the least-squares procedure 245 W were kept fixed to refined values, appropriately truncated, see W , could not be determined and were fixed to zero. The values of the ε and t q coefficients of the v 2 = 1, v 4 = 1 state compare well, both in magnitude and sign, with the corresponding ones in v 4 = 1 [3,8,9]. The refined parameters in Tables 2 and3 Two or three times the reported uncertainties are more realistic estimates, comparable to those of (35,18) and (37,18).

Apart from the case mentioned above, all the refined parameters are free from internal correlations, with the exception of (35,[START_REF] Papousek | Molecular Vibrational-Rotational Spectra[END_REF]. The standard deviations of the fits in Tables 2 and3 are about 3 ÷ 5 times the estimated precision of the experimental data.

2 5 v 2,v 1 Aζ = = which is 96% correlated with 2 5 v 2,v 1 A = = in
Combining the depertubed band origins of the ν 2 + ν 4 bands with those of the ν 2 and ν 4 fundamentals [2][3][4]8,9] the anharmonicity constants 24

x have been calculated. The obtained values, listed in Tables 2 and3, allow a consistent comparison with those recalculated from ab inito methods including the anharmonic resonance. From the inspection of Table 4 it can be evidenced that, notwithstanding the differences between the experimental and recomputed values of x 24 decrease, the calculated values of x 24 are still considerably smaller than the experimental ones, a discrepancy due to a lack of accuracy of the computed force fields. This problem may be traced back to difficulties in the ab initio calculations of the anharmonicity of the Cl-O bond stretching, which is the dominant deformation involved in the v 4 mode, (see Table 7 of [14] ). Similar problems were noticed for the anharmonicity constants involving ν 1 , the other mode corresponding to the Cl-O bond stretching. Another possible explanation could be the neglect of important interactions between the states presently investigated and some nearby dark states. We are confident that this possibility can be ruled out for several reasons. First, for each isotopologue, a large number of transitions has been fitted to a consistent set of parameters which include only three interaction constants, with a standard deviation of the fits close to the experimental uncertainty. Second, the v 2 = v 4 = 1 and v 2 = 2, v 5 = 1 states are well isolated, since the closest state which could interact via Fermi resonance, v 2 = 2, v 6 = 1, is 200 cm -1 lower in energy. Moreover, the presence of a Fermi interaction between v 4 = 1 and v 2 = 1, v 6 = 1 has not been evidenced in earlier analyses [3,8,9]. Finally, the values of 24 x are similar in all isotopologues and close to those of the 25 x anharmonicity constants. In addition, the values of the corresponding parameters presently calculated are very similar for all isotopologues. Thus, the inclusion of the Fermi resonance in the calculations is essential to remove the irregular trend evident in Table 10 of [14]. More noticeably, the x 25 values are in much better agreement with their experimental counterparts, the only exception being x 25 of (37,[START_REF] Mills | Theoretical Chemistry[END_REF]. -2.00 -1.43 -1.50983(6) c,d -2.02 -1.42 -2.21516(7) c,e -2.66 -1.50 -1.5845(2) c,d -3.19 -1 and 2 1986.84863 1986.84979 -116 2+4 PP(65,45) 64 44 1988.42673 1988.42739 -67 2+4 RP(50,12) 49 13 1986.94898 1986.94894 4 2+4 RP(49, 3) 48 4 1988.43609 1988.43571 39 2+4 PP(59,18) 58 17 1986.99586 1986.99435 151 2+4 PP(66,48) 65 47 1988.44138 1988.44375 -237 2+4 PP(60,21) 59 20 1986.99586 1986.99502 85 2+4 RP(47, 9) 46 10 1988.56544 1988.56601 -57 2+4 PP(58,15) 57 14 198657 14 .99586 1986.99728 -142 2+4 RP(46,12) .99728 -142 2+4 RP(46,12) + + , respectively), the branch, ( J lower and K lower ) are reported.

v 2 = v 4 = 1 v 2 = 2, v 5 = 1 v 2 = v 4 = 1 v 2 = 2, v 5 =
v 2 = v 4 = 1 v 2 = 2, v 5 = 1 v 2 = v 4 = 1 v 2 = 2, v 5 = 1 0 v E 1983.
ν ν ν ν ± ± + + Bands of F 35 Cl 16 O 3 ( I ) ( II ) ( III ) ( IV ) ( V ) ( I ) ( II ) ( III ) ( IV ) ( V ) F o r P e e r R e v i e w O n l y 2 Continued ( I ) ( II ) ( III ) ( IV ) ( V ) ( I ) ( II ) ( III ) ( IV ) ( V ) F o r P e e r R e v i e w O n l y 3 Continued ( I ) ( II ) ( III ) ( IV ) ( V ) ( I ) ( II ) ( III ) ( IV ) ( V ) F o r P e e r
( I ) ( II ) ( III ) ( IV ) ( V ) ( I ) ( II ) ( III ) ( IV ) ( V ) F o r P e e r R e v i e w O n l y Continued ( I ) ( II ) ( III ) ( IV ) ( V ) ( I ) ( II ) ( III ) ( IV ) ( V ) F o r P e e r R e v i e w O n l y ( I ) ( II ) ( III ) ( IV ) ( V ) ( I ) ( II ) ( III ) ( IV ) ( V ) 2+4 PQ(
( II ) J upper and K upper ( III) Observed wavenumbers (in cm --1 ).

( IV) Calculated wavenumbers (in cm --1 ).

(.V ) Observed -Calculated values (in 10 5 cm --1 ).

Observed wavenumbers identified by a W in column ( III ) have not been included in the fit.

( I ) ( II ) ( III ) ( IV ) ( V ) ( I ) ( II ) ( III ) ( IV ) ( V )
and 2

ν ν ν ν ± ± + + , respectively), the branch, ( J lower and K lower ) are reported.

( II ) J upper and K upper ( III) Observed wavenumbers (in cm --1 ).

( IV) Calculated wavenumbers (in cm --1 ).

(.V ) Observed -Calculated values (in 10 5 cm --1 ).

Observed wavenumbers identified by a W in column ( III ) have not been included in the fit. + + , respectively), the branch, ( J lower and K lower ) are reported.

( II ) J upper and K upper ( III) Observed wavenumbers (in cm --1 ).

( IV) Calculated wavenumbers (in cm --1 ).

(.V ) Observed -Calculated values (in 10 5 cm --1 ).

Observed wavenumbers identified by a W in column ( III ) have not been included in the fit.

  in v 4 and in v 2 plus v 5 . This was evidenced in the analysis of the 2ν 4

  whose intensities are increased by the interactions between the v 2 = v 4 = 1

Figure 1 .

 1 Figure captions

Figure 2 .

 2 Figure 2. Detail of the ν 2 + ν 4 band of F 35 Cl 16 O 3 . Two patterns of p P K (J) transitions and part of the r P K (J) sub-branch are evidenced. The J and K assignments are indicated.

Figure 3 .

 3 Figure 3. Detail of the ν 2 + ν 4 and 2ν 2 + ν 5 bands of F 37 Cl 18 O 3 . One pattern of p P K (J) transitions of the ν 2 + ν 4 band and two r R K (J) sub-branches of 2ν 2 + ν 5 are identified. The J and K assignments are indicated. Lines of 2ν 2 + ν 5 have been evidenced by an asterisk.

Figure 4 .

 4 Figure 4. Detail of the ν 2 + ν 4 band of F 35 Cl 16 O 3 . One pattern of r R K (J) transitions is

Figure 5 .

 5 Figure 5. Detail of the ν 2 + ν 4 band of F 37 Cl 16 O 3 . Two patterns of r R K (J) transitions are

Figure 6 .

 6 Figure 6. Detail of the ν 2 + ν 4 band of F 35 Cl 18 O 3 . Two patterns of r R K (J) transitions are

Figure 7 .

 7 Figure 7. Detail of the ν 2 + ν 4 band of F 37 Cl 18 O 3 . Three r R K (J) sub-branches are evidenced.

Table 2 ,

 2

	columns M
	F o r
	P
	e
	e r
	R
	e
	v i e
	w
	O n l
	y

Table 2

 2

	. The final values of 245 W for the four isopologues are very close to each other and to
	their initial values. The J and K dependences of the anharmonic resonance coefficients, 245 J W
	and 245 K

Table 1 .

 1 Ground state molecular constants ( in cm -1 ) of FClO 3 a .

	parameter	F 35 Cl 16 O 3	F 37 Cl 16 O 3	F 35 Cl 18 O 3	F 37 Cl 18 O 3
	0 A							0.1871217(42)	0.1871364(29)	0.166325417(100) 0.166331592(59)
	0 B							0.17541085(8)	0.17532656(6)	0.16379784(5)	0.16374819(4)
	0 D × J	10	7		0.49979(9)	0.49939(9)	0.42716(7)	0.42654(7)
	0 D × JK	10	7	0.5534(5)	0.5591(6)	0.3937(4)	0.3977(4)
	0 D × K	10	7	-0.76 b	-0.76 b	-0.5916(1)	-0.6288(1)
	0 10 ε ×	8				-0.307(60)	-0.320(64)	-0.531(8)	-0.547(9)

Table 2 .

