Projections in enlargements of filtrations under Jacod's hypothesis and examples

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Projections in enlargements of filtrations under Jacod’s hypothesis and examples

Pavel Gapeev, Monique Jeanblanc, Dongli Wu

To cite this version:

Pavel Gapeev, Monique Jeanblanc, Dongli Wu. Projections in enlargements of filtrations under Jacod’s hypothesis and examples. 2019. �hal-02120343�

Projections in enlargements of filtrations under Jacod’s hypothesis and examples ^∗

Pavel V. Gapeev

Monique Jeanblanc

Dongli Wu

In this paper, we consider two kinds of enlargements of a Brownian filtration F: the initial enlargement with a random time τ, denoted by F^(τ), and the progressive enlargement with τ, denoted by G. We assume Jacod’s equivalence hypothesis, that is, the existence of a positiveF-conditional density for τ. Then, starting with the predictable representation of an F^(τ)-martingale Y(τ) in terms of a standard F^(τ)-Brownian motion, we consider its projection on G, denoted by Y^G, and on F, denoted by y. We show how to obtain the coefficients which appear in the predictable representation property for Y^G (and y) in terms of Y(τ) and its predictable representation. In the last part, we give examples of conditional densities.

1 Introduction

In this paper, we consider two kinds of enlargement of a Brownian filtration F generated by a Brownian motion W: the initial enlargement with a random time (a positive random variable) τ, denoted by F^(τ), and the progressive enlargement with τ, denoted by G. We assume Jacod’s

∗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 2000: Primary 91B70, 60J60, 62M20. Secondary 91B28, 60J65, 60J25.

Key words and phrases: Brownian motion, conditional probability density.

Date: May 5, 2019

equivalence hypothesis, that is, the existence of a positive conditional density for τ (see (2.1) below in Section 2 for a precise definition). Processes considered in the filtration F^(τ) will be denoted as Y(τ) since, under Jacod’s condition, any F_t^(τ)-measurable random variable is of the form Y_t(ω, τ(ω)) for some F_t⊗ B(R⁺) measurable function (ω, u) → Y_t(ω, u) (see [1;

Proposition 4.22]), processes considered in G will be indicated with the superscript G, as Y^G. Processes denoted without these symbols are F-adapted as Y or y(u).

In Section 2, we recall basic facts on enlargements of filtrations. In Section 3, we start with an F^(τ)-martingale Y(τ), which admits a decomposition

Yt(τ) =Y0(τ) + Z t

0

ys(τ)dWs(τ) (1.1)

for all t ≥ 0, where W(τ) is an F^(τ)-Brownian motion and, for any u ≥ 0 the process y(u) is F-predictable and for any t the map u → y_t(u) is Borelian, and we study its G-optional projection Y^G and its F-optional projection y. From the predictable representation properties [5], we obtain the existence of two G-predictable processes β^G and γ^G such that

Y_t^G =E Y_t(τ)

G_t

=Y₀^G+ Z t

0

β_s^GdW_s^G+ Z t

0

γ_s^GdM_s^G (1.2)

for all t ≥ 0, where M^G is the compensated G-martingale of the default process 11{τ≤t}, and the existence of an F-predictable process σ such that

Y_t =E[Y_t(τ)|F_t] =y₀+ Z t

0

σ_sdW_s.

We show how to compute β^G, γ^G and σ in terms of Y(τ) and y(τ). We apply these compu- tations to compare equivalent martingale measures for a financial market with price process following an F-geometric Brownian motion considered in the two enlarged filtrations in Section 4. In the last section, we give a family of example of conditional densities.

In the whole paper, ”positive” means ”strictly positive”.

2 Enlargement of filtrations

We recall that, if H ⊂ K, the H-optional projection of a K-martingale µ is the H-optional process ν such that E[µ_ϑ11{ϑ<∞}|H_ϑ) = ν_ϑ11{ϑ<∞} for any H-stopping time ϑ. This optional projection satisfies E(µ_t|H_t) = ν_t. By abuse of language, we shall call E(µ_t|H_t) the H-optional projection of µ.

We consider a probability space (Ω,G,P) endowed with a standard Brownian motion W = (Wt)t≥0 and a positive random variable τ with the positive density g on R⁺ under P. We denote by F = (F_t)_t≥0 the natural right-continuous and completed filtration of W and G is the progressive enlargement of F with τ. We assume Jacod’s equivalence hypothesis (see [1;

Chapter IV; Section 5]), that is, the existence of a positive function (ω, t, u) → p_t(ω, u) such that

(i): the map u7→p_t(u;ω) is Borelian, (dt×dP) a.s.

