The Market - Application to a robust utility maximization problem

6.3 Application to a robust utility maximization problem

6.3.1 The Market

y^P₀^,n−y^P₀

2≤Ckξ_n−ξk²

L^2,κH

This shows that the convergence of y₀^P,n to y^P₀ is uniform with respect to P ∈ P^κ_H. Hence we can pass to the limit in (6.2.16) and exchange the limit and the suprema to obtain the desired result.

We finish this Section by recalling a result from [110] (see Proposition 5.4) which gives a sufficient condition for the density condition (6.2.14)

Lemma 6.2.3 Assume that the domain ofF does not depend ontand thatD¹_F contains a countable dense subset. Then (6.2.14) holds.

Proof. It suffices to notice that in our framework, all the constant mappings belong to Ae0. Then

Proposition5.4 in [110] applies.

6.3 Application to a robust utility maximization problem

We study in this part a robust exponential utility maximization problem in the spirit of [59] and [87]. We consider the case of an agent maximizing the utility of his terminal wealth, over all possible trading strategies. The agent also have at the terminal date T a claimξ, which is a F_T-measurable random variable. We work in a setting with model ambiguity, due to the presence of the set P_H^κ, composed of non dominated probability measures, accounting for the volatility uncertainty and jump measure uncertainty (see also remark6.3.2). In [51] and [59], the utility maximization problems are solved by means of stochastic control techniques, and the value function is linked with some particular BSDEs. In our case, since we work with a whole set of non dominated probability measures, our value function will be linked to some second order BSDE with jumps, and the general martingale optimality principle described in [51] is treated via the use of non-linear martingales.

6.3.1 The Market

In this section, we will always assume that the matricesa:=a^P and a:=a^P are uniformly bounded inP. In particular, this implies that we can restrict ourselves to the case where the parameter ain

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the definition of a generatorF is bounded. We consider a financial market consisting of one riskless asset, whose price is assumed to be equal to1for simplicity, and one risky asset whose price process (St)0≤t≤T is assumed to follow a mixed-diffusion

dS_t

S_t⁻ =b_tdt+dB_t^c+ Z

β_t(x)µ_Bd(dt, dx), P^κ_H−q.s., (6.3.1) where we assume that

Assumption 6.3.1 (i) (b_t) is a bounded F-predictable process uniformly continuous in ω for the Skorohod topology.

(ii) (β_t) is a bounded F-predictable process which is uniformly continuous in ω for the Skorohod topology, verifies

sup

ν∈N

Z _T

|β_t(x)|ν_t(dx)dt <+∞, P^κ_H −q.s., and satisfies

C1(1∧ |x|)≤βt(x)≤C2(1∧ |x|), P^κ_H −q.s., for all (t, x)∈[0, T]×E, where C2 ≥0≥C1 >−1 +δ, where δ >0 is fixed.

Remark 6.3.1 The uniform continuity assumption onω is here to ensure that the 2BSDEs we will encounter in the sequel indeed have solutions. The assumption onβ is classical [82] and implies that the price process S is positive.

Remark 6.3.2 The volatility is implicitly embedded in the model. Indeed, under each P∈ P_H^κ, we have dB^c_s ≡ ba^1/2_t dW_t^P where W^P is a Brownian motion under P. Therefore, ba^1/2 plays the role of volatility under each P and thus allows us to model the volatility uncertainty. Similarly, we have incertitude on the jumps of our price process, since the predictable compensator associated to the jumps of the discontinuous part of the canonical process changes with the probability considered. This allows us to have incertitude not only about the size of the jumps but also about their laws.

We then denote π= (πt)0≤t≤T a trading strategy, which is a1-dimensionalF-predictable process, supposed to take its value in some compact set C. The process π_t describes the amount of money invested in the stock at time t. The number of shares is _S^π^t

t−. So the liquidation value of a trading strategyπ with positive initial capitalx is given by the following wealth process:

X_t^π =x+ Z t

πs

dB_s^c+bsds+ Z

βs(x)µ_Bd(ds, dx)

, 0≤t≤T, P^κ_H−q.s.