 2 Spectroscopic parameters (in cm -1 ) for the v 2 = v 4 = 1 and v 2 = 2, v 5 = 1 states of F 35 Cl 16 O 3 and

	F 37 Cl 16 O 3	a .
	parameter	F 35 Cl 16 O 3	F 37 Cl 16 O 3

Table 3 .

 3 Spectroscopic parameters (in cm -1 ) for the v 2 = v 4 = 1 and v 2 = 2, v 5 = 1 states of F 35 Cl 18 O 3 and

	1

Table 4 .

 4 Effects of the inclusion of the ν 4 / ν 2 + ν 5 Fermi resonance on the computed anharmonicity constants ( in cm -1 ) for FClO 3 .

	parameter	F 35 Cl 16 O 3	F 37 Cl 16 O 3	F 35 Cl 18 O 3	F 37 Cl 18 O 3
			calc. a calc.	experimental	calc. a calc.	experimental	calc. a calc.	experimental calc. a calc.	experimental
	x	24	-0.05 -0.65 -2.12663(6) c	0.07 b -0.55 -1.97993(9) c	0.50 b -0.68	-2.0539(3) c	1.15 b -0.56	-1.9985(2) c
	x	25

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	Molecular Physics Continued ( V ) ( I ) Molecular Physics Molecular Physics Molecular Physics Continued ( V ) ( I ) 70 2+4 PP(14, 9) 13 76 2+4 RQ(50,15) 50 16 ( II ) ( II ) 8 35 2+4 RQ(41,24) 41 25 Molecular Physics Molecular Physics Molecular Physics Molecular Physics Molecular Physics Molecular Physics Molecular Physics Molecular Physics Molecular Physics Table III 15 Molecular Physics Observed and Calculated Transition Wavenumbers in the ( I ) ( II ) ( III ) ( IV ) Page 53 of 106 1 2 3 4 5 6 7 1 2 3 4 5 6 7 1 2 3 4 5 6 7 1 2 3 4 5 6 7 1 2 3 4 5 6 7 ( I ) ( II ) ( III ) ( IV ) 2+4 RP(13, 0) 12 1 2003.21722 2003.21653 2+4 RQ(50,27) 50 28 2003.24205 2003.24129 2+4 PP(14, 3) 13 2 2003.24205 2003.24170 2+4 RQ(56,18) 56 19 2003.25612 2003.25597 1 4 5 6 7 1 4 5 6 7 Page 59 of 106 1 4 5 6 7 1 4 5 6 7 Page 61 of 106 1 4 5 6 7 1 4 5 6 7 1 4 5 6 7 1 4 5 6 7 1 4 5 6 7 1 4 5 6 7 2 3 2 3 2 3 2 3 2 3 2 3 2 3 2 3 2 3 2 3 1 2 4 and 2 ν ν ± +	( III ) ( III ) 2004.06197 2004.06783 2004.07470 1 2 5 ν ν ± + Bands of F 35 Cl 18 O 3 ( IV ) ( V ) ( IV ) ( V ) 2004.06248 -52 2004.06653 130 2004.07467 3 Page 60 of 106 Page 64 of 106
	8 8 8 8 8 8 8 8 8 8 8 8 8 8 8
	9 9 9 9 9 9 9 9 9 9 9 9 9 9 9
	10 10 10 10 10 10 10 10 10 10 10 10 10 10 10
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	16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 F o r P e e r R e v i e w O n l y 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 F o r P e e r R e v i e w O n l y 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 F o r P e e r R e v i e w O n l y 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 F o r P e e r R e v i e w O n l y 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 F o r P e e r R e v i e w O n l y 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 F o r P e e r R e v i e w O n l y 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 F o r P e e r R e v i e w O n l y -5 2+4 RQ(22, 0) 22 1 2007.49734 2+4 PQ(47, 6) 47 5 2006.86770 2006.86756 13 2+4 RQ( 5, 3) 5 4 2007.49734 2+4 RQ(18, 6) 18 7 2006.87570 2006.87629 -59 2+4 RQ( 4, 3) 4 4 2007.50707 2+4 PP( 4, 3) 3 2 2006.88764 2006.88683 81 2+4 PQ(44, 9) 44 8 2007.50707 2+4 RQ(35, 0) 35 1 2006.89337 2006.89231 106 2+4 RQ(21, 0) 21 1 2007.53357 2+4 PQ(41, 3) 41 2 2006.89337 2006.89359 -21 2+4 PQ(30, 3) 30 2 2007.53947 2+4 RQ(27, 3) 27 4 2006.90348 2006.90226 122 2+4 PQ(49,12) 49 11 2007.55619 2+4 RQ(17, 6) 17 7 2006.90348 2006.90599 -252 2+4 PQ(37, 6) 37 5 2007.55619 2+4 RQ(16, 6) 16 7 2006.93377 2006.93405 -28 2+4 RQ(20, 0) 20 1 2007.56693 2+4 PQ(46, 6) 46 5 2006.94651 2006.94386 265 2+4 PQ(43, 9) 43 8 2007.57875 2+4 RQ(26, 3) 26 4 2006.94651 2006.94662 -11 2+4 PQ(29, 3) 29 2 2007.58829 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 F o r P e e r R e v i e w O n l y 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 F o r P e e r R e v i e w O n l y 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 F o r P e e r R e v i e w O n l y 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 F o r P e e r R e v i e w O n l y 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 F o r P e e r R e v i e w O n l y 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 F o r P e e r R e v i e w O n l y 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 F o r P e e r R e v i e w O n l y 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 o r P e e r R e v i e w O n l F o r P e e r R e v i e w O n l 59 F y y	2007.49787 2007.49922 2007.50745 2007.50894 2007.53325 2007.53905 2007.55354 2007.55824 2007.56702 2007.58032 2007.58802	-53 -189 -39 -188 32 42 265 -205 -9 -157 27
	2+4 RQ(34, 0) 34	1	2006.94651 2006.94847	-196	2+4 RQ(19, 0) 19	1	2007.59938	2007.59919	19
	2+4 PQ(40, 3) 40	2	2006.96046 2006.96033	12	2+4 PQ(36, 6) 36	5	2007.61862	2007.61845	17
	2+4 RQ(15, 6) 15	7	2006.96046 2006.96046	0	2+4 RQ(18, 0) 18	1	2007.63020	2007.62976	45
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	16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 F o r P e e r R e v i e w O n l y 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 F o r P e e r R e v i e w O n l y 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 F o r P e e r R e v i e w O n l y 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 F o r P e e r R e v i e w O n l y 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 F o r P e e r R e v i e w O n l y 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 F o r P e e r R e v i e w O n l y 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 F o r P e e r R e v i e w O n l y 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 F o r P e e r R e v i e w O n l y 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 F o r P e e r R e v i e w O n l y 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 F o r P e e r R e v i e w O n l y 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 F o r P e e r R e v i e w O n l y 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 F o r P e e r R e v i e w O n l y 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 F o r P e e r R e v i e w O n l y 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 F o r P e e r R e v i e w O n l y 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 F o r P e e r R e v i e w O n l y 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 F o r P e e r R e v i e w O n l y 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 F o r P e e r R e v i e w O n l y 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 F o r P e e r R e v i e w O n l y 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 F o r P e e r R e v i e w O n l y 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 o r P e e r R e v i e w O n l 59 F y

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Acknowledgments E.C. and L.F. acknowledge the financial support from the Università di Bologna and from the Ministero della Ricerca e dell'Università, PRIN "Trasferimenti di energia e di carica: dalle collisioni ai processi dissipativi" as well as Dr. G. Pawelke for providing the pure isotopic sample.

2+4 *R (43,15) 44 19 1996(43,15) 44 19 .32260 1996.32182 78 2+4 PR(38, 6) .32182 78 2+4 PR(38, 6) 39


Stéphane Goutte
email: goutte@math.univ-paris-diderot.fr

Armand Ngoupeyou
email: armand.ngoupeyou@univ-paris-diderot.fr

Defaultable bond pricing using regime switching intensity model

Keywords: Defaultable bond, Regime switching, Conditional Laplace Transform, Credit rating, Markov copula. MSC Classification (2010): 60H10 91G40 91G60 91B28 65C40

In this paper, we are interested in finding explicit numerical formulas for the defaultable bonds prices of firms which fit well with real financial data. For this purpose, we use a default intensity whose values depend on the credit rating of these firms. Each credit rating corresponds to a regime of the default intensity. Then, this regime switches as soon as the credit rating of the firms also changes. This regime switching default intensity model allows us to capture well some market features or economics behaviors. We obtain two explicit different formulas to evaluate the conditional Laplace transform of a regime switching Cox Ingersoll Ross model. One using the property of semi-affine of this model and the other one using analytic approximation. We conclude by giving some numerical illustrations of these formulas and real data estimation results.