(ii): for each u≥0, the process p(u) = (p_t(u))t≥0 is a continuous F-martingale, (iii): for each t ≥0, for every bounded Borel function f, we have

E f(τ)

F_t

= Z ∞

0

f(u)p_t(u)g(u)du . (2.1)

Note that R∞

0 p_t(u)g(u)du = 1, for all t ≥ 0, and p₀(u) = 1, for each u ≥ 0. Moreover, it follows from the predictable representation theorem in a Brownian filtration that, for each u≥0, there exists anF-predictable process ϕ(u) = (ϕ_t(u))t≥0 such that the positive martingale p(u) admits the representation

p_t(u) =p₀(u) + Z t

0

p_s(u)ϕ_s(u)dW_s (2.2)

so that it takes the form of the Dol´eans-Dade stochastic exponential pt(u) = p0(u) exp

Z t

0

ϕs(u)dWs−1 2

Z t

0

ϕ²_s(u)ds

(2.3) for all t≥0. It can be shown that u7→ϕt(u) is a Borel function on [0,∞), for all t≥0.

We introduce the supermartingale G defined as G_t :=P(τ > t| F_t) which can be written in terms of the conditional density as

G_t = Z ∞

t

p_t(u)g(u)du , t≥0. (2.4) Note that G is a positive continuous F-supermartingale called the Az´ema supermartingale of τ which admits a Doob-Meyer decomposition G_t = N_t− A_t, where N = (N_t)t≥0 is an F-martingale under P and A = (A_t)t≥0 is a nondecreasing F-predictable process. Then, by means of Jacod’s equivalence hypothesis, we have

A_t= Z t

0

p_u(u)g(u)du and N_t= 1− Z t

0

p_t(u)−p_u(u)

g(u)du (2.5)

for all t ≥0 (see [7; Subsection 4.2.1]). Moreover, by means of Itˆo-Wentzell’s lemma (see, e.g.

Kunita [19], Wentzell [22], or [14; Theorem 1.5.3.2]), we have dNt=−

Z t

0

dtpt(u)g(u)du=− Z t

0

ϕt(u)pt(u)g(u)du

dWt (2.6)

where d_t means that one makes use of the stochastic differential, and the associated predictable covariation processes have the form

dhW, Ni_t=−Z t 0

ϕ_t(u)p_t(u)g(u)du

dt and dhW, p(u)i_t=p_t(u)ϕ_t(u)dt (2.7) for all t, u≥0.

It follows from Jacod’s theorem for initial enlargements of filtrations ([13; Corollaire 1.11]

or [1; Chapter IV, Proposition 4.40]) that the process W(τ) = (W_t(τ))t≥0 defined by W_t(τ) :=W_t−

Z t

0

dhp(u), Wi_s|_u=τ

p_s(τ) =W_t− Z t

0

ϕ_s(τ)ds (2.8)

for all t ≥ 0, is a P-Brownian motion with respect to the initially enlarged filtration F^(τ) defined by F^(τ) = (Ft ∨ σ(τ))t≥0. Note that, according to the results of Fontana [9], the filtration F^(τ) is right-continuous.

It follows from the results of Grorud and Pontier [12] and also Amendinger [2], that the process (1/p_t(τ))t≥0 forms an F^(τ)-martingale under P, and thus, since p₀(τ) = 1 holds, we have E[1/p_t(τ)] = 1, for all t ≥0. Therefore, we can define a probability measure P^∗ on F_t^(τ) by

dP^∗ dP

_F(τ)

t

= 1

p_t(τ) = exp

− Z t

0

ϕ_s(τ)dW_s(τ)−1 2

Z t

0

ϕ²_s(τ)ds

(2.9) for all t≥0. It is proved in [12] (see also [1; Chapter IV, Proposition 4.37]) that the probability measure P^∗ defined in (2.9) coincides with P on F and on σ(τ), and the random time τ is independent of F under P^∗. This fact particularly implies that P^∗(τ > u| F_t) =P^∗(τ > u) = P(τ > u) holds, for all t, u≥ 0, as well as that the process W is a (P,F) standard Brownian motion with respect to F under P^∗. Note that, by Girsanov’s theorem, following the arguments of Callegaro et al. [5], we can recover that the process W(τ) from (2.8) is a (P,F^(τ))-Brownian motion.

Let us now introduce the progressively enlarged filtration G= (Gt)t≥0 which is the smallest right-continuous filtration containing F and making τ a stopping time.

Note that, according to the hypothesis that the positive random variable τ has a positive density with support R⁺, the σ-algebra G0 is trivial. It is known (see Jeulin [17; Chapter III]

or Callegaro et al. [5; Chapter I, Section 2]) that any G-predictable process Z^G can be written as

Z_t^G =Z_t⁰11_{t≤τ}+Z_t¹(τ) 11_{{τ <t}} (2.10) for all t ≥ 0, where the process Z⁰ is F-predictable and, for any u ≥ 0, the process Z¹(u) is F-predictable. In that case, using the key lemma for the computation of Z⁰ (see, e.g. [1;

Chapter II, Lemma 2.9]), we get

Z_t¹(u) =Z_t(u), fort≥u, and Z_t⁰ = 1 G_t

Z ∞

t

Z_t(u)p_t(u)g(u)du . (2.11) We shall say that Z⁰ is the F-predictable reduction of Z^G.