The problem of the investor in this financial market is to maximize his expected exponential utility under model uncertainty from his total wealth X_T^π −ξ where ξ is a liability at time T which is a random variable assumed to be F_T-measurable. Then the value function V of the maximization problem can be written as

V^ξ(x) : = sup

π∈C

inf

P∈P^κ_HE^P[−exp (−η(X_T^π−ξ))] =−inf

π∈C sup

P∈P^κ_H

E^P[exp (−η(X_T^π−ξ))]. (6.3.2)

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where

C:={(π_t) which are predictable and take values inC}, is our set of admissible strategies.

Before going on, we emphasize immediately, that in the sequel we will limit ourselves to probability measures in P_H^κ. We will recover the supremum over all probability measures in P^κ_H at the end by showing that Theorem6.2.2 applies.

To find the value function V^ξ and an optimal trading strategy π^∗, we follow the ideas of the generalmartingale optimality principle approach as in [51] and [59] but adapt it here to a nonlinear framework. This is the same approach as in [87].

Let{R^π}be a family of processes which satisfy the following properties Properties II.1 (i) R^π_T = exp(−η(X_T^π−ξ)) for all π∈ C.

(ii) R^π₀ =R0 is constant for allπ ∈ C.

(iii) We have

R^π_t ≤ ess sup^P

P⁰∈P_H^κ(t⁺,P)

E^P

t [R^π_T], ∀π∈ C R^π_t^∗ = ess sup^P

P⁰∈P_H^κ(t⁺,P)

E^P

t [R_T^π^∗] for someπ^∗ ∈ C, P−a.s. for all P∈ P_H^κ. Then it follows

sup

P∈P_H^κE^P[U(X_T^π −ξ)]≥R0 = sup

P∈P_H^κE^P[U(X_T^π^∗−ξ)] =−V^ξ(x). (6.3.3) 6.3.2 Solving the optimization problem with a Lipschitz 2BSDEJ

To constructR^π, we set

R^π_t = exp(−ηX_t^π)Y_t, t∈[0, T], π ∈ C,

where(Y, Z, U)∈D^2,κ_H ×H^2,κ_H ×J^2,κ_H is the unique solution of the following 2BSDEJ Yt=e^ηξ +

Z T t

Fbs(Ys, Zs, Us)ds− Z T

ZsdB_s^c− Z T

Us(x)µe_B^d(ds, dx) +KT −Kt. (6.3.4) The generator Fb is chosen so that R^π satisfies the Properties II.1. Let us apply Itô’s formula to exp(−ηX_t^π)Yt under some P∈ P_H^κ. We obtain after some calculations

d e^−ηX^t^πYt

=e^−ηX^t^π⁻

−ηπ_sbsYsds+η²

2 π_s²basYsds−ηπsbasZsds−Fbs(Ys, Zs, Us)ds +

e^−ηπ^s^β^s^(x)−1

(Ys+Us(x))bνs(dx)ds+ (Zs−ηπsYs)dB_s^c +

e^−ηπ^s^β^s^(x)−1

(Y_s⁻+U_s(x)) +U_s(x)µe_Bd(ds, dx)−dK_t

. (6.3.5)

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Hence the appropriate choice forF First, because of Assumption 6.3.1, F is uniformly Lipschitz in (y, z), uniformly continuous in ω.

It is also continuous in a and sinceD¹_F = [a, a], it is even uniformly continuous in a. Besides, it is convex inaandν (since it is the infimum of a family of linear functions) and hence can be written as a Fenchel-Legendre transform. Moreover, its domain clearly does not depend on(ω, t, y, z, u)by our boundedness assumptions. Besides, D_F¹ clearly contains a countable dense subset. This in particular shows that Theorem 6.2.2 applies here. Finally,

π∈Cinf

Since C is compact and β is bounded, it is therefore clear from the above inequalities that As-sumption5.3.1(iv)is satisfied. Therefore, if we assume thate^ηξ ∈ L^2,κ_H (for instance ifξ ∈ L^∞,κ_H ), the 2BSDEJ (6.3.4) indeed has a unique solution andR^π is well defined. Let us now prove that it satisfies the propertiesII.1. The property(i)is clear by definition and(ii)holds because of Proposition 6.2.3 together with the Bluementhal 0−1 law. Now for any 0≤t≤T, anyπ ∈ C, anyP∈ P_H^κ and any

This is similar to what we did in the proof of Theorem 5.4.1, and therefore we know that it is sufficient to prove that for anyp >1

E^P

for some positive constant C_p depending only on p and the bounds for π, b and β. Let M be such that −M ≤π≤M, we have

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where we used in the last inequality the fact that π,baandβ are uniformly bounded and that sup

ν∈N

Z T 0

|β_t(x)|νt(dx)dt <+∞ and |β_s(x)−ln(1 +βs(x))| ≤C|β_s(x)|².