Introduction

In an economic crisis situation where the credit ratings of countries or firms have a big impact in the general financial market, we need to understand and capture the change of these ratings in the dynamic of a the firm bond price. Moreover, we also have to model the contagion risk due to a bad rating of a firm on other one. For example, the Bond of countries in the Euro zone are affected by the Greek bad rating. In the literature, models for pricing defaultable securities have been introduced by Merton [START_REF] Merton | On the pricing of corporate debt: the risk structure of interest rates[END_REF]. It consists of explicitly linking the risk of firm default and the value of the firm. Although this model is a good issue to understand the default risk, it is less useful in practical applications since it is too difficult to capture all the macroeconomics factors which appear in the dynamics of the value of the firm. Hence, Duffie and Singleton [START_REF] Singleton | Credit risk: Pricing, Measurement and Management[END_REF] introduced the reduced form modeling, followed by Madan and Unal [START_REF] Madan | Pricing the risks of default[END_REF], Jeanblanc and Rutkowski [START_REF] Jeanblanc | Mathematical Finance-Bachelier Congress 2000. Chapter Default risk and hazard process[END_REF] and others. The main tool of this approach is the "default intensity process" which describes in short terms the instantaneous probability of default. To deal with contagion risk, the most popular approach is copula. The credit rating of each firm is modeled by a Markov chain on which we will construct our copula. In this regard, we use a continuous time Markov chain called credit migration process studied by Bielecki and Rutkowski in [START_REF] Bielecki | Credit Risk: Modeling, Valuation and Hedging[END_REF]. Hence, our copula which depends on the credit ratings will affect the dynamic of the default intensity. In fact, we define default intensity process by a Cox-Ingersoll-Ross (CIR) model whose parameters values depend on this copula.

The Cox-Ingersoll-Ross model was first considered to model the term structure of interest rate by Cox and al. in [START_REF] Cox | A theory of the term structure of interest rates[END_REF]. The study of this class of processes was caution by the fact that it allows us a closed form expression of Laplace transform (see Duffie and al. [START_REF] Duffie | Affine processes and applications in finance[END_REF]) and model well the default intensity (Alfonsi and Brigo [START_REF] Alfonsi | Credit default swap calibration and derivatives pricing with the SSRD stochastic intensity model[END_REF]). Moreover, Choi in [START_REF] Choi | Regime-Switching Univariate Diffusion Models of the Short-Term Interest Rate[END_REF] shows that regime switching CIR process captures more short term interest rate than standard models. In a econometric point of view, regime switching model were introduced by Hamilton in [START_REF] Hamilton | Rational-expectations econometric analysis of changes in regime[END_REF].

In this framework, we obtain explicit formulas to evaluate defaultable bond prices. More precisely, we obtain two different formulas to evaluate the Laplace transform of defaultable intensity. In a first time, we use the semi affine property of the regime switching Cox Ingersoll Ross model and then solve a system of Riccati's equations. In a second time, we extend the analytic approximation found in Choi and Wirjanto [START_REF] Choi | An analytic approximation formula for pricing zero-coupon bonds[END_REF]. Indeed Choi and Wirjanto in [START_REF] Choi | An analytic approximation formula for pricing zero-coupon bonds[END_REF] give an analytic approximation of the value of bond price with constant CIR parameter and with constant time step model discretization. We extend this result in three ways: firstly to evaluate conditional Laplace transform of a regime switching Cox Ingersoll Ross, secondly to evaluate defaultable regime switching bond price and thirdly in the case of non uniform deterministic time step model discretization (in our case, the time step model discretization depends on the regime switching stopping time). We apply these two formulas to price defaultable bond. We illustrate the efficiency of our new modelization of regime switching intensity firstly by comparing the computing time of each formulas, secondly by showing (using real historical data based on the Greece spread CDS) that our model estimates well data and that each regime captures well some market features or economics behaviors.

In Section 1, we introduce the Markov copula, the credit migration process and the regime switching Cox-Ingersoll-Ross model. In Section 2, we give the two formulas to evaluate the conditional Laplace transform in this framework. Finally, in Section 3, we show some simulations to compare the formula results, illustrate the model and then we give some estimation on real data.

The defaultable model 1.Credit migration model

Let T > 0 be a fixed maturity time and denote by (Ω, F := (F t ) [0,T ] , P) an underlying probability space.

Definition 1.1. A notation is a label given by an entity which measures the viability of a firm. This graduate notation goes from 1 to K. 1 for the best economic and financial situation and K for the worst. We will call an indicator of notation a continuous time homogeneous Markov chain on the finite space S = {1, . . . , K}.

Let A and B be two firms with their own indicator of notation (X A ) t∈[0,T ] and (X B ) t∈[0,T ] . Hence X A and X B are Markov chains with generator matrix Π A and Π B . We recall that the generator matrix of C ∈ {A, B} is given by Π C ij ≥ 0 if i = j for all i, j ∈ S and Π C ii =j =i Π ij otherwise. We can remark that Π C ij represents the intensity of the jump from state i to state j. Moreover, we denote by F A t := {σ(X A s ); 0 ≤ s ≤ t} and F B t := {σ(X B s ); 0 ≤ s ≤ t} the natural filtrations generated by X A and X B .

Markov Copula

Let denote by X the bivariate process X = (X A , X B ), which is a finite continuous time Markov chain with respect to its natural filtration F X = F A,B . We recall now the Corollary 5.1 of Bielecki and al. [START_REF] Bielecki | Study of dependence for some stochastic processes[END_REF], applied to our case, which gives the condition that the components of the bivariate processes X are themselves Markov chain with respect to their own natural filtration.

Corollary 1.1. Consider two Markov chains X A and X B , with respect to their own filtrations F A and F B , and with values in S. Suppose that their respective generators are Π A ij and Π B hk with i, j, h and k are in S. Consider the system of equations in the unknown Π X ij,hk where i, j, h, k ∈ S and

(i, h) = (j, k): k∈S Π X ij,hk = Π A ij ∀h, i, j ∈ S, i = j and j∈S Π X ij,hk = Π B hk ∀i, h, k ∈ S, h = k (1.1)
Suppose that the above system admits a solution such that the matrix

Π Z := Π Z ij,hk i,j,h,k∈S with Π X ii,hh = - (j,k)∈S×S,(j,k) =(i,h) Π X ij,hk (1.2)
properly defines an infinitesimal generator of a Markov chain with values in S × S. Consider, the bivariate Markov chain X = (X A , X B ) on S × S with generator matrix Π X . Then, the components X A and X B are Markov chains with respect to their own filtrations, their generators are Π A and Π B .

Hence we can now formulate the Definition of a Markov copula.

Definition 1.2. A Markov copula between the Markov chains X A and X B is any solution to system (1.1) such that the matrix Π X , with Π X ii,hh given in (1.2), properly defines an infinitesimal generator of a Markov chain with values in S × S.

Moreover, the infinitesimal generator process of X which is a matrix with N := K 2 rows and columns, since the cardinal of the state of notation is K, can be written as

Π X =       π (1,1) . . . π (1,K) π (2,1 . . . π (2,K) . . . . . . π (K,1) . . . π (K,K)      
Then the possible states are N couples which are given by E := {(1, 1), (1, 2), . . . , (1, K), (2, 1), (2, 2), . . . , (2, K), . . . (K, 1), (K, 2), . . . , (K, K)}

Markov copula in the hazard rate framework

We denote by F := (F t ) t∈[0,T ] the filtration such that F t = F t ∨ F X t . Let τ A and τ B be the two default times of firms A and B. Let define for all t ∈ [0, T ]:

H A t = 1 {τ A ≤t} and H B t = 1 {τ B ≤t} (1.3)
We define now some others filtrations

G A t = F t ∨ H B t , , G B t = F t ∨ H A t and G t = F t ∨ H A t ∨ H B t
where H A (resp. H B ) is the natural filtration generated by H A (resp. H B ) and we will denote

G := (G t ) t∈[0,T ] , G A := G A t t∈[0,T ] and G B := G B t t∈[0,T ] . Let now consider λ i := λ i (X)
, for i ∈ {A, B} two F-progressively non-negative processes defined on (Ω, G, P) endowed with the filtration F. We assume that ∞ 0 λ i (X s )ds = +∞ and we set:

τ i = inf t ∈ R + , t 0 λ i (X s )ds ≥ -ln(U i ) , i ∈ {A, B}.
where U i are mutually independent uniform random variables defined on (Ω, G, P) which are independent of λ i . The stopping times τ A and τ B are totally inaccessible and conditionally independent given the filtration F, moreover the (H)-hypothesis is satisfied (i.e. that every local F-martingale is a local G-martingale too). The process λ i is called the F-intensity of the firm i and we have that

M i t = H i t - t∧τ i 0 λ i (X s )ds
are G-martingales. In general case, processes λ i are F ∨ G (i) -adapted which jump when any default occurs. This jump impacts the default of the firm and makes some correlation between the firms. In our case, the correlation is constructed using the F-Markov chain X = (X A , X B ). Since from the explicit formula of the intensity given the survey probability for each i ∈ {A, B}:

λ i t = - 1 P(τ i ≥ t|G i t )
dP(τ i ≥ θ|G i t ) dθ θ=t we can find, from Bielecki and al. [START_REF] Bielecki | Credit risk[END_REF] (Example 4.5.1 p 94), that the formula of the conditional survey probability is given by:

P(τ i ≥ θ|G t ) = 1 {τ i ≥t} E e -θ t λ i (Xs)ds |F t (1.4)
for i ∈ {A, B}. The Markov chain X will explain how the curve of the default bond moves with different states (regimes) of the financial market.