The decomposition (2.10), proved for any progressive enlargement in Jeulin does not extend in general to optional processes (see Barlow [3] for a counter example). However, under Jacod’s equivalence hypothesis, any G-optional process U^G can be written as

U_t^G =U_t⁰11_{t<τ}+U_t¹(τ) 11_{τ≤t} (2.12) for all t ≥ 0, where U⁰ is F-optional and, for each u ≥ 0, U¹(u) is F-optional (see Song [21]). In particular, we shall use that result for the G-optional projection of an integrable F^(τ)-optional process U(τ), i.e., for

U_t^G =E U_t(τ)

G_t

(2.13) In that case, similarly to (2.11), we have

U_t¹(u) =Ut(u), fort≥u, and U_t⁰ = 1 G_t

Z ∞

t

Ut(u)pt(u)g(u)du (2.14) for all t ≥ 0. The process U⁰ is called the optional reduction of U. Note that, working in a Brownian filtration under Jacod’s equivalence hypothesis, U⁰ and U¹(u) being continuous are also F-predictable.

Then, it follows from the results of Jeanblanc and Le Cam [16] that the process W^G = (W_t^G)t≥0 defined by

W_t^G :=Wt− Z t∧τ

0

dhW, Nis

G_s − Z t

t∧τ

dhW, p(u)is|u=τ

p_s(τ)

=W_t+ Z t∧τ

0

1 G_s

Z s

0

ϕ_s(u)p_s(u)g(u)du

ds− Z t

t∧τ

ϕ_s(τ)ds (2.15)

=W_t+ Z t

0

α^G_s ds (2.16)

is a (P,G)-standard Brownian motion, where α^G is the G-predictable process with the decom- position

α^G_t = 11_{t≤τ} 1 G_t

Z t

0

ϕ_t(u)p_t(u)g(u)du11_{{τ <t}}ϕ_t(u) := 11_{t≤τ}α⁰_t + 11_{{τ <t}}α¹_t(τ) (2.17) for all t≥0. Observe from (2.8) and (2.15) that

W_t(τ) = W_t^G− Z t∧τ

0

α⁰_s+ϕ_s(τ)

ds (2.18)

for all t≥0.

We define the G-martingale

M_t^G := 11_{τ≤t}− Z t

0

λ^G_s ds= 11_{τ≤t}− Z t∧τ

0

λ⁰_sds (2.19)

for all t ≥0, where λ^G_s = 11_{s≤τ}λ⁰_s and λ⁰_s =p_s(s)g(s)/G_s (see, e.g. [5; Subsection 1.2] or [1;

Chapter II]).

We recall that the process W(τ) enjoys the F^τ-predictable representation property and that the pair (W^G, M^G) enjoys the G-predictable representation property (see, e.g. [5; Proposition 4.3]):

Any F^(τ)-martingale Y(τ) has the form

Y_t(τ) =Y₀(τ) + Z t

0

y_s(τ)dW_s(τ) (2.20)

for all t≥0, whereY₀ is a (deterministic) function andy(u), u≥0, is a family ofF-predictable processes. In particular, any (P,F^(τ))-martingale is continuous, and if Y(τ) is square integrable on [0, T], then E[RT

0 (y_s(τ))²ds]<∞.

Any G-martingale admits the representation Y_t^G =Y₀^G+

Z t

0

β_s^GdW_s^G+ Z t

0

γ_s^GdM_s^G (2.21)

where β^G and γ^G are G-predictable processes.

3 Martingales and projections

3.1 Projections of F

martingales on G

Proposition 3.1 Let Y(τ) be an F^(τ)-martingale with representation (2.20). Its G-optional projection Y^G admits the following representation

Y_t^G =E Y_t(τ)

G_t

=Y₀^G+ Z t

0

β_s^GdW_s^G+ Z t

0

γ_s^GdM_s^G (3.1)

where Y₀^G = E[Y₀(τ)], and the G-predictable processes β^G and γ^G admit, from (2.10), the decomposition

β_t^G = β_t⁰11_{t≤τ}+β_t¹(τ) 11_{{τ <t}} (3.2) γ_t^G = γ_t⁰11{t≤τ} +γ_t¹(τ) 11{τ <t}. (3.3) Then,

β_t⁰ = 1 G_t

Z ∞

t

y_t(u) + (α⁰_t +ϕ_t(u))Y_t(u)

p_t(u)g(u)du, β_t¹(u) =y_t(u), fort ≥u ,(3.4)

γ_t⁰ = Y_t(t)−Y_t⁰, (3.5)

for all t ≥ 0, and Y⁰ is given by (2.11). Note that for any choice of γ¹, the property Rt

0 γ_s^GdM_s^G =Rt

0 γ_s⁰dM_s^G holds.

