First and second-to-default options in models with various information flows *

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First and second-to-default options in models with various information flows *

Pavel Gapeev, Monique Jeanblanc

To cite this version:

Pavel Gapeev, Monique Jeanblanc. First and second-to-default options in models with various infor-

mation flows *. 2021. �hal-03119469�

First and second-to-default options in models with various information flows ^∗

Pavel V. Gapeev ^† Monique Jeanblanc ^‡

We continue to study the credit risk model of a financial market introduced in [19]

in which the dynamics of intensity rates of two default times are described by linear combinations of three independent geometric Brownian motions. The dynamics of two default-free risky asset prices are modeled by two geometric Brownian motions which are dependent of the ones describing the default intensity rates. We obtain closed form expressions for the no-arbitrage prices of some first- and second-to-default European style contingent claims given the reference filtration initially and progressively enlarged by the two successive default times. The accessible default-free reference filtration is generated by the standard Brownian motions driving the model.

1 Introduction

In this paper, we derive closed form expressions for the (no-arbitrage) prices of first and second- to-default European style contingent claims in a model of a financial market introduced in [19]

given the flows of information which are expressed by the reference filtration progressively and initially enlarged by means of the successive default times. It is assumed that the option payoffs depend on the default times and the current prices of the underlying default-free risky assets taken at the times of defaults. The dynamics of market prices of the two risky assets are described by geometric Brownian motions driven by constantly correlated standard Brownian motions. The default times are given by the first times at which linear combinations of three integral processes of independent geometric Brownian motions hit certain random thresholds which are independent of each other and of the standard Brownian motions driving the model.

∗

This research benefited from the support of the ’Chaire March´ es en Mutation’, French Banking Federation and ILB, Labex ANR 11-LABX-0019.

†

London School of Economics, Department of Mathematics, Houghton Street, London WC2A 2AE, United Kingdom; e-mail: p.v.gapeev@lse.ac.uk

‡

Laboratoire de Math´ ematiques et Mod´ elisation d’ ´ Evry (LaMME), UMR CNRS 8071; Univ Evry-Universit´ e Paris Saclay, 23 Boulevard de France, 91037 ´ Evry cedex France; e-mail: monique.jeanblanc@univ-evry.fr

Mathematics Subject Classification 2010: Primary 91G40, 60G44, 60J65. Secondary 91B70, 60J60, 91G20.

Key words and phrases: Successive default times, first- and second-to-default options, geometric Brownian motion, initial and progressive enlargements of filtrations.

Date: January 13, 2021

The dependence between the default times is then expressed by means of the dynamics of their intensity rates given by linear combinations of the three independent geometric Brownian motions which are driven by standard Brownian motions constantly correlated with the ones related to the risky asset prices. The default-free reference filtration accessible from the market is generated by the standard Brownian motions driving the model. The prices of the resulting defaultable European style contingent claims are explicitly expressed through the transition densities of the marginal distributions of the geometric Brownian motions and their integral processes describing the model.

The credit risk models in which the default times are defined as the first times at which the associated cumulative intensity processes reach certain random thresholds were initiated by Lando [22]. The computations of conditional distributions of the default times given the observable filtrations in such a first passage intensity model with independent default intensi- ties and correlated thresholds were presented in Sch¨ onbucher [24; Chapter X, Proposition 10.9].

Brigo and Chourdakis [7] studied the problem of pricing of credit default swaps (CDSs) in such a model with counterparty risk in which the intensities of the default times are independent of each other, but the associated random thresholds are correlated. Brigo, Capponi and Pallavicini [6] developed the pricing framework for bilateral counterparty credit risk models and specified the credit and debit valuation adjustments (CVAs and DVAs) in the cases in which the default intensity rates are expressed by means of the (strictly positive) Feller’s square root diffusion processes, and the associated thresholds are correlated through a Gaussian copula. Bielecki et al. [3] provided the analytic basis for the quantitative methodology of dynamic hedging of the counterparty risk and developed the main theoretical issues of dynamic hedging of credit valuation adjustments. Assefa et al. [1] derived a model-free general counterparty risk repre- sentation formula for the credit valuation adjustment of a netted and collateralised portfolio.

Some related discussions on modelling and computational aspects regarding managing of ex- posure to counterparty risk are provided in the monographs by Gregory [21], Cesari, Aquilina and Charpillon [10], Brigo, Morini and Pallavicini [8], and Cr´ epey, Bielecki and Brigo [11].

El Karoui, Jeanblanc and Jiao [15]-[16] emphasised the roles of conditional distributions of

several default times in the intensity credit risk models given the appropriate filtrations and

presented general expressions for the prices of various defaultable European style contingent

claims. In this paper, we consider a model in which the default intensity rates are explicitly given

as linear combinations of three independent geometric Brownian motions which are dependent

of the ones describing the dynamics of the risky asset price processes. We then use the Markov

property of the resulting multi-dimensional process describing the model and apply the explicit

formula from Yor [26] for the joint marginal density of a geometric Brownian motion and its

integral process to derive closed form expressions for the prices of first- and second-to-default

options given the reference filtration progressively and initially enlarged by means of the default

times. We also note that the model proposed in the paper keeps its Markovian feature in the

filtrations which are obtained by means of the progressive and initial enlargements of the initial

Brownian reference filtration. The results of this paper can naturally be extended to the case

of credit risk models with more than two default times and more than two underlying risky

assets of a similar dependence structure. The prices of first and second-to-default options

and other European style defaultable contingent claims can then be expressed through the

transition densities of the marginal distributions of the resulting multi-dimensional continuous

Markov process describing the model. The prices of other defaultable contingent claims in some

switching models with partial information were recently computed in [18].

The paper is organised as follows. In Section 2, we reproduce the credit risk model of a financial market introduced in [19; Section 2] with the dependence structure of the dynamics of prices of two risky assets and two default intensity rates described above. In Section 3, we derive explicit expressions for the conditional distributions of the two successive default times given the accessible default-free reference filtration and the observable filtrations. In Section 4, we compute closed form expressions for the prices of first- and second-to-default options in the model with two underlying risky assets given the reference filtration progressively and initially enlarged by the ordered default times. In Section 5, we recall explicit expressions from [19;

Sections 3 and 4] for the conditional distributions of the two non-successive default times given the accessible default-free reference filtration and the observable filtrations, these results being used in the previous sections. The main results of the paper are stated in Propositions 4.1-4.3.