Construction of the Markov chain

We are now going to present the canonical construction of a conditional Markov chain X, based on a given filtration F and a stochastic infinitesimal generator Π X . This construction can be found in Bielecki and Rukowski [START_REF] Bielecki | Credit Risk: Modeling, Valuation and Hedging[END_REF] or Eberlein and Ozkan [START_REF] Eberlein | The defaultable Lévy term structure: ratings and restructuring[END_REF], which we follow closely in the exposition. Each component Π X ij : Ω × [0, T ] → R + are bounded, F-progressively measurable stochastic processes. We recall that for every i, j ∈ S, i = j, processes Π X ij are non-negative and Π X ii (t) =j =i Π X ij(t) . The process X is constructed from an initial distribution µ and the F-conditional adapted infinitesimal generator Π X by enlarging the underlying probability space (Ω, F, P T ) to a probability space denoted in the sequel by (Ω, F, Q T ). The new probability space is obtained as a product space of the underlying one with a probability space supporting the initial distribution µ of X and a probability space supporting a sequence of uniformly distributed random variables, which control, together with the entries of the infinitesimal generator Π X , the laws of jump times (τ k ) k∈N of X and jump heights. We denote by F its trivial extension from the original probability space (Ω, F, P T ) to (Ω, F, Q T ). We refer to [START_REF] Bielecki | Credit Risk: Modeling, Valuation and Hedging[END_REF] or [START_REF] Grbac | Credit Risk in Levy Libor Modeling: Rating Based Approach[END_REF] for details of this construction. However an important step of this construction is that they construct a discrete time process (X k ) k∈N which allows us to construct the credit migration process X as

X t := X k-1 for all t ∈ [τ k-1 , τ k [, k ≥ 1 (1.5)
where τ k are the jump times. An important result is that the progressive enlargement of filtration F t := F t ∨F X t , t ∈ [0, T ] satisfies the (H)-hypothesis. In the sequel, we will work under the enlarging probability space (Ω, F, Q T ). The expectations will be taken under the probability measure Q T but for simplicity of notation, we will write E for E Q T .

Pricing defaultable bond with Markov copula

Defaultable Model

Let W be a standard real Brownian motion with filtration F t = σ{W s ; 0 ≤ s ≤ t}. We recall that a Cox Ingersoll Ross (CIR) process is the solution of the stochastic differential equation given by

dλ t = κ(θ -λ t )dt + σ λ t dW t , t ∈ [0, T ] (1.6)
where κ, θ and σ are constants which satisfy the condition σ > 0 and κθ > 0. We will assume that λ 0 ∈ R + and that 2κθ ≥ σ 2 . This is to ensure that the process (λ t ) is positive. We will now define the notion of CIR process with each parameters values depend on the value of a Markov chain.

Definition 1.3. Let (X) t be a two-dimensional continuous time Markov chain on finite space S 2 := {1, . . . , K} 2 for all t ∈ [0, T ]. We will call a Regime switching CIR the process (λ t ) which is the solution of the stochastic differential equation given by

dλ t = κ(X t )(θ(X t ) -λ t )dt + σ(X t ) λ t dW t , t ∈ [0, T ]. (1.7)
For all j ∈ {1, . . . , K} 2 , we have that κ(j)θ(j) > 0 and 2κ(j)θ(j) ≥ σ(j) 2

For simplicity, we will denote the values κ(X t ), θ(X t ) and σ(X t ) by κ t , θ t and σ t .

Assumption 1.1. We assume that both intensities processes λ A and λ B follow a regime switching CIR given for i = {A, B} by

dλ i t = κ(X t )(θ(X t ) -λ i t )dt + σ(X t ) λ i t dW t . (1.8)
Remark 1.1. We have that the intensity process (λ i t ) depends on the value of the credit migration process X = (X A , X B ). Hence each firm A and B has an increasing sequence of F X -stopping times given by:

-for the firm A it is 0 ≤ τ A 1 < τ A 2 < • • • < τ A n ≤ T . -for the firm B it is 0 ≤ τ B 1 < τ B 2 < • • • < τ B m ≤ T .
Hence with these two sequences, we construct another sequence by a rearrangement of these two sequences in one where we put every stopping time τ A i , i ∈ {1, . . . n} and τ B j , j ∈ {1, . . . , m} in an increasing order. We obtain a new increasing sequence of stopping times of size M ∈ N given by 0

≤ τ 1 < τ 2 < • • • < τ M ≤ T . As an example of this construction - 0 τ A 1 τ 1 τ A 2 τ 2 τ B 1 τ 3 τ A 3 τ 4 τ B 2 τ 5 τ A 4 τ 6 T Remark 1.2.
By this construction, we have that on each interval t ∈ [τ k , τ k+1 [ that the regime switching CIR process λ i defined in (1.8) is a classical CIR with constant parameters.

Zero coupon bond price

We can now define the defaultable Zero coupon bond price.

Definition 1.4. We will denote by D i t,T t∈[0,T ]

, i = {A, B} the price of a defaultable discounted bond price which pays $1 at the maturity T.

Using the partitioning time, the notation defined in the previous subsection and the general asset pricing theory in Harrison and Pliska [START_REF] Harrison | Martingales and stochastic integrals in the theory of continuous trading[END_REF] and [START_REF] Harrison | A stochastic calculus model of continuous time trading: complete markets[END_REF], the conditional defaultable discounted bond price D t,T is given by Proposition 1.1. For i = {A, B}, we have for all t ∈ [0, T ] that

D i t,T = (1 -H i t )E exp - T t (r s + λ i s )ds |F X t , λ 0 .
(1.9)

Remark 1.3. The quantity r t + λ i t t∈[0,T ] can be seen as a default-adjusted interest rate process. The part λ i t t∈[0,T ] is the risk-neutral mean loss rate of the instrument due to the default of the firm i ∈ {A, B}. The quantity r t + λ i t t∈[0,T ] therefore represents the probability and the timing of default, as well as for the effect of losses on default. This model allows us to capture an economic health of each firm since for each firm i ∈ {A, B}, the stochastic process (λ i t ) has parameters whose values depend on the credit notation of the firm. And by the construction of the migration process X, we have correlation between each firm notation. This allows the model to capture financial health correlation between each firm, like the impact of the default of one firm against the others.

Our aim is so to obtain explicit formulas of (1.9). This is done by the following Theorem using two different methods to evaluate the conditional Laplace transform of λ i . The first one uses a Ricatti approach and the second one an analytical approximation.

Theorem 1.1. Under Assumptions 1.1 and assuming that X is independent of W and that the risk-free interest rate r is a deterministic function, then we have for i ∈ {A, B} that the defaultable bond price can be obtained by two formulas:

1. Riccati Approach:

D i 0,T = E   exp - T 0 r s ds exp    - M j=1 B M -j (∆ t j-1 )    exp (-A 0 (∆ t 0 , i 0 )λ)   (1.10)
where

A 0 (∆ t 0 ) = 2 γ 1 + κ 1 - 4γ 1 γ 1 + κ 1 1 (γ 1 + κ 1 ) exp(γ 1 ∆ t 0 ) + γ 1 -κ 1 B M -j (∆ t j-1 ) = - κ M -j+1 θ M -j+1 (γ M -j+1 + κ M -j+1 ) (σ M -j+1 ) 2 ∆ t j-1 +2 κ M -j+1 θ M -j+1 (σ M -j+1 ) 2 ln (γ M -j+1 + κ M -j+1 ) exp(γ M -j+1 ∆ t j-1 ) + γ M -j+1 -κ M -j+1 -2 κ M -j+1 θ M -j+1 (σ M -j+1 ) 2 ln 2γ M -j+1 γ M -j+1 = (κ M -j+1 ) 2 + 2(σ M -j+1 ) 2
where we denote for simplicity κ j = κ(X t j ), θ j = θ(X t j ) and σ j = σ(X t j )

2. Analytic Approximation:

D i 0,T = E(exp - T 0 r s ds exp - u 2 n k=1 h 2 n-k+1 a n-k+1 κ n-k θ n-k - u 2 h 1 λ 0 [1 + a 1 (1 -κ 0 h 1 )] × exp n k=1 ln E t 0 λ 0 ,X exp h 3 n-k+1 8 u 2 σ 2 n-k a 2 n-k+1 λ 0 + n-k i=0 κ i (θ i -λ i )h i+1 + n-k i=0 σ i λ i ∆W i )(1.11)
where the sequence a is given by

a n-1 = 1 + h n h n-1 + h n h n-1 a n (1 -h n κ n-1
) and a n = 1.