Proof: In a first step, we assume that Y(τ) (hence Y^G), are square integrable on [0, T] We start to determine β^G (which is square integrable). For this purpose, we observe that for any bounded G-predictable process n^G, using the tower property and the fact that W^G is a G-martingale orthogonal to M^G, we have

E

Y_t(τ) Z t

0

n^G_s dW_s^G

=E

Y_t^G Z t

0

n^G_sdW_s^G

(3.6)

=E Z t

0

β_s^GdW_s^G Z t

0

n^G_s dW_s^G

=E hZ t

0

β_s^Gn^G_s ds i

for all t≥0. Recall that, from (2.18), we have W_t^G =W_t(τ) +

Z t∧τ

0

α⁰_s+ϕ_s(τ)

ds (3.7)

for all t≥0. We introduce the continuous G-martingale V_t^G =

Z t

0

n^G_s dW_s^G = Z t

0

n^G_s dW_s(τ) + Z t

0

α⁰_s+ϕ_s(τ)

11{s≤τ}n^G_s ds (3.8) for all t≥0. By means of the integration by parts, we have

E

Y_t(τ)V_t^G

=E

+ Z t

0

Y_s(τ)dV_s^G+hY(τ), V^Gi^F_t^(τ) Z t

0

V_s^GdY_s(τ)

(3.9) for all t≥0.

A martingale X is square integrable if sup_t≤TE(X_t²)<∞.

We recall that a local martingale M satisfying E[(hMi_T)^1/2]< ∞ is a martingale. Let us prove that the F^(τ)-local martingale Mt=Rt

0 V_s^GdYs(τ) is a martingale. One has E[(hMi_T)^1/2) = E

Z T

0

(V_s^G)²(y_s(τ))²ds1/2

ds

≤E h

sup

s≤T

|V_s^G|Z T 0

(y_s(τ))²ds1/2i

≤E[sup

s≤T

|V_s^G|²] +E Z T

0

y_s(τ)²ds .

From Burkholder’s inequality {footnote Burkholder’s inequality states that for any p ≥ 1, E[sup_t≤T |Mt|^p)] ≤ CpE[hMiT)^p/2] where Cp is a constant depending only on p. the quan- tity E[sup_s≤T |V_s^G|²] is smaller than C₂E[RT

0 (n^G_s)²ds] which is bounded. Furthermorcdote, E[RT

0 y_s(τ)²ds] is bounded. Hence, R·

0V_s^GdY_s(τ) is a martingale and its expectation is null.

The boundeness of n^G and the square integrability of Y(τ) imply that Rt

0 Y_s(τ)n^G_sdW_s(τ) is a martingale, so that

E Z t

0

V_s^GdY_s(τ) + Z t

0

Y_s(τ)dV_s^G

=E Z t

0

Y_s(τ) α⁰_s+ϕ_s(τ)

11{s≤τ}n^G_s ds

. (3.10) Recall that hY(τ), V^Gi^F_t^(τ) =Rt

0 y_s(τ)n^G_sds, for all t≥0. Then, we have E

Y_t(τ)V_t^G

=E Z t

0

Y_s(τ) α⁰_s+ϕ_s(τ)

11{s≤τ}+y_s(τ) n^G_s ds

(3.11) hence, using the tower property

E hZ t

0

β_s^Gn^G_s ds i

=E hZ t

0

E

ys(τ) +Ys(τ) α⁰_s+ϕs(τ) 11{s≤τ}

Gs

n^G_s ds i

(3.12) for all t≥0 and for any bounded n^G It follows that

β_t^G =E

y_t(τ) +Y_t(τ) (α⁰_t +ϕ_t(τ) 11{t≤τ}

G_t

, (3.13)

and the F-predictable reduction of β^G is β_t⁰ = 1

G_t Z ∞

t

y_t(u) + α⁰_t +ϕ_t(u)

Y_t(u)

p_t(u)g(u)du (3.14) and on {t > τ}, one has, as expected, that β_t^G =E[y_t(τ)| G_t] =y_t(τ) so that β_t¹(u) =y_t(u).

In the second step, we determine γ^G. For this purpose, on the one hand, for any bounded G-predictable process n^G, using the fact that M^G is a G-martingale orthogonal to W^G and

that M^G is flat after τ (M_t^G = Mt∧τ), and that the equality dhM^Gi_t =λ⁰_t11{t≤τ}dt holds, we have

E

Y_t(τ) Z t

0

n^G_s dM_s^G

=E

Y_t(τ) Z t

0

n⁰_sdM_s^G

=E

Y_t^G Z t

0

n⁰_sdM_s^G

(3.15)

=E Z t

0

γ_s⁰dM_s^G Z t

0

n⁰_sdM_s^G

=E Z t

0

γ_s⁰λ⁰_s11_{s<τ}n⁰_sds

=E Z t

0

γ_s⁰λ⁰_sG_sn⁰_sds

for all t ≥0, where n⁰ is the F-predictable reduction of n^G and where the last equality comes from the tower property.