2 The model

In this section, we reproduce the model of a financial market with two defaultable risky assets introduced in [19; Section 2]. We also define the accessible default-free reference filtration as well as the observable filtrations and refer some known results and distribution laws.

2.1 The dynamics of default intensities and firm values

Let (Ω, G, P ) be a probability space supporting independent standard Brownian motions W

^l

= (W

_t^l

)

_t≥0

and B

^l

= (B

^l_t

)

_t≥0

, l = 0, 1, 2, as well as the random variables U

_i

, i = 1, 2, which are uniformly distributed on (0, 1). Suppose that the variables U

_i

, i = 1, 2, are independent of each other and of the processes W

^l

and B

^l

, l = 0, 1, 2. We define the random times τ

_i

, i = 1, 2, by

τ

_i

= inf t ≥ 0

δ

_i

A

⁰_t

+ λ

_i

A

ⁱ_t

≥ − ln U

_i

(2.1) where the processes A

^l

= (A

^l_t

)

t≥0

, l = 0, 1, 2, are given by

A

^l_t

= Z

t

0

Y

_s^l

ds (2.2)

for all t ≥ 0, and some δ

_i

, λ

_i

≥ 0, i = 1, 2, fixed, so that the processes (δ

_i

A

⁰_t

+ λ

_i

A

ⁱ_t

)

_t≥0

, i = 1, 2, form the cumulative intensities, and the processes (δ

_i

Y

_t⁰

+ λ

_i

Y

_tⁱ

)

t≥0

, i = 1, 2, are the intensity rates of the random times τ

_i

, i = 1, 2. These notions mean that the processes (I(τ

_i

≤ t) − δ

_i

A

⁰_t∧τ

i

− λ

_i

A

ⁱ_t∧τ

i

)

_t≥0

, i = 1, 2, are martingales in their natural filtrations. Assume that the processes Y

^l

= (Y

_t^l

)

t≥0

, j = 0, 1, 2, admit the representations

Y

_t^l

= exp

β

_l

− γ

_l²

2

t + γ

_l

W

_t^l

(2.3)

for all t ≥ 0, and some constants β

_l

∈ R and γ

_l

> 0, l = 0, 1, 2. Note that the random times

τ

_i

, i = 1, 2, defined in (2.1) with (2.2) and (2.3) can occur simultaneously only with probability

zero, and thus, the property P (τ

₁

= τ

₂

) = 0 holds, by construction, that we take into account

in the sequel, that is, noting that the events {τ

_i

< τ

3−i

} and {τ

_i

≤ τ

3−i

} are equal ( P -a.s.),

for every i = 1, 2.

Suppose that the random times τ

_i

, i = 1, 2, represent the default times of two firms (ref- erence credits) with the value dynamics described by the processes X

ⁱ

= (X

_tⁱ

)

_t≥0

, i = 1, 2, given by X

_tⁱ

= (Y

_tⁱ

)

^αi

(Z

_t⁰

)

^ζi

Z

_tⁱ

, for some α

_i

and ζ

_i

∈ R , i = 1, 2, fixed. Here, the processes Z

^l

= (Z

_t^l

)

t≥0

, l = 0, 1, 2, are defined by

Z

_t^l

= exp

η

_l

− θ

_l²

2

t + θ

_l

B

_t^l

(2.4) for all t ≥ 0, and some constants η

l

∈ R and θ

l

> 0, l = 0, 1, 2. We further assume that the discounted firm value processes (e

^−rt

X

_tⁱ

)

t≥0

, i = 1, 2, are martingales with respect to the pricing measure P under which the processes Y

^l

and Z

^l

, l = 0, 1, 2, admit the representations in (2.3) and (2.4), where r ≥ 0 is the interest rate of a riskless bank account. Thus, taking into account the independence of the driving processes W

^l

and B

^l

, l = 0, 1, 2, we may conclude that the equality

β

_i

α

_i

+ γ

_i²

2 α

_i

(α

_i

− 1) + η

₀

ζ

_i

+ θ

₀²

2 ζ

_i

(ζ

_i

− 1) + η

_i

= r (2.5) should hold, for every i = 1, 2.

2.2 Some filtrations and distribution laws

Let us denote by (F

_t

)

t≥0

the natural filtration of the processes Y

^l

and Z

^l

, l = 0, 1, 2, defined by F

t

= σ(Y

_t^l

, Z

_t^l

| 0 ≤ s ≤ t, l = 0, 1, 2), for all t ≥ 0, which coincides with the one of the driving standard Brownian motions W

^l

and B

^l

, l = 0, 1, 2, given by σ(W

_t^l

, B

_t^l

| 0 ≤ s ≤ t, l = 0, 1, 2), for all t ≥ 0. We define the progressively enlarged filtrations (G

_tⁱ

)

t≥0

, i = 1, 2, by G

_tⁱ

= F

t

∨ σ(τ

i

∧ t), and (G

t

)

t≥0

by G

t

= F

t

∨ σ(τ

1

∧ t) ∨ σ(τ

2

∧ t), for all t ≥ 0. Let us also introduce the initially enlarged filtrations (F

_tⁱ

)

t≥0

, i = 1, 2, by F

_tⁱ

= F

_t

∨ σ(τ

_i

), for all t ≥ 0. We actually consider the smallest right-continuous completed filtrations that contain the appropriate filtrations defined above. The default-free reference filtration (F

t

)

t≥0

reflects the information flow which is accessible for the investors trading in the market, while the filtrations (G

_tⁱ

)

t≥0

, i = 1, 2, and (G

_t

)

t≥0

reflect the accessible information including the one about the appearance of the default times. Note that, by virtue of the independence of the random variables U

_i

, i = 1, 2, and the filtration (F

_t

)

t≥0

, it follows that (F

_t

)

t≥0

is immersed in the filtrations (G

_tⁱ

)

t≥0

, i = 1, 2, and (G

_t

)

t≥0

(see, e.g., [5] and [17]). Similarly, we also have that (G

_tⁱ

)

t≥0

is immersed in the filtration (G

_tⁱ

∨ σ(U

3−i

))

t≥0

, and hence, in (G

t

)

t≥0

, for every i = 1, 2.

We recall that the immersion of a filtration in a larger filtration, also known as the (H)- hypothesis for the two nested filtrations, means that any martingale for the smaller filtration is a martingale for the larger one (see, e.g., [5], [23; Chapter V, Section 4], [4; Chapter VIII, Section 3], or [2; Chapter III]). Note that the immersion of (F

_t

)

_t≥0

in (G

_tⁱ

)

_t≥0

is equivalent to the conditional independence of G

_tⁱ

and F

∞

with respect to F

_t

, for all t ≥ 0, for i = 1, 2, while the immersion of (F

_t

)

t≥0

in (G

_t

)

t≥0

is equivalent to the conditional independence of G

_t

and F

_∞

with respect to F

_t

, for all t ≥ 0 (see, e.g., [13]).