Remark 1.4. The hypothesis that X is independent of W has an economic sense since for example X = (X A , X B ) could represent the credit notation of two countries given by an exogenous entity like a credit rating agencies.

Conditional Laplace transform formulas

We are now going to prove the Theorem 1.1. More precisely, we will find two explicit formulas to evaluate the conditional Laplace transform of λ with respect to X denoted by Φ. It is given, for all u ∈ C, by

Φ 0,T,λ,X (u) = E exp -u T 0 λ s ds |λ 0 = λ, F X T = E λ,X exp -u T 0 λ s ds .
(2.12)

Hence, our defaultable bond price formulas will be obtained as a particular case of this equation by taking u = 1.

A Ricatti approach

By Remark 1.1, there exists an increasing sequence of F X -stopping times in interval [0, T ], where the value of the Markov chain changes. We denote by Γ this subdivision

0 = τ 0 < τ 1 < • • • < τ M = T So in each time interval [τ k , τ k+1 [, k ∈ {1, . . . n}
the process X is constant. And so the CIR regime switching process λ has constant parameters on this each time interval.

Proposition 2.2. The conditional Laplace transform of the regime switching CIR process (for

u = 1) between time [τ k , τ k+1 [ with λ τ k = λ and X τ k+1 = j ∈ S d is given by Φ τ k ,τ k+1 ,j := E exp - τ k+1 τ k λ s ds |λ τ k = λ, X τ k+1 = j = exp {-A(∆ t k , j)λ -B(∆ t k , j)} (2.13)
where ∆ t k = τ k+1 -τ k and A(∆ t k , j) = 2 γ j + κ j - 4γ j γ j + κ j 1 (γ j + κ j ) exp(γ j ∆ t k ) + γ j -κ j , (2.14)
B(∆ t k , j) = - κ j θ j (γ j + κ j ) σ 2 j ∆ t k + 2 κ j θ j σ 2 j ln ((γ j + κ j ) exp(γ j ∆ t k ) + γ j -κ j ) -2 κ j θ j σ 2 j ln (2γ j ) , (2.15)
γ j = κ 2 j + 2σ 2 j .
(2.16)

Proof. We recall that the constant parameter CIR process defined in (1.6) is an affine process (see Duffie and al. [START_REF] Duffie | Affine processes and applications in finance[END_REF]). So as in each step of time [τ k , τ k+1 [, the stochastic process X is constant. So the process λ is a classical CIR with constant parameters on each step. So on each time interval [τ k , τ k+1 [, the process λ is affine, hence we can assume that the expression of Φ τ k ,τ k+1 ,j is given by the form exp

{-A(∆ t k , j)λ τ k -B(∆ t k , j)} (2.17)
for some functions A(∆ t k , j) and B(∆ t k , j) solution of a system of Riccati equation. Then the expected result is well known and can be found for instance in Cox and al. [START_REF] Cox | A theory of the term structure of interest rates[END_REF].

We would like now to give an explicit form of the conditional Laplace transform of the CIR process between time 0 and T. This is done by the following Theorem. Theorem 2.2. Assume that the intensity process (λ t ) follows a regime switching CIR, then we have for all λ 0 = λ > 0 and

X τ 1 = i 0 ∈ S d that Φ 0,T,λ,X (1) = E exp - T 0 λ s ds |λ 0 = λ, F X T := E λ,X exp - T 0 λ s ds = exp    - M j=1 B M -j (∆ t j-1 )    exp (-A 0 (∆ t 0 , i 0 )λ)
where

A 0 (∆ t 0 ) = 2 γ 1 + κ 1 - 4γ 1 γ 1 + κ 1 1 (γ 1 + κ 1 ) exp(γ 1 ∆ t 0 ) + γ 1 -κ 1 , (2.18)
B M -j (∆ t j-1 ) = - κ M -j+1 θ M -j+1 (γ M -j+1 + κ M -j+1 ) (σ M -j+1 ) 2 ∆ t j-1 (2.19) +2 κ M -j+1 θ M -j+1 (σ M -j+1 ) 2 ln (γ M -j+1 + κ M -j+1 ) exp(γ M -j+1 ∆ t j-1 ) + γ M -j+1 -κ M -j+1 -2 κ M -j+1 θ M -j+1 (σ M -j+1 ) 2 ln 2γ M -j+1 , γ M -j+1 = (κ M -j+1 ) 2 + 2(σ M -j+1 ) 2 , (2.20)
where we denote for simplicity κ j = κ(X t j ), θ j = θ(X t j ) and σ j = σ(X t j ).

Proof. We have a sequence of increasing times 0 = τ 0 < τ 1 < • • • < τ M = T where the Markov chain X changes its value. Hence

E λ,X exp - T 0 λ s ds = E λ,X exp - M -1 k=0 τ k+1 τ k λ s ds = E λ,X exp - M -1 k=0 τ k +∆t k τ k λ s ds = E λ,X M -1 k=0 exp - τ k+1 τ k λ s ds
By hypothesis, X is independent of W , then conditioning with respect to

F τ M -1 := F M -1 , we obtain E λ,X exp - T 0 λ s ds = E λ,X E M -1 k=0 exp - τ k+1 τ k λ s ds |F M -1 = E λ,X M -2 k=0 exp - τ k+1 τ k λ s ds E exp - τ M τ M -1 λ s ds |F M -1 .(2.21)
Moreover, we know that E exp -

τ M τ M -1 λ s ds |F M -1 is equal to Φ(τ M -1 , τ M , X M )
, where X M means X τ M . So applying Proposition 2.2, we get

E exp - τ M τ M -1 λ s ds |F M -1 = exp -A M -1 (∆ t M -1 , X M )λ τ M -1 -B M -1 (∆ t M -1 , X M ) .
We recall that the quantities A M -1 (∆ t M -1 , X M ) and B M -1 (∆ t M -1 , X M ) are constants. Hence replacing this result in the expectation (2.21) gives

E λ,X exp - T 0 λsds = E λ,X M -2 k=0 exp - τ k+1 τ k λsds exp -AM-1(∆t M -1 , XM )λτ M -1 -BM-1(∆t M -1 , XM ) = exp -BM-1(∆t M -1 , XM ) E λ,X M -2 k=0 exp - τ k+1 τ k λsds -AM-1(∆t M -1 , XM )λτ M -1 .
To simplify the notation of the calculus we will denote by A k-1 (resp. B k-1 ) the quantity

A k-1 (∆ t k-1 , X k ) (resp. B k-1 (∆ t k-1 , X k )). Hence E λ,X exp - T 0 λ s ds = exp {-B M -1 } E λ,X M -2 k=0 exp - τ k+1 τ k λ s ds -A M -1 λ τ M -1 .
We condition again with respect to F M -2 to obtain

E λ,X exp - T 0 λsds = exp {-BM-1} E λ,X E M -2 k=0 exp - τ k+1 τ k λsds -AM-1λτ M -1 |FM-2 = exp {-BM-1} E λ,X M -3 k=0 exp - τ k+1 τ k λsds E exp - τ M -1 τ M -2 λsds -AM-1λτ M -1 |FM-2 .
To continue, we need to evaluate the conditional expectation:

ϕ τ M -2 ,∆t M -2 := E exp - τ M -1 τ M -2 λ s ds -A M -1 λ τ M -1 |F M -2 .
Lemma 2.1. Assume for all k ∈ {1, . . . , M } that the conditional expectation ϕ τ M -k ,∆t M -k has an exponential affine structure form given by

ϕ τ M -k ,∆t M -k = exp -A M -k (∆ t M -k , X M -k+1 )λ τ M -k -B M -k (∆ t M -k , X M -k+1 ) . ( 2

.22)

Then we can find explicit forms for functions

A M -k (∆ t M -k , X M -k+1 ) and B M -k (∆ t M -k , X M -k+1 )
which are given explicitly by equations (2.14) and (2.15) under the conditions that

A M -k (0) = A M -k+1 and B M -k (0) = 0. Proof. Let ϕ τ M -k ,∆t M -k := E exp - τ M -k+1 τ M -k λ s ds -A M -k+1 λ τ M -k+1 |F M -k then ϕ τ M -k ,∆t M -k = E M -k exp - τ M -k+1 τ M -k λ s ds -A M -k+1 λ τ M -k +∆t M -k .
Taking a small time interval dt

∆ t M -2 to obtain E M -k exp - τ M -k+1 τ M -k λ s ds -A M -k+1 λ τ M -k +∆t M -k = E M -k E M -k+dt exp - τ M -k+1 τ M -k λ s ds -A M -k+1 λ τ M -k +∆t M -k . Thus E M -k+dt exp - τ M -k+1 τ M -k λ s ds -A M -k+1 λ τ M -k+1 = E M -k ϕ(τ M -k + dt, ∆ t M -k -dt) exp - τ M -k +dt τ M -k λ s ds .
We now use the hypothesis on the form of ϕ to get