Recall thatM^G is a predictable bounded variation process in the initially enlarged filtration.

Let U_t^G = Rt

0 n⁰_sdM_s^G. By integration by parts, using the fact that Y(τ) is continuous, its bracket (in F^G with the F^(τ)-predictable bounded variation process U^G is null

E[Y_t(τ)U_t^G] =E[ Z t

0

Y_s(τ)n⁰_sdM_s^G+ Z t

0

U_s^GdY_s(τ)]. The same methodology than in the first part forR

V^GdY_s(τ) proves that the processR·

0U_s^GdY_s(τ) is a martingale. Furthermore, setting H_t = 11{t≤τ}, we have

E[ Z t

0

Y_s(τ)n⁰_sdM_s^G] = E[ Z t

0

Y_s(τ)n⁰_sdH_s− Z t

0

Y_s(τ)n⁰_sλ⁰_s11{s<τ}ds

= E[11_{τ≤t}Y_τ(τ)n⁰_τ − Z t

0

E[Y_s(τ)G_s]n⁰_sλ⁰_s11_{s<τ}ds]

= E[ Z t

0

Ys()n⁰_upt(s)g(s)ds− Z t

0

Y_s⁰n⁰_sλ⁰_s11{s<τ}ds

= E[ Z t

0

[Y_s(s)p_s(s)−Y_u⁰λ⁰_uG_u]n⁰_udu]

where in the last equality, we used the fact that p(u) is an F-martingale. To conclude, we have E

Z t

0

γ_s⁰λ⁰_sGsn⁰_sds

=E Z t

0

(Ys(s)ps(s)g(s)−λ⁰_sGsY_s⁰)n⁰_sds

(3.16) and using the fact that λ⁰_sG_s =p_s(s)g(s), one has, for any F adapted bounded process n⁰

E Z t

0

(γ_s⁰−Y_s(s) +Y_s⁰)λ⁰_sG_sn⁰_sds

= 0 (3.17)

for all t ≥ 0, so that the expression in (3.5) holds. The result obtained for square integrable

martingales Y(τ) extends to all martingales.

3.2 Projection of F

-martingales on F

Proposition 3.2 Let Y(τ) be an F^(τ)-martingale of the form dY_t(τ) = y_t(τ)dW_t(τ). Its F- optional projection is Y_t =E[Y_t(τ)| F_t] =Y₀+Rt

0 σ_sdW_s, where the F-predictable process σ is given by

σ_t= Z ∞

0

y_t(u) +Y_t(u)ϕ_t(u)

p_t(u)g(u)du (3.18)

for all t≥0.

Proof: It follows from the predictable representation property in the filtration F that there exists an F-predictable process σ such that

E Y_t(τ)

F_t

=Y₀+ Z t

0

σ_sdW_s (3.19)

holds for all t≥0. On the one hand, for any bounded F-adapted process n, we have E

Y_t(τ)

Z t

0

n_sdW_s

=E

Y_t Z t

0

n_sdW_s

=E Z t

0

σ_sn_sds

(3.20) for all t≥0. On the other hand, using (2.8) and an integration by parts on the left-hand side lead to, assuming that Y(τ) is square integrable

E

Y_t(τ) Z t

0

n_sdW_s

=E

Y_t(τ) Z t

0

n_sdW_s(τ) + Z t

0

n_sϕ_s(τ)ds

(3.21)

=E Z t

0

y_s(τ) +Y_s(τ)ϕ_s(τ) n_sds

=E Z t

0

E

y_s(τ) +Y_s(τ)ϕ_s(τ)| F_s]n_sds

for all t≥0. Hence, we have σ_t=E

y_t(τ) +Y_t(τ)ϕ_t(τ)| F_t

= Z ∞

0

y_t(u) +Y_t(u)ϕ_t(u)

p_t(u)g(u)du (3.22)

for all t≥0.

3.3 Projection of G -martingales on F

Proposition 3.3 Consider the G-martingale Y^G = (Y_t^G)t≥0 defined by Y_t^G =Y₀^G+

Z t

0

β_s^GdW_s^G+ Z t

0

γ_s^GdM_s^G =Y_t⁰11{t<τ} +Y_t¹(τ)11{τ≤t}. (3.23)

Then, its F-optional projection is Y_t = E[Y_t^G| F_t] = Y₀ +Rt

0 η_sdW_s, where the F-predictable process η satisfies

η_t =E

β_t^G−Y_t^Gα^G_t | F_t

= (β_t⁰−Y_t⁰α⁰_t)G_t+ Z t

0

β_t¹(u)−Y_t¹(u)α¹_t(u)

p_t(u)g(u)du (3.24) for all t≥0.