We now define the random times κ

1

= τ

₁

∧ τ

₂

and κ

2

= τ

₁

∨ τ

₂

. Along with the filtrations

introduced above, let us define the progressively enlarged filtrations (H

ⁱ_t

)

t≥0

, for i = 1, 2, by

H

_tⁱ

= F

_t

∨ σ( κ

i

∧ t), for all t ≥ 0. We introduce H

_t

= F

_t

∨ σ( κ

1

∧ t) ∨ σ( κ

2

∧ t), so that

H

_t

( G

_t

, for all t ≥ 0, since the filtration (H

_t

)

t≥0

does not contain information which default

time τ

_i

, i = 1, 2, occurred the first and which occurred the second, except in the trivial case

in which the default times τ

_i

, i = 1, 2, are ordered. We also consider the initially enlarged filtrations H

ⁱ_t

∨ σ( κ

3−i

), for i = 1, 2. By virtue of the same arguments as before, we conclude that (F

_t

)

t≥0

is immersed in (H

ⁱ_t

)

t≥0

, i = 1, 2, and in (H

_t

)

t≥0

.

2.3 Some implications of the key lemma

Let us now consider a filtration (K

_t

)

t≥0

larger than the filtration (F

_t

)

t≥0

, that is, F

_t

⊆ K

_t

, for all t ≥ 0. Then, if K

t

coincides with F

t

on the event J

t

∈ K

t

such that P (J

t

) > 0, that is, if for any K

_t

∈ K

_t

, there exists an event F

_t

∈ F

_t

such that J

_t

∩ K

_t

= J

_t

∩ F

_t

, then, on the event J

_t

, the conditional expectation E[V | K

_t

] of an integrable random variable V is equal to an F

t

-measurable random variable. Hence, denoting by I(J ) the indicator function of the set J , according to the results in [12; page 122] and [4; Section 5.1], this fact leads to the equality

I(J

t

) E V

K

t

P (J

t

| F

t

) = I(J

t

) E

V I(J

t

) F

t

(2.6)

and thus, taking into account the fact that P (J

_t

| F

_t

) > 0 on the event J

_t

, we have I(J

_t

) E

V K

_t

= I(J

_t

) E[V I(J

_t

) | F

_t

]

P (J

_t

| F

_t

) (2.7)

for any integrable random variable V and all t ≥ 0. We further refer to the result in (2.6)- (2.7) as to the generalised key lemma for the filtrations (K

_t

)

t≥0

and (F

_t

)

t≥0

. Observe that G

_tⁱ

coincides with F

_t

on the event {τ

_i

> t}, and G

_t

coincides with F

_t

on the event {τ

_i

∧ τ

3−i

> t}, while G

_tⁱ

∨ σ(τ

3−i

) coincides with F

_t³⁻ⁱ

≡ F

t

∨ σ(τ

3−i

) on the event {τ

i

> t} , for all t ≥ 0 and every i = 1, 2. In these cases, the expressions in (2.6)-(2.7), together with the tower property for conditional expectations, imply that, for each F

_T

-measurable integrable random variable V

_Tⁱ

, the equality

I(τ

_i

> t) E V

_Tⁱ

G

_tⁱ

= I(τ

_i

> t) E[V

_Tⁱ

P (τ

_i

> t | F

_T

) | F

_t

]

P (τ

_i

> t | F

_t

) (2.8) holds, for all t ≥ 0 and every i = 1, 2 (see, e.g., [2; Lemma 2.9]). Moreover, it follows that, for each (F

_t

)

t≥0

-progressively measurable process V

ⁱ

= (V

_tⁱ

)

t≥0

, the equality

E V

_τⁱ

i

I(τ

_i

> t) | G

_tⁱ

= I(τ

_i

> t) E Z

∞

t

V

_uⁱ

P (τ

_i

∈ du | F

_u

) P (τ

i

> t | F

t

)

F

_t

(2.9) holds, for all t ≥ 0 and every i = 1, 2 (see, e.g. [2; Corollary 2.10]). We further refer to the results in (2.8) and (2.9) as to the first and the second part of the key lemma for the filtrations (G

_tⁱ

)

t≥0

and (F

_t

)

t≥0

, for every i = 1, 2.

For any Borelian bounded function ψ

_i

, let us now compute the conditional expectation E[ψ

_i

(τ

_i

) | F

_t

∨ σ(τ

3−i

)], for all t ≥ 0 and every i = 1, 2. For this purpose, we apply the result of [9; Proposition 2.7] to conclude that any (F

_t

∨ σ(τ

_3−i

))

_t≥0

-progressively measurable process can be written as Φ

ⁱ_t

(τ

3−i

), where Φ

ⁱ

(v) = (Φ

ⁱ_t

(v))

t≥0

is (F

_t

)

t≥0

-progressively measurable, for any v ≥ 0 fixed, while the function v 7→ Φ

ⁱ_t

(v) is Borel measurable, for all t ≥ 0 and every i = 1, 2. In particular, there exists Ψ

ⁱ

with the above measurability properties such that

E

ψ

_i

(τ

_i

) | F

_t

∨ σ(τ

3−i

)

= Ψ

ⁱ_t

(τ

3−i

) (2.10)

for all t ≥ 0 and every i = 1, 2. Then, we observe that, by definition of conditional expectations, for any event F

_t

∈ F

_t

, and any Borelian bounded function ϕ, the equality

E

Ψ

ⁱ_t

(τ

3−i

) I(F

_t

) ϕ(τ

3−i

)

= E

I (F

_t

) ψ

_i

(τ

_i

) ϕ(τ

3−i

)

(2.11) holds, and thus, we have

E Z

∞

v=0

Ψ

ⁱ_t

(v) I(F

_t

) ϕ(v ) P (τ

3−i

∈ dv | F

_t

)

(2.12)

= E

I(F

_t

) Z

∞

u=0

Z

∞

v=0

ψ

_i

(u) ϕ(v) P (τ

_i

∈ du, τ

3−i

∈ dv | F

_t

)

for all t ≥ 0 and every i = 1, 2. Hence, the equality in (2.12) being valid for any Borelian bounded function ϕ and the conditional law of τ

_3−i

being absolutely continuous with respect to Lebesque’s measure imply that the equality

Ψ

ⁱ_t

(v) = Z

∞

u=0

ψ

_i

(u)P (τ

_i

∈ du, τ

3−i

∈ dv | F

_t

)

P (τ

3−i

∈ dv | F

t

) (2.13)

is satisfied, for all t, v ≥ 0, and every i = 1, 2.