E M -k exp -A M -k (∆t M -k -dt, X M -k+1 )λ τ M -k +d t -B M -k (∆t M -k -dt, X M -k+1 ) exp - τ M -k +dt τ M -k λsds .
Then using our simplified notations we obtain

= E M -k exp -A M -k (∆ t M -k -dt)λ τ M -k +dt -B M -k (∆ t M -k -dt) exp - τ M -k +dt τ M -k λ s ds .
For small dt and using the stochastic differential equation of λ, we get

= E M -k [exp{-A M -k (∆ t M -k -dt) λ τ M -k + κ M -k+1 θ M -k+1 -λ τ M -k dt + σ M -k+1 λ τ M -k dW t -B M -k (∆ t M -k -dt) -λ τ M -k dt}]
where

κ M -k+1 = κ(X τ M -k+1 ), θ M -k+1 = θ(X τ M -k+1 ) and σ M -k+1 = δ(X τ M -k+1 ). = exp -A M -k (∆ t M -k -dt)λ τ M -k -A M -k (∆ t M -k -dt)κ M -k+1 θ M -k+1 -λ τ M -k dt × exp -B M -k (∆ t M -k -dt) -λ τ M -k d t E M -k exp -A M -k (∆ t M -k -dt)σ M -k+1 λ τ M -k dW t = exp -A M -k (∆ t M -k -dt)λ τ M -k -A M -k (∆ t M -k )κ M -k+1 θ M -k+1 -λ τ M -k dt × exp -B M -k (∆ t M -k -dt) -λ τ M -k dt exp 1 2 A 2 M -k (∆ t M -k )(σ M -k+1 ) 2 λ τ M -k dt .
By identifying with the assumed expression of ϕ in (2.22), we get

     A M -k (∆ t M -k ) = A M -k (∆ t M -k -dt) -A M -k (∆ t M -k )κ M -k+1 dt -1 2 A 2 M -k (∆ t M -k )(σ M -k+1 ) 2 dt + dt B M -k (∆ t M -k ) = B M -k (∆ t M -k -dt) + A M -k (∆ t M -k )κ M -k+1 θ M -k+1 dt
Taking dt close to zero,

   ∂A M -k (∆t M -k ) ∂∆t M -k = -A M -k (∆ t M -k )κ M -k+1 -1 2 A 2 M -k (∆ t M -k )(σ M -k+1 ) 2 + 1 ∂B M -k (∆t M -k ) ∂∆t M -k = A M -k (∆ t M -k )κ M -k+1 θ M -k+1 with conditions for ∆ t M -k ≡ 0, A M -k (0) = A M -k+1 and B M -k (0) = 0.
Hence by Proposition 2.2, we know the explicit forms of A M -k (∆ t M -k ) and B M -k (∆ t M -k ) which are given by equations (2.14), (2.15) with the recursive condition that A M -k (0) = A M -k+1 and initial condition B M -k (0) = 0

We continue the proof of the Theorem 2.2, by applying the Lemma 2.1 with k = 2, we obtain

E exp - τ M -1 τ M -2 λ s ds -A M -1 λ τ M -1 |F M -2 = exp -A M -2 (∆ t M -2 )λ τ M -2 -B M -2 (∆ t M -2 ) with deterministic function A M -2 (∆ t M -2 ) and B M -2 (∆ t M -2 ). Hence E λ,X exp - T 0 rsds = exp {-BM-1} E λ,X M -3 k=0 exp - τ k+1 τ k λsds exp -AM-2(∆t M -2 )λt M -2 -BM-2(∆t M -2 ) = exp {-BM-1 -BM-2} E λ,X M -3 k=0 exp - τ k+1 τ k λsds -AM-2(∆t M -2 )λτ M -2
Conditioning an other time with respect to F M -3 , we obtain

E λ,X exp - T 0 λsds = exp {-B M -1 -B M -2 } E λ,X E M -3 k=0 exp - τ k+1 τ k λsds -A M -2 λτ M -2 |F M -3 = exp    - 2 j=1 B M -j    E λ,X M -4 k=0 exp - τ k+1 τ k λsds E exp - τ M -2 τ M -3 λsds -A M -2 λτ M -2 |F M -3
We can now again apply Lemma 2.1 with k = 3 and we obtain again that

E exp - τ M -2 τ M -3 λ s ds -A M -2 λ τ M -2 |F M -3 = exp -A M -3 (∆ t M -3 )λ τ M -3 -B M -3 (∆ t M -3 )
And so

E λ,X exp - T 0 λ s ds = exp    - 3 j=1 B M -j    E λ,X M -4 k=0 exp - τ k+1 τ k λ s ds -A M -3 (∆ t M -3 )λ τ M -3
By iterating the conditioning with respect to F M -k , k going to 4 to M and applying the Lemma 2.1 we finally obtain

E λ,X exp - T 0 λ s ds = exp    - M j=1 B M -j    exp (-A 0 (∆ t 0 )λ τ 0 )
with by hypothesis λ τ 0 = λ and A 0 (

∆ t 0 ) = A 0 (∆ t 0 , X τ 1 ) with X τ 1 = i 0 ∈ S d .
We can obtain the general expression of the conditional Laplace transform of the regime switching CIR process using Theorem 2.2.

Corollary 2.2. For all u ∈ C, we have that the conditional Laplace transform of the regime switching CIR process with λ 0 = λ and X τ 1 = i 0 ∈ S d is given by

Φ 0,T,λ,X (u) := E λ,X exp -u T 0 λ s ds = exp    - M j=1 BM-j ∆ t M -j    exp -Ã0 (∆ t 0 , i 0 ) λ (2.23)
where the functions BM-j for j = {1, . . . , M } and Ã0 are given by equations (2.18) and (2.19) taking parameters κj := κ(X τ j ) = κ j , θj := θ(X τ j ) = uθ j and σj := σ(X τ j ) = √ uσ j .

Proof. Since E exp -u

T 0 λ s ds |λ 0 = λ, F X T = E λ,X exp - T 0 (uλ s
)ds . This is the conditional Laplace transform of a process (uλ) t which is still a CIR process with new parameters κt = κ t , θt = uθ t and σt = √ uσ t , for all t ∈ [0, T ]. Hence applying Theorem 2.2 with this set of parameters gives the expected result.

Analytic approximation

We give now a second way to evaluate the defaultable bond. In fact, we give now an analytical approximation to evaluate the conditional Laplace transform of a regime switching CIR.

Construction of the new times grid

Let ∆ t be a fixed time step, then starting in time 0 we partition the time interval [0, T ] in time steps of -size ∆ t if there is no jump of the Markov process between time 0 to ∆ t .

-size τ 1 if there is the first jump of the Markov process at stopping time τ 1 less than ∆ t .

Hence we denote by h 1 the first time step of size ∆ t or τ 1 . Then we will proceed as the following: at time t k , corresponding of the time after the step h k , we construct the step h k+1 of size -∆ t if there is no jump of the Markov process between time t k to t k + ∆ t .

-τ i if there is the i jumps of the Markov process at stopping time τ i less than t k + ∆ t .

As an example of this construction

- ∆ t 2∆ t 3∆ t 4∆ t 0 τ 1 τ 2 τ 3 T t 1 - h 1 t 2 - h 2 t 3 - h 3 t 4 - h 4 t 5 - h 5 t 6 - h 6 t 7 - h 7 - h 8
This construction implies that h k = t k -t k-1 ≤ ∆ t and that the parameters of the regime switching CIR are constants (and bounded) in these each time intervals [t k , t k+1 [, k ∈ {0, 1, . . . , n -1}. It follows as an application of the tree property of conditional expectation that the conditional Laplace transform of λ is given by

Φ 0,T,λ,X (u) := E exp -u T 0 λ s ds |λ 0 , F X T = E λ 0 ,X exp -u T 0 λ s ds = E t 0 λ 0 ,X E t 1 λ,X . . . E t n-1 λ,X exp -u T 0 λ s ds .
(2.24) Proposition 2.3. Let for all k ∈ {1, . . . , n -1},

F k = exp h 3 n-k+1 8 u 2 σ 2 n-k a 2 n-k+1 λ n-k . (2.25)
Then, we have

E t 0 λ 0 ,X   exp -u T 0 λ s ds n-1 i=1 F i   = exp - u 2 n k=1 h 2 k a k κ k-1 θ k-1 - u 2 h 1 λ 0 [1 + a 1 (1 -κ 0 h 1 )] F n (2.26)
where

a n-1 = 1 + h n h n-1 + h n h n-1 a n (1 -h n κ n-1
) and a n = 1.