Proof: We proceed as before and consider, for a bounded F-adapted process n, the quantity E

h Y_t^GRt

0 n_sdW_si

, for all t ≥0. On the one hand, using the same procedure as in the analysis of (3.12), we get

E

Y_t^G Z t

0

nsdWs

=E Z t

0

ηsnsds

(3.25) for all t ≥0. On the other hand, setting Ut =Rt

0nsdWs, by means of integration by parts and (2.15), we obtain, using the same methodology as before

E

Y_t^G Z t

0

n_sdW_s

=E

Y_t^GU_t

=E Z t

0

U_sdY_s^G+ Z t

0

Y_s^Gn_sdW_s+ Z t

0

β_s^Gn_sds

(3.26)

=E Z t

0

β_s^G−Y_s^Gα^G_s n_sds

=E Z t

0

E

β_s^G−Y_s^Gα^G_s | F_s n_sds

for all t≥0.

Remark 3.4 Since E[Y_t(τ)|F_t] =E[E[Y_t(τ)|G_t]|F_t] we have σ =η where these quantities are defined in (3.18) and (3.24). Indeed, from (3.13),

E[β_t^G−α_t^GY_t^G|F_t] = E[y_t(τ) +Y_t(τ)(α⁰_t +ϕ_t(τ))11_{t≤τ}−α⁰_tY_t⁰11_{t≤τ}−α¹(τ)Y_t¹(τ)11_{{τ <t}}|F_t]

= E[y_t(τ) +Y_t(τ)ϕ_t(τ)|F_t]. (3.27)

3.4 Change of probability measures

3.4.1 F^(τ) versus G

Note that, from PRP, and positive F^(τ)-martingale has the form L_t(τ) =L₀(τ)E(ζ(τ)·W(τ))_t,=L₀(τ) exp

Z t

0

ζ_s(τ)dW_s(τ)− 1 2

Z t

0

(ζ_s(τ))²ds

, t ≥0 (3.28) where L₀ is a positive function and ζ(τ) is F^(τ)-predictable, or in a closed form

L_t(τ) =L₀(τ) expZ t

ζ_s(τ)dW_s(τ)− 1 2

Z t

(ζ_s(τ))²ds

and any positive G-martingale has the form

CE(µ^G·W^G)tE(ψ^G·M^G)t, (3.29) where C is a positive constant and µ^G and ψ^G are G predictable and ψ^G > −1. We recall that

E(ψ^G·M^G)_t= exp

Z t∧τ

0

ψ^G_sλ⁰_sds

(1 +ψ_τ^G)^Ht.

Proposition 3.5 Let L(τ) = (L_t(τ))t≥0 be of the form (3.28). Then, its G-optional projection L^G satisfies

L^G_t =E

L_t(τ)| G_t

=E L₀(τ)

+ Z t

0

L^G_s−µ^G_s dW_s^G+ Z t

0

L^G_s−ψ_s^GdM_s^G (3.30) where the G-predictable processes µ^G and ψ^G are given by

µ^G_t = 11{t≤τ}

1 L⁰_tG_t

Z ∞

t

L_t(u) ζ_t(u) +ϕ_t(u) +α⁰_t

p_t(u)g(u)du+ 11{τ <t}ζ_t(τ) (3.31) ψ_t^G = L_t(t)

L⁰_t −1 (3.32)

where L⁰ is the optional reduction of L^G.

Proof: This is an application of Proposition 3.2.

Note that ψ_t^G > −1, as it must be. If E[L₀(τ)] = 1, and L(τ) is a positive (P,F^(τ))- martingale with initial value `(τ), one can associate to it the change of probability measure defined by deP = L_t(τ)dP on F_t^(τ), for any t ≥ 0. The choice L₀ ≡ 1 is equivalent to Pe(τ > u) = P(τ > u), for each u≥0. Indeed, since τ is F₀∨σ(τ)-measurable, we have, using tower property eP(τ > u) = E

`(τ)11{τ >u}

and the equality E

L₀(τ)11{τ >u}

= E 11{τ >u}

holds, for each u≥0, which implies that L₀ ≡1.

In particular, in the case Lt(τ) =p0(tau)/pt(τ), which satisfies dLt(τ) =−Lt(τ)ϕt(τ)dWt(τ), one obtains (recall that P^∗ is defined in (2.9)) dP^∗|Gt =L^G_t^,∗dP|Gt with

L^G,∗_t =E(p₀(τ)

p_t(τ)|G_t) =E(µ^G,∗ ·W^G)_tE(ψ^G,∗·M^G)_t (3.33) with

µ^G_t^,∗ = 11{t≤τ}α_t⁰−11{τ <t}ϕ_t(τ) and ψ_t^G,∗ = 11{t≤τ}

Gt

p_t(t)(1−F(t)) −1

(3.34)

where F(t) = Rt

0 g(u)du.