Similarly, for any Borelian bounded functions ψ e

_i

and ξ e

_i

, let us now compute the conditional expectations E[ ψ e

_i

(τ

_i

) I(τ

_i

> τ

3−i

) | F

_t

∨ σ( κ

1

)] and E[e ξ

_i

(τ

_i

) I(τ

_i

< τ

3−i

) | F

_t

∨ σ( κ

1

)], for all t ≥ 0 and every i = 1, 2. We apply again the result of [9; Proposition 2.7] to conclude that any (F

_t

∨ σ( κ

1

))

t≥0

-progressively measurable process can be written as Φ e

ⁱ_t

( κ

1

), where Φ e

ⁱ

(u) = (e Φ

ⁱ_t

(u))

t≥0

is (F

_t

)

t≥0

-progressively measurable, for any u ≥ 0 fixed, while the function u 7→ Φ e

ⁱ_t

(u) is Borel measurable, for all t ≥ 0 and every i = 1, 2. In particular, there exist Ψ e

ⁱ

and e Ξ

ⁱ

such that

E

ψ e

_i

(τ

_i

) I(τ

_i

> τ

3−i

) | F

_t

∨ σ( κ

1

)

= Ψ e

ⁱ_t

( κ

1

) (2.14) and

E

ξ e

_i

(τ

_i

) I(τ

_i

< τ

3−i

) | F

_t

∨ σ( κ

1

)

= Ξ e

ⁱ_t

( κ

1

) (2.15) for all t ≥ 0 and every i = 1, 2. Then, we observe that, by definition of conditional expectations, for any event F

_t

∈ F

_t

, and any (positive measurable) bounded function ϕ, the equalities e

E Z

∞

u=0

Z

∞

v=0

ψ e

_i

(v) I(F

_t

) ϕ(u) e I(u < v) P (τ

3−i

∈ du, τ

_i

∈ dv | F

_t

)

(2.16)

= E

I(F

_t

) Z

∞

u=0

Z

∞

v=0

Ψ e

ⁱ_t

(u) ϕ(u) e I(u < v) P (τ

₁

∈ du, τ

₂

∈ dv | F

_t

) + P (τ

₂

∈ du, τ

₁

∈ dv | F

_t

)

= E

I(F

_t

) Z

∞

u=0

Ψ e

ⁱ_t

(u) ϕ(u) e P (τ

₁

> u, τ

₂

∈ du | F

_t

) + P (τ

₂

> u, τ

₁

∈ du | F

_t

)

and E

Z

∞

u=0

Z

∞

v=0

ξ e

i

(u) I(F

t

) ϕ(u) e I(u < v) P (τ

i

∈ du, τ

3−i

∈ dv | F

t

)

(2.17)

= E

I (F

t

) Z

∞

u=0

Z

∞

v=0

Ξ e

ⁱ_t

(u) ϕ(u) e I(u < v) P (τ

1

∈ du, τ

2

∈ dv | F

t

) + P (τ

2

∈ du, τ

1

∈ dv | F

t

)

= E

I (F

t

) Z

∞

u=0

Ξ e

ⁱ_t

(u) ϕ(u) e P (τ

1

> u, τ

2

∈ du | F

t

) + P (τ

2

> u, τ

1

∈ du | F

t

)

hold, for all t ≥ 0 and every i = 1, 2. Hence, the equality in (2.16) being valid for any Borel function ϕ, and the conditional law e P (τ

_i

> du, τ

_3−i

> u | F

_t

) being absolutely continuous with respect to the Lebesque measure (see Subsection 5.2 below), the equalities

Ψ e

ⁱ_t

(u) = Z

∞

v=0

ψ e

i

(v)I(u < v)P (τ

3−i

∈ du, τ

i

∈ dv | F

t

)

P (τ

₁

> u, τ

₂

∈ du | F

_t

) + P (τ

₂

> u, τ

₁

∈ du | F

_t

) (2.18) and

Ξ e

ⁱ_t

(u) = ξ e

i

(u)P (τ

3−i

> u, τ

i

∈ du | F

_t

)

P (τ

₁

> u, τ

₂

∈ du | F

_t

) + P (τ

₂

> u, τ

₁

∈ du | F

_t

) (2.19) are satisfied, for all t, u ≥ 0, and every i = 1, 2. Here, we note that the equality P (τ

₁

> u, τ

₂

∈ du | F

_t

) + P (τ

₂

> u, τ

₁

∈ du | F

_t

) = P ( κ

1

∈ du | F

_t

) holds, for all t, u ≥ 0, which explain the meaning of the denominator.

Finally, for any Borelian bounded function ψ b

_i

and ξ b

_i

, let us now compute the conditional expectations E[ ψ b

_i

(τ

_i

) I(τ

_i

< τ

3−i

) | F

_t

∨ σ( κ

2

)] and E[b ξ

_i

(τ

_i

) I(τ

_i

> τ

3−i

) | F

_t

∨ σ( κ

2

)], for all t ≥ 0 and every i = 1, 2. We apply again the result of [9; Proposition 2.7] to conclude that any (F

t

∨ σ( κ

2

))

t≥0

-progressively measurable process can be written as Φ b

ⁱ_t

( κ

2

), where Φ b

ⁱ

(v ) = (b Φ

ⁱ_t

(v))

t≥0

is (F

_t

)

t≥0

-progressively measurable, for any v ≥ 0 fixed, while the function v 7→ Φ b

ⁱ_t

(v ) is Borel measurable, for all t ≥ 0 and every i = 1, 2. In particular, there exist Ψ b

ⁱ

and b Ξ

ⁱ

such that

E

ψ b

_i

(τ

_i

) I(τ

_i

< τ

3−i

) | F

_t

∨ σ( κ

2

)

= Ψ b

ⁱ_t

( κ

2

) (2.20) and

E

ξ b

_i

(τ

_i

) I(τ

_i

> τ

_3−i

) | F

_t

∨ σ( κ

2

)

= Ξ b

ⁱ_t

( κ

2

) (2.21) for all t ≥ 0 and every i = 1, 2. Then, we observe that, by definition of conditional expectations, for any event F