Then (6.3.8) comes from the fact that above Doléans-Dade exponentials have moments of any order (it is clear for the continuous one, and we refer again to Lemma B.2.4 for the discontinuous one).

Using (6.3.7) in (6.3.6), we obtain

R^π_t ≤ ess sup^P

P⁰∈P_H^κ(t⁺,P)

E^P

t [R^π_T].

Now, using a classical mesurable selection argument (see [36] (chapitre III) or [45] or Lemma 3.1 in [49]) we can define a predictable processπ^∗ ∈ C such that

Fb_s(Y_s, Z_s, U_s) = (−ηb_s+η²

2 π_s^∗ba_s)π^∗_sY_s−ηπ^∗_sba_s+ Z

e^−ηπ^s^∗^β^s^(x)−1

(Y_s+U_s(x))bν_s(dx).

Using the same arguments as above, we obtain R_t^π^∗ = ess sup^P

P⁰∈P_H^κ(t⁺,P)

E^P

t [R^π_T^∗], which proves (iii) of PropertyII.1holds.

We summarize everything in the following proposition

Proposition 6.3.1 Assume thatexp(ηξ)∈ L^2,κ_H . Then, under Assumption6.3.1, the value function of the optimization problem (6.3.2) is given by

V^ξ(x) =−e^−ηxY0,

where Y0 is defined as the initial value of the unique solution (Y, Z, U) ∈ D^2,κ_H ×H^2,κ_H ×J^2,κ_H of the following 2BSDEJ

Yt=e^ηξ + Z T

Fbs(Ys, Zs, Us)ds− Z T

ZsdB_s^c− Z T

Us(x)µe_Bd(ds, dx) +KT −Kt, (6.3.9) where the generator is defined as follows

Fbt(ω, y, z, u) :=Ft(ω, y, z, u,bat,bνt), where

F_t(y, z, u, a, ν) := inf

π∈C

(−ηb_t+η²

2 πa)πy−ηπaz+ Z

e^−ηπβ^t^(x)−1

(y+u(x))ν(dx)

. Moreover, there exists an optimal trading strategy π^∗ realizing the infimum above.

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6.3.3 A link with a particular quadratic 2BSDEJ

In the case without uncertainty on the parameters, coming back to the paper of El Karoui and Rouge [51], utility maximization problems in a continuous framework with constrained strategies have usually been linked to BSDEs with a quadratic growth generator. The same type of results were later proved by Morlais [92] in a discontinuous setting, that is to say when the assets in the market are assumed to have jumps. More recently, Lim and Quenez [82] showed that when the strategies are constrained in a compact set, then one only needed to consider a BSDE with a Lipschitz generator, thus simplifying the approach of Morlais. Since we provided in this chapter a wellposedness theory for Lipscitz 2BSDEs with jumps, we used the ideas of [82] in the previous section. Our aim in this section is to show that with the result of Proposition 6.3.1 and by making a change of variables, we can prove the existence of a solution to a particular 2BSDEJ, similar to the one in [92], whose generator satisfies a quadratic growth condition.

Before proceeding with the proof, we recall that as for quadratic BSDEs and2BSDEs, we always have a deep link between the martingale part of the solution and the BMO spaces. So we need to introduce the following spaces.

J^2,κ,H_{BM O} denotes the space of predictable andE-mesurable applications U : Ω×[0, T]×E such that kUk²

H^2,κ,H_{BM O} denotes the space of all F⁺-progressively measurableR^d-valued processes Z with kZkH_{BM O}^2,κ,H := sup

We refer the reader to the book by Kazamaki [72] and the references therein for more information on the BMO spaces.