From the definition of L^G,∗ one has L^G,∗_t = 11{t≤τ}

1−F(t)

G_t + 11{τ <t}L_t(τ) (3.35) for all t≥0.

3.4.2 G versus F

From PRP, any positive (P,G)-martingale L^G = (L^G_t )t≥0 can be written as L^G_t =L^G₀ +

Z t

0

L^G_s µ^G_s dW_s^G+ Z t

0

L^G_s−ψ_s^GdM_s^G =L⁰_t11{t<τ}+L¹_t(τ) 11{τ≤t}. (3.36) Let L = (L_t)t≥0 be its F-optional projection L_t = E

L^G_t | F_t

, for all t ≥ 0. Then, we have L_t =L₀+Rt

0 L_sη_sdW_s, where L₀ =E[L₀(τ)] and η is the F-predictable process η_t = L⁰_t

L_t (µ⁰_t −α⁰_t)G_t+ Z t

0

L¹_t(u) µ¹_t(u)−α¹_t(u)

p_t(u)g(u)du (3.37) for all t≥0.

3.4.3 F^(τ) versus F

Let L(τ) be a positive F^(τ)-martingale of the form (3.28), then its F-optional projection is L_t =E[L_t(τ)| F_t] =L₀+Rt

0L_sσ_sdW_s where L₀ =E[L₀(τ)] and σ is the F-predictable process σ_t = 1

`t

Z ∞

0

L_t(u) ζ_t(u) +ϕ_t(u)

p_t(u)g(u)du (3.38)

for all t≥0.

3.5 Stability of the Brownian property

As we mentioned before, due to the fact that the processes W^G and M^G enjoy the predictable representation property in G, any locally equivalent probability measure Q which is equivalent to P on G_t is given by

dQ dP _G

t

=E(µ^G·W^G)tE(ψ^G·M^G)t (3.39) for all t ≥ 0, where µ^G and ψ^G are G predictable and ψ^G > −1. Note that the change of probability measure does not affect the intensity if and only if ψ^G = 0.

For a probability measure Q which is locally equivalent to P on G, let us denote by Q^F its restriction to F, which is locally equivalent to P on F, with its density being of the form dQ^F/dP=E(η·W)_t, for allt ≥0. UnderQ^F, the processW^Q,F defined by W_t^Q,F =W_t−Rt

0η_sds is a (Q^F,F)-standard Brownian motion. Let us provide a set of probability measures Q which are equivalent to P on G such that the (Q^F,F)-standard Brownian motion W^Q,F is a (Q,G)- standard Brownian motion.

Proposition 3.6 Let L^G be a positive (P,G)-martingale of the form E(µ^G·W^G), and define the probability measure Q by

dQ dP Gt

=L^G_t (3.40)

for all t ≥0. The process W^Q,F is a (Q,G)-standard Brownian motion if and only if µ^G−α^G is F-adapted. Then, we have η_t =µ^G_t −α^G_t , for all t≥0.

Proof: On the one hand, we have seen in (2.15) that W_t^G = W_t + Rt

0 α^G_sds, for all t ≥ 0. Hence, we have W_t^G = W_t^Q,F + Rt

0 σ_sds +Rt

0 α^G_sds, for all t ≥ 0. On the other hand, we see that W_t^G − Rt

0µ^G_sds is a (Q,G)-standard Brownian motion. It follows that W_t^Q,F+Rt

0 σ_sds+Rt

0 α^G_sds−Rt

0 µ^G_sds is a G-standard Brownian motion. Therefore, W^Q,F is a (Q,G)-standard Brownian motion if and only if σ_t+α^G_t −µ^G_t = 0, for all t ≥ 0. Note that indeed, if α^G_t −µ^G_t is F adapted, then, from (3.24) L_tσ_t =E(L^G_t(α_t^G−µ^G_t)|F_t) =L_t(α_t^G−µ^G_t)

where L_t =E(L^G_t|F_t).