_t

∈ F

_t

, and any Borelian bounded function ϕ, the equalities b

E Z

∞

u=0

Z

∞

v=0

ψ b

_i

(u) I(F

_t

) ϕ(v) b I (u < v) P (τ

_i

∈ du, τ

3−i

∈ dv | F

_t

)

(2.22)

= E

I (F

_t

) Z

∞

u=0

Z

∞

v=0

Ψ b

ⁱ_t

(v) ϕ(v) b I(u < v) P (τ

₁

∈ du, τ

₂

∈ dv | F

_t

) + P (τ

₂

∈ du, τ

₁

∈ dv | F

_t

)

= E

I (F

_t

) Z

∞

v=0

Ψ b

ⁱ_t

(v) ϕ(v) b P (τ

₁

≤ v, τ

₂

∈ dv | F

_t

) + P (τ

₂

≤ v, τ

₁

∈ dv | F

_t

)

and E

Z

∞

u=0

Z

∞

v=0

ξ b

_i

(v) I(F

_t

) ϕ(v) b I(u < v) P (τ

3−i

∈ du, τ

_i

∈ dv | F

_t

)

(2.23)

= E

I (F

_t

) Z

∞

u=0

Z

∞

v=0

Ξ b

ⁱ_t

(v) ϕ(v) b I(u < v) P (τ

₁

∈ du, τ

₂

∈ dv | F

_t

) + P (τ

₂

∈ du, τ

₁

∈ dv | F

_t

)

= E

I (F

_t

) Z

∞

v=0

Ξ b

ⁱ_t

(v) ϕ(v) b P (τ

₁

≤ v, τ

₂

∈ dv | F

_t

) + P (τ

₂

≤ v, τ

₁

∈ dv | F

_t

)

hold, for all t ≥ 0 and every i = 1, 2. Hence, the equality in (2.22) being valid for any Borelian bounded function ϕ, and the conditional law b P (τ

_i

≤ v, τ

_3−i

∈ dv | F

_t

) being absolutely continuous with respect to the Lebesque measure (see Subsection 5.2 below), the equalities

Ψ b

ⁱ_t

(v ) = Z

∞

u=0

ψ b

i

(u)I(u < v)P (τ

i

∈ du, τ

3−i

∈ dv | F

t

)

P (τ

₁

≤ v, τ

₂

∈ dv | F

_t

) + P (τ

₂

≤ v, τ

₁

∈ dv | F

_t

) (2.24) and

Ξ b

ⁱ_t

(v) = ξ b

i

(v)P (τ

3−i

≤ v, τ

i

∈ dv | F

_t

)

P (τ

₁

≤ v, τ

₂

∈ dv | F

_t

) + P (τ

₂

≤ v, τ

₁

∈ dv | F

_t

) (2.25) are satisfied, for all t, v ≥ 0, and every i = 1, 2. Here, we note that the equality P (τ

₁

≤ v, τ

₂

∈ dv | F

_t

) + P (τ

₂

≤ v, τ

₁

∈ dv | F

_t

) = P ( κ

2

∈ dv | F

_t

) holds, for all t, v ≥ 0, which explain the meaning of the denominator.

3 Conditional distributions of the default times

In this section, we derive explicit expressions for the conditional distributions of two successive default times given the accessible filtration generated by the market prices of the risky assets as well as given the observable filtrations.

3.1 Conditional distributions of κ ^j , j = 1, 2, under (F _t ) _t≥0

Let us now compute the conditional distributions P ( κ

¹

> u, κ

²

> v | F

t

) of the successive default times κ

j

, j = 1, 2, given the reference filtration (F

_t

)

t≥0

, for all t, u, v ≥ 0. We first observe that the equalities

P ( κ

¹

> u, κ

²

> v | F

_t

) (3.1)

= Z

∞

u

Z

∞

v

I(u

⁰

< v

⁰

) P (τ

₁

∈ du

⁰

, τ

₂

∈ dv

⁰

| F

_t

) + P (τ

₂

∈ du

⁰

, τ

₁

∈ dv

⁰

| F

_t

)

hold, for all t, u, v ≥ 0, where the conditional probabilities P (τ

_i

∈ du

⁰

, τ

3−i

∈ dv

⁰

| F

_t

), for all t, u

⁰

, v

⁰

≥ 0 every i = 1, 2, are given in Subsection 5.2 below in the expressions of (5.17), (5.19) and (5.22), according to the positions of u

⁰

, v

⁰

with respect to t . Moreover, it follows from the expressions in (5.17) that the equalities

P ( κ

¹

∈ du, κ

²

∈ dv | F

_∞

) (3.2)

= e

^{−δ1A0u−λ1A1u−δ2A0v−λ2A2v}

(δ

₁

Y

_u⁰

+ λ

₁

Y

_u¹

) (δ

₂

Y

_v⁰

+ λ

₂

Y

_v²

) + e

^{−δ2A0u−λ2A2u−δ1A0v−λ1A1v}

(δ

₂

Y

_u⁰

+ λ

₂

Y

_u²

) (δ

₁

Y

_v⁰

+ λ

₁

Y

_v¹

)

dudv

= P ( κ

1

∈ du, κ

2

∈ dv | F

_t

) for 0 ≤ u < v ≤ t

are satisfied. Furthermore, according to the tower property for conditional expectations, it follows from the representation in (3.2) that the equalities

P ( κ

1

∈ du, κ

2

∈ dv | F

_t

) = E

P ( κ

1

∈ du, κ

2

∈ dv | F

_v

) F

_t

(3.3)

= E

e

^{−δ1A0u−λ1A1u−δ2A0v−λ2A2v}

(δ

₁

Y

_u⁰

+ λ

₁

Y

_u¹

) (δ

₂

Y

_v⁰

+ λ

₂

Y

_v²

) + e

^{−δ2A0u−λ2A2u−δ1A0v−λ1A1v}

(δ

₂

Y

_u⁰

+ λ

₂

Y

_u²

) (δ

₁

Y

_v⁰

+ λ

₁

Y

_v¹

) F

_t

dudv

= e

^{−δ1A0u−λ1A1u−δ2A0t−λ2A2t}

(δ

1

Y

_u⁰

+ λ

1

Y

_u¹

) D

_v−t²

(Y

_t⁰

, Y

_t²

) + e

^{−δ2A0u−λ2A2u−δ1A0t−λ1A1t}

(δ

₂

Y

_u⁰

+ λ

₂

Y

_u²

) D

¹_v−t

(Y

_t⁰

, Y

_t¹

)

du dv for 0 ≤ u ≤ t < v

hold, where D

¹_v−t

(y

₀

, y

₁

) and D

²_v−t

(y

₀

, y

₂

), are given as in (5.15) below. Finally, taking into account the representation in (3.3), according to the tower property for conditional expectations, we obtain that the equalities