(2.27)

Proof. Using trapezoidal rule, we obtain that the expectation at time t n-1 is given by

E t n-1 λ 0 ,X exp -u T 0 λ s ds = E t n-1 λ 0 ,X exp -u n i=1 λ i + λ i-1 2 h i = exp -u n-2 i=1 λ i + λ i-1 2 h i -u λ n-2 2 h n-1 E t n-1 λ,X exp - u 2 [h n λ n + h n λ n-1 + h n-1 λ n-1 ] .
Using the approximation

λ n λ n-1 +κ n-1 (θ n-1 -λ n-1 ) h n +σ n-1 λ n-1 ∆W n-1 where ∆W n-1 = W n -W n-1 and denote by G n-2 the quantity exp -u n-2 i=1 λ i +λ i-1 2 h i -u λ n-2 2 h n-1 . We obtain that E t n-1 λ 0 ,X exp -u T 0 λ s ds is equal to G n-2 E t n-1 λ 0 ,X exp - u 2 h n λ n-1 + κ n-1 (θ n-1 -λ n-1 ) h n + σ n-1 λ n-1 ∆W n-1 + h n λ n-1 + h n-1 λ n-1 = G n-2 exp - u 2 h n λ n-1 + h 2 n κ n-1 θ n-1 -h 2 n κ n-1 λ n-1 + h n λ n-1 + h n-1 λ n-1 ×E t n-1 λ 0 ,X exp - u 2 h n σ n-1 λ n-1 ∆W n-1 .
Moreover we have that ∼ N (0, 1) then for a constant K we know that

E exp K √ T = exp K 2 T 2 .
(2.28) Applying (2.28) and factorize by -uλ n-1 h n-1

2

, we obtain that E

t n-1 λ 0 ,X exp -u T 0 λ s ds is equal to G n-2 exp u 2 8 h 3 n σ 2 n-1 λ n-1 a 2 n exp - uλ n-1 h n-1 2 1 + h n h n-1 + h n h n-1 a n (1 -h n κ n-1 ) - u 2 h 2 n a n κ n-1 θ n-1 = G n-2 exp - u 2 h 2 n a n κ n-1 θ n-1 exp - uλ n-1 h n-1 2 a n-1 F 1 .
Hence

E t n-1 λ 0 ,X   exp -u T 0 λ s ds F 1   = G n-2 exp u 2 8 h 3 n σ 2 n-1 λ n-1 exp - uλ n-1 h n-1 2 a n-1
Then denoting

G n-3 = exp -u n-3 i=1 λ i +λ i-1 2 h i -u λ n-3 2 h n-2
, we get the conditional expectation based on the information until t n-2

E t n-2 λ 0 ,X   E t n-1 λ,X   exp -u T 0 λ s ds F 1     = G n-3 exp - u 2 h 2 n a 2 n κ n-1 θ n-1 ×E t n-2 λ 0 ,X exp - u 2 [λ n-1 h n-1 a n-1 + h n-2 λ n-2 + h n-1 λ n-2 ] = G n-3 exp - u 2 h 2 n a n κ n-1 θ n-1 exp u 2 8 h 3 n-1 σ 2 n-2 λ n-2 a 2 n-1 exp - u 2 λ n-2 h n-1 a n-1 - u 2 κ n-2 (θ n-2 -λ n-2 ) h 2 n-1 a n-1 -h n-2 λ n-2 -h n-1 λ n-2 = G n-3 exp - u 2 h 2 n a n κ n-1 θ n-1 exp u 2 8 h 3 n-1 σ 2 n-2 λ n-2 a 2 n-1 exp - u 2 h 2 n-1 a n-1 κ n-2 θ n-2 × exp - u 2 λ n-2 h n-1 a n-1 + u 2 κ n-2 λ n-2 h 2 n-1 a n-1 -h n-2 λ n-2 -h n-1 λ n-2 = G n-3 exp - u 2 h 2 n a n κ n-1 θ n-1 - u 2 h 2 n-1 a n-1 κ n-2 θ n-2 F 2 × exp - u 2 λ n-2 h n-2 1 + h n-1 h n-2 + h n-1 h n-2 a n-1 (1 -κ n-2 h n-1 ) = G n-3 exp - u 2 h 2 n a n κ n-1 θ n-1 - u 2 h 2 n-1 a n-1 κ n-2 θ n-2 exp - u 2 λ n-2 h n-2 a n-2 F 2 .
Hence by iterations, we obtain

E t n-k λ 0 ,X   exp -u T 0 λ s ds k-1 i=1 F i   = G n-k-1 exp - u 2 k i=1 h 2 n-k+i a n-k+i κ n-k+i-1 θ n-k+i-1 × exp - u 2 λ n-k h n-k a n-k F k .
Then until time t 0 , we finally obtain the expected result.

Theorem 2.3. For all u ∈ C, the conditional Laplace transform Φ of the regime switching CIR process is given by

ln (Φ 0,T,λ,X (u)) = ln E t 0 λ 0 ,X exp -u T 0 λ s ds = - u 2 n k=1 h 2 k a k κ k-1 θ k-1 - u 2 h 1 λ 0 [1 + a 1 (1 -κ 0 h 1 )] + n k=1 ln E t 0 λ 0 ,X exp h 3 n-k+1 8 u 2 σ 2 n-k a 2 n-k+1 λ 0 + n-k i=0 κ i (θ i -λ i )h i+1 + n-k i=0 σ i λ i ∆W i (2.29)
where the sequence a is defined in Proposition 2.3.

Proof. As in [START_REF] Choi | An analytic approximation formula for pricing zero-coupon bonds[END_REF], we see that it would be difficult to compute the expression E

t n-k-1 λ 0 ,X [F k ] explicitly. That is why we simply approximate the expression F k at time t n-k by E 0 λ 0 ,X [F k ].
Firstly, we can use the following approximation

λ n-k λ 0 + n-k i=0 κ i (θ i -λ i )h i+1 + n-k i=0 σ i λ i ∆W i .
Then

F k = exp h 3 n-k+1 8 u 2 σ 2 n-k a 2 n-k+1 λ n-k = exp h 3 n-k+1 8 u 2 σ 2 n-k a 2 n-k+1 λ 0 + n-k i=0 κ i (θ i -λ i )h i+1 + n-k i=0 σ i λ i ∆W i . (2.30)
Approximate the expression of F k at time t n-k by the expectation at time 0, we obtain ln

  E t 0 λ 0 ,X   exp -u T 0 λ s ds n-1 i=1 F i     ln   E t 0 λ 0 ,X   exp -u T 0 λ s ds n k=1 E t 0 λ 0 ,X [F k ]     = ln   E t 0 λ 0 ,X exp -u T 0 λ s ds n k=1 E t 0 λ 0 ,X [F k ]   = ln E t 0 λ 0 ,X exp -u T 0 λ s ds -ln n k=1 E t 0 λ 0 ,X [F k ] = ln (Φ 0,T,λ,X (u)) - n k=1 ln E t 0 λ 0 ,X [F k ] .
We conclude using (2.30).

Remark 2.5. The analytic approximation formula given in previous Theorem could be also used as a way to simulate conditional Laplace transform.

3 Simulations

Pricing zero coupon Bond in the two firms case with two regimes

We fixe the time maturity of zero coupon bond T equal to 10 (i.e. a ten years ahead maturity). We take a deterministic risk free interest rate equals to zero.

The model parameters and heuristic

The heuristic of the calculus of the defaultable Bond price is then done by a Monte Carlo approach with M C ∈ N steps:

1. We know the value of the infinitesimal generator Π X of the credit migration process X. This one is given or estimated on some historical datas.

2. We generate a sequence of increasing stopping times and the times corresponding trajectory of X.

(a)

We apply the formula (1.10) to calculate the price of this defaultable Bond price for the firm A or B.

(b) We apply the construction of the time grid studied in subsection 2.2.1. Then we applied the formula (1.11).

4. We come back to step 2. until we will have done MC times this methods.

5. We evaluate the means of the MC values obtained in points 3 (a) and (b).

Hence assume that we have 2 regimes which represent a "normal" economic regime (regime 0) and a "crisis" regime (regime 1), then the credit migration process X is done in a set of four states: {(0, 0); (1, 0); (0, 1); (1, 1)}. For our simulation, in this part, we fix the infinitesimal generator Π X of the credit migration process X equals to In other words, if we are in a state where only the firm A is on "crisis" (i.e. state (1,0)) the probability that the firm B goes into "crisis" in the next time step is 0.01. Figure 1 gives an example of the trajectory of the credit migration process X and of the sequence of stopping times τ where the credit migration process jumps. So we need to have four sets of CIR default intensity parameters. Let for i ∈ {A, B}, ν i , ξ i and ρ i be real valued such that the set of parameters are given by Table Remark 3.6.

Π X =      -0.
-For i ∈ {A, B}, the constant ν i , ξ i and ρ i are choosen such that the CIR condition holds, i.e. 2κ X θ X ≥ σ 2 X .

-The state (0, 0) can be seen as a standard economic state where nor firm A nor firm B are in crisis.

Comparison of the different formulas to evaluate defaultable bond price.

Convergence: We know that the formula of the conditional survey probability with respect to G is given by equation (1.4). We would like now to compare the different formulas to pricing defaultable zero coupon bond (i.e. formulas (1.4), (1.10) and (1.11)). In tables 2, 3 and 4, we resume the convergence results in the case of a four states regime parameters defined as in Table 1.