Examples 3.7 (i) If the process µ^G is such that

µ⁰_t = 0 (3.41)

then Q is equal to P on G_τ and the immersion holds for F and G under Q. The choice µ⁰_t = 0 leads to η=α⁰, hence µ¹_t(u) = α¹_t(u)−α⁰_t =−ϕ_t(u)−α_t⁰, for all t≥u. Therefore, we have

E(µ¹·W^G)_t (3.42)

= 11{t<τ}+ 11{τ≤t} exp Z t

τ

α¹_s(τ)dW_s^G−1 2

Z t

τ

(α¹_s)²ds

exp

− Z t

τ

α⁰_sdW_s^G−1 2

Z t

τ

(α⁰_s)²ds

= 11_{t<τ}+ 11_{τ≤t}p_τ(τ) p_t(τ)

Z_t Z_τ

where we have used the fact that, on {τ < t}, one hasp_τ(τ)/p_t(τ) = exp Rt

τ α¹_sdW_s+ ¹₂Rt

τ(α¹_s)²ds and Z_t=E(−α⁰·W)_t=G_te^Λt, for all t≥0. More precisely, we have

L^G_t = 11{t<τ}+ 11{τ≤t}

p_τ(τ)e^ΛtG_t

p_t(τ)e^ΛτG_τ (3.43)

and

η_t = Gt

1−F(t) (3.44)

This example was presented in [1; Chapter V].

(ii) the case µ⁰ =α⁰ leads to η = 0 and µ¹ =α¹, that is, this is the case in which Q=P^∗.

4 Equivalent Martingale Measures

consider the case where ν and σ are processes Let us consider a model of a financial market in which the stock price process S = (S_t)t≥0 started at S₀ = 1 satisfies the stochastic differential equations (written in various filtrations)

dS_t=S_t ν dt+σ dW_t

=S_t ν_t^Gdt+σ dW_t^G

=S_t ν+σϕ_t(τ)

dt+σ dW_t(τ)

(4.1) for some σ >0 fixed, where, from (2.15), we have

ν_t^G =ν+σα^G_t =ν+σ α⁰_t11_{t≤τ}−ϕ_t(τ) 11_{{τ <t}}. (4.2) We assume that a riskless asset is traded with a null interest rate. It is straightforward to show that, for positive function ` satisfying E[`(τ)] = 1, the positive martingale

Y_t(τ) =`(τ) exp Z t

0

y_s(τ)dW_s(τ)− 1 2

Z t

0

y_s²(τ)ds

(4.3) is a Radon-Nikod´ym density of an equivalent martingale measure on F^(τ) (i.e. such that SY(τ) is an F^(τ)-martingale) if and only if

y_t(τ) =−ϕ_t(τ)− ν

σ (4.4)

for all t≥0. We denote by L(τ) = (Lt(τ))t≥0 such a positive martingale defined by L_t(τ) = L₀(τ) exp

− Z t

0

ϕ_s(τ) + ν σ

dW_s(τ)−1 2

Z t

0

ϕ_s(τ) + ν σ

2

ds

(4.5)

for all t ≥0. Then, we define a new probability measure eP which is locally equivalent to P on F^(τ) by

deP dP _F(τ)

t

=L_t(τ) (4.6)

for all t ≥ 0. Actually, there exists infinitely many such probabilities, which differ from each other by the choice of the initial value L₀, that is, by the choice of the law of τ (under eP):

Pe(τ > u) = E

L₀(τ)11_{{τ >u}}

=R∞

u L₀(v)g(v)dv, for all u≥0. The case L₀ ≡1 corresponds to the situation in which eP(τ > u) =P(τ > u), for any u≥0. Note that, by virtue of Girsanov’s theorem, the process Wf(τ) defined as

Wft(τ) =Wt(τ) + Z t

0

ϕs(τ) + ν σ

ds (4.7)

is a (eP,F^(τ))-standard Brownian motion.

Working in the filtration G, it easily seen that the set of G-equivalent martingale measures (i.e. the set of positive G-martingales Y^G such that SY^G is a G-martingale) is

dY_t^G =Y_t−^G

− ν_t^G

σ dW_t^G+γ_t^GdM_t^G

(4.8) so that

Y_t^G =E(−(ν^G/σ)·W^G)_tE(γ^G·M^G)_t (4.9) where γ is any G-predictable process, with γ >−1.

Let us now consider L^G, the G-optional projection of Lt(τ). The process L^G defines an equivalent martingale measure for S and can be written as in Section 3.4

L^G_t = 1 + Z t

0

L^G_s−µ^G_s dW_s^G+ Z t

0

L^G_s−ψ^G_s dM_s^G =E(µ^G·W^G)_tE(ψ^G·M^G)_t (4.10) where the processes µ^G andψ^G are given in (3.31). Using the fact that L⁰_t = _G¹

t

Rt

0 L_t(u)p_t(u)g(u)du, one has

µ^G_t = 11_{t≤τ} 1 G_tL⁰_t

Z t

0

− ν

σ −α⁰_t

L_t(u)p_t(u)g(u)du+ 11_{{τ <t}}ϕ_t(τ) (4.11)

=−11_{t≤τ}ν^G

σ +α⁰_t

+ 11_{{τ <t}}ϕ_t(τ) =−11_{t≤τ} ν_t^G

σ + 11_{{τ <t}}ϕ_t(τ) ψ^G_t = 11{t≤τ}

Lt(t) L⁰_t −1

. (4.12)