P ( κ

1

∈ du, κ

2

∈ dv | F

_t

) = E E

P ( κ

1

∈ du, κ

2

∈ dv | F

_v

)

F

_u

F

_t

(3.4)

= E

e

^{−(δ1+δ2)A0u−λ1A1u−λ2A2u}

(δ

₁

Y

_u⁰

+ λ

₁

Y

_u¹

) D

²_v−u

(Y

_u⁰

, Y

_u²

) + (δ

₂

Y

_u⁰

+ λ

₂

Y

_u²

) D

_v−u¹

(Y

_u⁰

, Y

_u¹

) F

_t

dudv

= e

^{−(δ1+δ2)A0t−λ1A1t−λ2A2t}

E

e

^{−(δ1+δ2)Yt0(A0u−A0t)/Yt0−λ1Yt1(A1u−A1t)/Yt1−λ2Yt2(A2u−A2t)/Yt2}

× δ

₁

Y

_t⁰

(Y

_u⁰

/Y

_t⁰

) + λ

₁

Y

_t¹

(Y

_u¹

/Y

_t¹

)

D

_v−u²

(Y

_t⁰

(Y

_u⁰

/Y

_t⁰

), Y

_t²

(Y

_u²

/Y

_t²

)) + δ

₂

Y

_t⁰

(Y

_u⁰

/Y

_t⁰

) + λ

₂

Y

_t²

(Y

_u²

/Y

_t²

)

D

_v−u¹

(Y

_t⁰

(Y

_u⁰

/Y

_t⁰

), Y

_t¹

(Y

_u¹

/Y

_t¹

)) F

_t

dudv

= e

^{−(δ1+δ2)A0t−λ1A1t−λ2A2t}

D

²_u−t,v−u

(Y

_t⁰

, Y

_t²

, Y

_t¹

) + D

¹_u−t,v−u

(Y

_t⁰

, Y

_t¹

, Y

_t²

) dudv for 0 ≤ t ≤ u < v

are satisfied, where D

¹_u−t,v−u

(y

₀

, y

₁

, y

₂

) and D

²_u−t,v−u

(y

₀

, y

₂

, y

₁

), are given as in (5.23) below.

3.2 Conditional distributions of κ ^j , j = 1, 2, under (H ^k _t ) t≥0 , k = 1, 2, and (H _t ) _t≥0

Let finally compute the conditional distributions P ( κ

j

> u | H

^k_t

) of the successive default times κ

j

, j = 1, 2, given the filtration (H

^k_t

)

t≥0

, k = 1, 2, for all t, u ≥ 0. In this case, we apply the first part of the key lemma in (2.8) for the filtrations (H

¹_t

)

_t≥0

and (F

_t

)

_t≥0

, where H

¹_t

coincides with F

_t

∨ σ( κ

1

) on the event { κ

1

≤ t} and with F

_t

on { κ

1

> t}, for all t ≥ 0, to get

P ( κ

¹

> u | H

¹_t

) = I(u < κ

¹

≤ t) + I( κ

¹

> t) P ( κ

¹

> u ∨ t | F

t

)

P ( κ

1

> t | F

_t

) for t, u ≥ 0 (3.5) where the conditional probability P ( κ

1

> u ∨ t | F

_t

) is computed as in (3.1) above, while

P ( κ

2

> v | H

¹_t

) = I( κ

1

≤ t) P ( κ

2

> v | F

_t

∨ σ( κ

1

)) + I( κ

1

> t) P ( κ

1

> t, κ

2

> v | F

_t

)

P ( κ

1

> t | F

_t

) (3.6)

for t, v ≥ 0

where the conditional probability P ( κ

1

> t, κ

2

> v | F

_t

) is computed as in (3.1) above and, by means of the arguments applied for derivation of equalities in (2.14)-(2.19), we have

P ( κ

2

> v | F

_t

∨ σ( κ

1

))[= P ( κ

2

> v ∨ κ

1

| F

_t

∨ σ( κ

1

))]] (3.7)

= P (τ

₁

> v, τ

₁

> τ

₂

| F

_t

∨ σ( κ

1

)) + P (τ

₂

> v, τ

₂

> τ

₁

| F

_t

∨ σ( κ

1

))

= P (τ

1

> u ∨ v, τ

2

∈ du | F

t

) + P (τ

2

> u ∨ v, τ

1

∈ du | F

t

) P (τ

₁

> u, τ

₂

∈ du | F

_t

) + P (τ

₂

> u, τ

₁

∈ du | F

_t

)

u=κ1

for t, v ≥ 0

where the conditional densities P (τ

_i

∈ du, τ

3−i

∈ dv | F

_t

), for every i = 1, 2, are given in the expressions of (5.17), (5.19), (5.22) below.

Now, we apply the first part of the key lemma in (2.8) for the filtrations (H

²_t

)

_t≥0

and (F

_t

)

_t≥0

, where H

²_t

coincides with F

_t

∨ σ( κ

2

) on the event { κ

2

≤ t} and with F

_t

on { κ

2

> t}, for all t ≥ 0, to get

P ( κ

¹

> u | H

²_t

) = I( κ

²

≤ t) P ( κ

¹

> u | F

t

∨ σ( κ

²

)) + I( κ

²

> t) P ( κ

¹

> u, κ

²

> t | F

_t

)

P ( κ

2

> t | F

_t

) (3.8) for t, u ≥ 0

where the conditional probability P ( κ

1

> u, κ

2

> t | F

_t

) is computed as in (3.1) above and, by means of the arguments applied for derivation of equalities in (2.20)-(2.25), we have

P ( κ

1

> u | F

_t

∨ σ( κ

2

))[= P ( κ

2

> κ

1

> u | F

_t

∨ σ( κ

2

))]] (3.9)

= P (τ

₁

> u, τ

₁

< τ

₂

| F

_t

∨ σ( κ

2

)) + P (τ

₂

> u, τ

₂

< τ

₁

| F

_t

∨ σ( κ

2

))

= P (u ∧ v < τ

₁

≤ v, τ

₂

∈ dv | F

_t

) + P (u ∧ v < τ

₂

≤ v, τ

₁

∈ dv | F

_t

) P (τ

1

≤ v, τ

2

∈ dv | F

t

) + P (τ

2

≤ v, τ

1

∈ dv | F

t

)

v=κ2

for t, u ≥ 0

and the conditional densities P (τ

_i

∈ du, τ

3−i

∈ dv | F

_t

), for every i = 1, 2, are given in the expressions of (5.17), (5.19), (5.22) below, while

P ( κ

2

> v | H

_t²

) = I(v < κ

2

≤ t) + I( κ

2

> t) P ( κ

2

> v ∨ t | F

_t

)

P ( κ

²

> t | F

_t

) for t, v ≥ 0 (3.10) where the conditional probability P ( κ

2

> v ∨ t | F

_t

) is computed as in (3.1) above.