Remark 3.7. We take for the time step parameter ∆ t (appearing in subsection 2.2.1 for the calculus of (1.11)) the value 0.01. Parameters: In Table 4, we can see that all formulas converge when the number of Monte Carlo simulations increases. Whereas the bond price value given by formula (1.10) based on Riccati approach or formula (1.11) based on analytic approach converges quicker than the value given by formula (1.4). Indeed, it is sufficient to take 400 Monte Carlo simulations to converge while it is necessary to take at least 2000 Monte Carlo simulations with formula (1.4). The difference of 10 -1 on the value given by (1.10) and (1.11) could be due to the error approximation of the conditional expectation at time t n-k of F k (see. proof of Theorem 2.3). Hence our two formulas need less simulations than formula (1.4) to converge. Moreover we observe that the Riccati approach formula (1.10) need a smaller computation time. Only 6.87 sec while formula (1.11) needs 60.48 sec and formula (1.4) needs 96.78 sec. Hence formula based on Riccati approach needs ten times less times than Analytic approach to converge. Whereas, we know that we used CIR model for intensity modeling since there exists explicit formula for bond. Hence as we said before, the analytic approach is interesting to obtain an explicit easy scheme to simulate defaultable bond.

ν A ν B ξ A ξ B ρ A ρ B Values: 0.2 0 0 0.3 0 0.1
Bond price with respect to the maturity T: We observe in Table 5 and Figure 2 (1.4)). Secondly, we remark, when the maturity T is greater than 10, that the result given by the analytic approximation is not better than the other. This relative mispricing was observed in the non regime switching case and uniform step time model discretization in [START_REF] Choi | An analytic approximation formula for pricing zero-coupon bonds[END_REF] as soon as the maturity T is greater than 10.

Other simulations with Riccati approach formula.

Bond Price all over time t ∈ [0, T ] Taking parameters as in Table 2, we can draw the value of a defaultable zero coupon bond price over time t ∈ [0, T ] using formula (1.10). An example is given in Figure 3.

Bond Price in function of probability that B goes to crisis: Taking parameters as in Table 2, we evaluate the price of a defaultable zero coupon bond in function of the probability P (X t+∆t = (0, 1)|X t = (0, 0)) and P (X t+∆t = (1, 1)|X t = (0, 0)). This

Regime switching defaultable intensity estimation

We work now on real data. We would like to focus more on the modeling issue. We will show that our regime switching model could capture some market features or economics behavior. Hence, we can use this algorithm to estimate the intensities of the two firms and then construct the Markov copula as explained before. and Zou [START_REF] Goutte | Continuous time regime switching model applied to foreign exchange rate[END_REF] for regime switching Cox Ingersoll Ross process applied to foreign exchange rate data. We assume that there are two regimes. This means that there is a "good" one and a "bad" one economies like a time crisis period and a "standard" economic period, or a spike time period and a non spike time period. Our results are stated in Table 6. Figure 6

Interpretations

We can see clearly in Figure 6 that there are two significant time periods. The first one between the 19/10/09 and april 2010, and the second one after april 2010. This period corresponds to the beginning of the economic world crisis . Hence we can see on the estimation results Table 6 that parameters values are very different in each regime. Before the crisis, we have a mean reverting parameter less than after crisis, κ1 = 0.02286 against κ2 = 0.117918. And in the time crisis period the volatility of the defaultable intensity is multiplied by 4 with respect to the volatility value before crisis.

We can see also in the right graph of Figure 7, the estimation process is fast, indeed only 15 iterations of our algorithm are sufficient for convergence to the true estimated parameters values.

Moreover, we can calculate the Regime classification measure (RCM) obtained by this regime switching model. In fact, let K > 0 be the number of regimes, the RCM statistics is then given by

RCM (K) = 100 1 - K K -1 1 T T t=1 K i=1 P X t = i|F λ T ; Θ - 1 K 2 (3.31)
where the quantity P X t = i|F λ T ; Θ is the smoothed probability given in the left graph on Figure 7 and Θ is the vector parameter estimation results i.e. Θ := κ, θ, σ, ΠX . The constant serves to normalize the statistic to be between 0 and 100. Good regime classification is associated with low RCM statistic value: a value of 0 means perfect regime classification and a value of 100 implies that no information about regimes is revealed. In our case we obtain a RCM equals to 8.41. Hence, it shows that this model with regime switching parameters captures very well two significant economics time period. And so this is a real add for the valuation of defaultable bond.

Methodology

Hence, we can apply this estimation method to find each estimated parameter for firms or countries A and B. Then using the copula construction theory defined in Corollary 1.1 and developed in section 1.2.1 we can apply pricing formulas given in Theorem 1.1 to obtain estimation of the price of defaultable bond regarding the correlation regime structure of each defaultable intensity regime switching estimations.

ACKNOWLEDGEMENTS:

The authors are grateful to Huyên Pham for his stimulating remarks and comments which allowed us to improve this paper.

Figure 1 :

 1 Figure 1: On left: Example of trajectory of the credit migration process X. On right: Example of instant of regime switching of the credit migration process X.

  that the three formulas give similar results. Whereas, firstly, we made this simulation taking 2000 Monte Carlo simulations for the Probabilistic approach (formula Bond Price Ricatti: 1.10 (M C = 400) Analytic: 1.11 (M C = 400) MC: 1.4 (M C = 2000)

Figure 2 :

 2 Figure 2: Graphs of the value of the Defaultable zero coupon bond price at time 0 with respect to the maturity time T with ∆ t = 0.01 (MC=400 for the two first formulas and 2000 for the Probability approach).

3. 2 . 1

 21 Calibration on Greek sovereign spread between 19/10/2009 to 13/05/2010 First, Figure 5 shows the plot of the Greek sovereign spread between the 19/10/2009 to 13/05/2010. For the estimation, we use the estimation procedure developed and studied in Goutte

Figure 5 :

 5 Figure 5: Greek Spread between the 19/10/2009 to 13/05/2010.

Figure 6 :

 6 Figure 6: Greek Spread regime calibration between the 19/10/2009 to 13/05/2010. (The color blue is when we are in regime 1 and red for regime 2).

Figure 7 :

 7 Figure 7: On left: Smoothed and Filtered probabilities. On right: Parameters convergence steps

Table 1 :

 1 A + ν B 0.15 + ξ A + ξ B 0.15 + ρ A + ρ B Parameters values of the CIR default intensity in the 2 regimes case.

	1

Table 2 :

 2 Values of the constant parameters defined in Table1.

	Regimes:	(0, 0)	(1, 0)	(0, 1)	(1, 1)
	Bond price values: 0.6086 0.3777 0.2740 0.0668

Table 3 :

 3 Values of the Bond price standard formula in t = 0 in each regime with a maturity T = 10 years.

	Bond Price Ricatti: 1.10 (std) C.T.(sec.) Analytic: 1.11 (Std) C.T.	MC: 1.4 C.T.
	M C = 100	0.5619 (0.1110)	1.94	0.5585 (0.1699)	15.98	0.6500	1.95
	M C = 300	0.5692 (0.1015)	5.34	0.5587 (0.1602)	52.52	0.6233	6.61
	M C = 400	0.5736 (0.0949)	6.87	0.5649 (0.1505)	60.48	0.6400	9.58
	M C = 500	0.5748 (0.0927)	7.97	0.5658 (0.1511)	78.32	0.6360 12.44
	M C = 1000	0.5738 (0.0961)	16.31	0.5654 (0.1533)	146.51	0.6220 33.23
	M C = 2000	0.5727 (0.0995)	27.33	0.5646 (0.1533)	221.22	0.5770 96.78

Table 4 :

 4 Results for the formulas convergence in t = 0 with initial regime the regime (0, 0) and maturity

	T = 10 years.
	Bond Price Analytic: 1.11
	∆ t = 1	0.5612
	∆ t = 0.1	0.5648
	∆ t = 0.01	0.5649

Table 5 :

 5 Value of the Defaultable zero coupon bond price at time 0 with respect to the maturity time T with ∆ t = 0.01.

	1
	0.9
	0.8
	0.7
	0.6
	0.5
	0.4
	1	2	5	7	10	15
				Time to maturity

Value of the zero coupon defaultable bond Riccatti Approach Analytic Approach Probability

Table 6 :

 6 Maximum Likelihood estimation results.of each time period in each regime.

			gives a graphical representation
		Regime 1	Regime 2
	κ	0.022860	0.117918
	θ	309.460660 620.721205
	σ	0.774675	3.092136
	ΠX ii	0.974977	0.934452
	π i	0.723722	0.276278

is p X 1,3 and p X 1,4 . Hence we take a parametric transition matrix of the form: grows up (i.e. the probability P (X t+∆t = (1, 1)|X t = (0, 0))), the price of the defaultable zero coupon bond price of the firm A decreases. This means that the economic status of the firm B (the probability to go in crisis) impacts the value of the defaultable zero coupon bond of the firm A.