Finally, we apply the first part of the key lemma in (2.8) for the filtrations (H

t

)

t≥0

and (F

_t

)

t≥0

, where H

_t

coincides with F

_t

∨σ( κ

1

)∨σ( κ

2

) on the event { κ

1

< κ

2

≤ t}, with F

_t

∨σ( κ

1

) on { κ

1

≤ t < κ

2

} , and with F

_t

on { κ

2

> κ

1

> t}, for all t ≥ 0, to get

P ( κ

1

> u, κ

2

> v | H

_t

) = I(u < κ

1

< κ

2

≤ t, κ

2

> v) (3.11) + I(u < κ

1

≤ t < κ

2

) P ( κ

2

> v ∨ t | F

_t

∨ σ( κ

1

))

P ( κ

2

> t | F

_t

∨ σ( κ

1

)) + I( κ

2

> κ

1

> t) P ( κ

1

> u ∨ t, κ

2

> v ∨ t | F

_t

)

P ( κ

²

> κ

¹

> t | F

_t

) for t, u, v ≥ 0

where the conditional probabilities P ( κ

1

> u ∨ t, κ

2

> v ∨ t | F

_t

) and P ( κ

2

> v ∨ t | F

_t

∨ σ( κ

1

))

are computed as in the expressions of (3.1) and (3.7) above, respectively.

4 The prices of first and second-to-default claims (Main results)

In this section, we derive explicit expressions for the prices of first and second-to-default options in the model defined above with some (non-negative measurable) deterministic recovery payoff functions R

_t

(x

₁

, x

₂

), for all 0 ≤ t ≤ T . In order to simplify the notations, without loss of generality, we further assume that the payoffs are already discounted by the dynamics of the bank account, that is equivalent to letting the interest rate r equal to zero. We compute the prices for the option holders in various particular cases of available information contained in the filtrations (H

^k_t

)

_t≥0

, or (H

_t

)

_t≥0

, or (H

^k_t

∨ σ( κ

^3−k

))

_t≥0

defined above, for every k = 1, 2.

In those cases, the option holders can observe only the default time κ

k

, or observe the both default times κ

k

, k = 1, 2, or observe the default time κ

k

but know the default time κ

^3−k

, for every k = 1, 2, from the beginning of observations, respectively.

Recall that the conditional probabilities P ( κ

1

> u ∨ t, κ

2

> v ∨ t | F

_t

), for all t, u, v ≥ 0, were computed in (3.1) above.

4.1 The case of filtrations (H ^k _t ) _t≥0 , k = 1, 2

Let us begin by computing the price P

^j,k

= (P

_t^j,k

)

t≥0

for the holder of a first- and second-to- default option in the model with the filtration (H

^k_t

)

t≥0

given by

P

_t^j,k

= E

R

_κj

(X

_κ¹_j

, X

_κ²_j

) I (t < κ

^j

≤ T ) H

_t^k

(4.1) for all 0 ≤ t ≤ T ∧ κ

^j

and every j, k = 1, 2.

In order to compute closed-form expressions for P

^1,k

in (4.1), we provide the decomposition P

_t^1,k

= E

R

_κ1

(X

_κ¹₁

, X

_κ²₁

) I(t < κ

1

≤ T ) H

^k_t

(4.2)

= E

R

_τi

(X

_τ¹

i

, X

_τ²

i

) I(τ

_i

< τ

3−i

, t < τ

_i

≤ T ) H

^k_t

+ E

R

τ3−i

(X

_τ¹_3−i

, X

_τ²_3−i

) I(τ

3−i

< τ

i

, t < τ

3−i

≤ T ) H

^k_t

= E

R

_τi

(X

_τ¹_i

, X

_τ²_i

) I(t < τ

_i

< τ

3−i

∧ T ) H

_t^k

+ E

R

_τ3−i

(X

_τ¹_3−i

, X

_τ²_3−i

) I(t < τ

3−i

< τ

_i

∧ T ) H

_t^k

for all 0 ≤ t ≤ T and every i, k = 1, 2. Then, we can apply the second part of the key lemma in (2.9) for the filtrations (H

_t^k

)

t≥0

and (F

t

)

t≥0

, where H

^k_t

coincides with F

t

on { κ

^k

> t}, for all t ≥ 0, and use Fubini’s theorem for interchanging the order of conditional expectation and integration along with the tower property for conditional expectations, and using the fact that, on the set {t > τ

k

}, the quantity I(t < τ

i

< τ

3−i

∧ T ) is equal to zero, to get the expression

E

R

_τi

(X

_τ¹

i

, X

_τ²

i

) I(t < τ

_i

< τ

_3−i

∧ T ) H

_t^k

(4.3)

= I( κ

^k

> t) E [R

_τi

(X

_τ¹

i

, X

_τ²

i

) I(t < τ

_i

< τ

3−i

∧ T ) | F

_t

] P ( κ

k

> t | F

_t

)

= I( κ

k

> t) E Z

T

t

Z

∞

t

I (u < v) R

_u

(X

_u¹

, X

_u²

) P (τ

_i

∈ du, τ

3−i

∈ dv | F

_v

) P ( κ

k

> t | F

_t

)

F

_t

= I( κ

k

> t) Z

T

t

Z

∞

t

I(u < v) E[R

_u

(X

_u¹

, X

_u²

) P (τ

_i

∈ du, τ

3−i

∈ dv | F

_v

) | F

_t

]

P ( κ

k

> t | F

_t

)

for all 0 ≤ t ≤ T and every i, k = 1, 2. Thus, taking into account the expressions in (5.19), according to the tower property for conditional expectations, we obtain that

E

R

_u

(X

_u¹

, X

_u²

) P (τ

_i

∈ du, τ

3−i

∈ dv | F

_u

) F

_t

(4.4)

= E

R

_u

(X

_u¹

, X

_u²

) e

^{−(δi+δ3−i)A0u−λiAiu−λ3−iA3−iu}

(δ

_i

Y

_u⁰

+ λ

_i

Y

_uⁱ

) D

_v−u³⁻ⁱ

(Y

_u⁰

, Y

_u³⁻ⁱ

) F

_t

dudv

= e

^{−(δi+δ3−i)A0t−λiAit−λ3−iA3−it}

× E

R

_u

X

_t¹

(Y

_uⁱ

/Y

_tⁱ

)

^αi

(Z

_u⁰

/Z

_t⁰

)

^ζi

(Z

_uⁱ

/Z

_tⁱ

), X

_t²

(Y

_u³⁻ⁱ

/Y

_t³⁻ⁱ

)

^α3−i

(Z

_u⁰

/Z

_t⁰

)

^ζ3−i

(Z

_u³⁻ⁱ

/Z

_t³⁻ⁱ

)

× e

^{−(δi+δ3−i)Yt0(A0u−A0t)/Yt0−λiYti(Aiu−Ait)/Yti−λ3−iYt3−i(A3−iu −A3−it )/Yt3−i}

× δ

_i

Y

_t⁰

(Y

_u⁰

/Y

_t⁰

) + λ

_i

Y

_tⁱ

(Y

_uⁱ

/Y

_tⁱ

)

D

³⁻ⁱ_v−u

(Y

_t⁰

(Y

_u⁰

/Y

_t⁰

), Y

_t³⁻ⁱ

(Y

_u³⁻ⁱ

/Y

_t³⁻ⁱ

)) F

_t

dudv

= e

^{−(δi+δ3−i)A0t−λiAit−λ3−iA3−it}

Q

^1,3−i_{t,u−t,v−u}

(X

_t¹

, X

_t²

, Y

_t⁰

, Y

_t³⁻ⁱ

, Y

_tⁱ

) dudv

holds, for each 0 ≤ t < u < v ≤ T , for every i = 1, 2. Here, by virtue of the Markov property of the processes (Y

^l

, A

^l

) and Z

^l

, l = 0, 1, 2, and the fact that the random variables Y

_u^l

/Y

_t^l

and Z

_u^l

/Z

_t^l

have the same laws as Y

_u−t^l

and Z

_u−t^l

, l = 0, 1, 2, for each 0 ≤ t < u, respectively, we have

Q

^1,3−i_{t,u−t,v−u}

(x

₁

, x

₂

, y

₀

, y

3−i

, y

_i

) (4.5)

= E

R

u

x

1

(Y

_u−tⁱ

)

^αi

(Z

_u−t⁰

)

^ζi

Z

_u−tⁱ

, x

2

(Y

_u−t³⁻ⁱ

)

^α3−i

(Z

_u−t⁰

)

^ζ3−i

Z

_u−t³⁻ⁱ

× e

^{−(δi+δ3−i)y0A0u−t−λiyiAiu−t−λ3−iy3−iA3−iu−t}

(δ

_i

y

₀

Y

_u−t⁰

+ λ

_i

y

_i

Y

_u−tⁱ

) D

³⁻ⁱ_v−u

(y

₀

Y

_u−t⁰

, y

3−i

Y

_u−t³⁻ⁱ

)

= Z

∞

0

Z

∞

0

Z

∞

0

Z

∞

0

Z

∞

0

Z

∞

0

Z

∞

0

Z

∞

0

Z

∞

0

R

_u

x

₁

(y

_i⁰

)

^αi

(z

₀⁰

)

^ζi

z

⁰_i

, x

₂

(y

_3−i⁰

)

^α3−i

(z

₀⁰

)

^ζ3−i

z

_3−i⁰

× e

^{−(δi+δ3−i)y0a0−λiyiai−λ3−iy3−ia3−i}

(δ

i

y

0

y

⁰₀

+ λ

i

y

i

y

⁰_i

) D

_v−u³⁻ⁱ

(y

0

y

₀⁰

, y

3−i

y

_3−i⁰

) g

⁰_u−t

(y

⁰₀

, a

0

)

× g

ⁱ_u−t

(y

⁰_i

, a

_i

) g

_u−t³⁻ⁱ

(y

_i⁰

, a

_i

) h

⁰_u−t

(z

₀⁰

) h

ⁱ_u−t

(z

_i⁰

) h

³⁻ⁱ_u−t

(z

_3−i⁰

) dy

⁰₀

da

₀

dy

_i⁰

da

_i

dy

_3−i⁰

da

_3−i

dz

₀⁰

dz

_i⁰

dz

_3−i⁰

for all 0 ≤ t < u < v ≤ T and every i = 1, 2, where the functions g

^l

, for l = 0, 1, 2, stand for g defined in (5.7) and the functions h

^l

, for l = 0, 1, 2, stand for h defined in (5.9). There, β and γ stand for β

_l

and γ

_l

, for l = 0, 1, 2, defined in (2.3), while η and θ stand for η

_l

and θ

_l

, for l = 0, 1, 2, defined in (2.4).

In order to compute closed-form expressions for P

^2,k

in (4.1), we provide the decomposition P

_t^2,k

= E

R

_κ2

(X

¹

κ2

, X

²

κ2

) I(t < κ

2

≤ T ) H

^k_t

(4.6)

= E

R

_τi

(X

_τ¹

i

, X

_τ²

i

) I(τ

_i

> τ

_3−i

, t < τ

_i

≤ T ) H

^k_t

+ E

R

_τ3−i

(X

_τ¹_3−i

, X

_τ²_3−i

) I(τ

3−i

> τ

_i

, t < τ

3−i

≤ T ) H

^k_t

= E

R

_τi

(X

_τ¹_i

, X

_τ²_i

) I(t ∨ τ

3−i

< τ

_i

≤ T ) H

^k_t

+ E

R

_τ3−i

(X

_τ¹_3−i

, X

_τ²_3−i

) I(t ∨ τ

_i

< τ

3−i

≤ T ) H

^k_t

for all 0 ≤ t ≤ T and every i, k = 1, 2. Then, we can apply the second part of the key lemma

in (2.9) for the filtrations (H

¹_t

)

t≥0

and (F

_t

)

t≥0

, where H

¹_t

coincides with F

_t

∨ σ( κ

1

) on the

event { κ

1

≤ t} and with F

_t

on { κ

1

> t}, for all t ≥ 0, as well as the arguments applied

for derivation of equalities in (2.14)-(2.19), and use Fubini’s theorem for interchanging the

order of conditional expectation and integration along with the tower property for conditional