Proposition 6.3.2 Assume that exp(ηξ) ∈ L^2,κ_H and that ξ is bounded quasi-surely. Then, there exists a solution (Y⁰, Z⁰, U⁰) ∈D^2,κ_H ×H^2,κ_H ×J^2,κ_H to the quadratic 2BSDEJ where the generator is defined as follows

Fb_t⁰(ω, z, u) :=F_t⁰(ω, z, u,ba_t,νb_t), (6.3.11)

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where j(u, ν) :=R

where y⁰^P is the solution to the quadratic BSDE with the same terminal condition ξ and generator Fb⁰. Besides, Y⁰ is also bounded quasi-surely.

Proof.

Step 1: As in Morlais [92], we can verify that the generatorFb⁰ satisfies the following conditions.

(i) Fb⁰ has the quadratic growth property. There exists(α, δ)∈R+×R^∗+such that for all(t, z, u),

(ii) We have a "local Lipschitz" condition in z, ∃µ > 0 and a progressively measurable process φ∈H^2,κ,H_{BM O} such that for all (t, z, z⁰, u),P_H^κ −q.s.

We know from [92], that under each P, the BSDEJ with terminal condition ξ and generator Fb⁰ has a solution, that we denote (y⁰^P, z⁰^P, u⁰^P). Moreover, y⁰^P is bounded and there exists a bounded version of u⁰^P (that we still denote u⁰^P) , with the following estimates

solution under Pof equation (6.3.9). This gives that

e^−C⁰ ≤y_t^P≤e^C⁰, t∈R⁺, C⁰∈R.

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In particular y^P is strictly positive and bounded away from0, and by representation (5.4.1), so is Y solving (6.3.9). Thus, we can make the following change of variables

Y_t⁰:= 1

ηlog(Yt).

Then by Itô’s formula and the fact thatK has only predictable jumps, we can verify that the triple (Y⁰, Z⁰, U⁰) satisties (6.3.10) where

In particular, K⁰ is predictable and nondecreasing withK⁰₀= 0. Finally, due to the monotonicity of the function _η¹log(x), we have the following representation forY⁰

Y_t⁰ = ess sup^P

P⁰∈P_H^κ(t⁺,P)

y_t⁰^P.

Step2: Next, we will prove the minimum condition forK⁰. As in [104] for 2BSDE with quadratic growth generator, we use the above representation of Y⁰ and the properties verified by Fb⁰ in z and u. By the "local Lipschitz" condition(ii) ofFb⁰ inz, there exist a processη with

|η_t| ≤µ of U⁰) is uniformly bounded. Actually, this can be proved using exactly the same arguments as in the proof of Lemma 3.3.1 in [104] on the one hand, that is to say applying Itô’s formula toe^−νY^t⁰ for a well chosen ν >0. This allows then to get rid of the non-decreasing process K⁰ in the estimates.

Then on the other hand, one can use the same calculations as in the proof of Lemma 4.2.1to obtain that

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Since c is bounded, this clearly implies thatU⁰ ∈J^2,κ,H_{BM O}.Then the process η defined above is also inH^2,κ,H_{BM O}. Since the process γ⁰ is bounded and has jumps greater than −1 +δ, Kazamaki criterion (see [70]) implies that we can define an equivalent probability measureQ⁰ such that

dQ⁰

By taking conditional expectations in (6.3.12) and using property(iii) satisfied byF⁰, we obtain Y_t⁰−y^0P Letr >1 be the number given by Lemma 3.2.2 in [104] applied to E¹. Then we estimate

E^P

where we used in the last inequality Lemma 3.2.2 in [104] for the term involving E¹ and where we used Lemma B.2.4 for the term involving E² (this is possible because of the properties verified by γ⁰).

With the same argument as in Step(iii) of the proof of Theorem5.4.1, the above inequality along with the representation forY⁰ shows that we have

ess inf^P

P⁰∈P_H^κ(t⁺,P)E^P

K_T⁰ −K_t⁰

= 0,

that is to say that the minimum condition5.3.5 is verified. Indeed, the only thing to verify is that ess sup^P

With the BMO properties satisfied by Z⁰ and U⁰ we can follow the proof of Theorem 3.2 in [104]

to show that

Then (6.3.13) can be obtained exactly as in the proof of Step(iii) of Theorem5.4.1.

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Technical Results for Quadratic BSDEs with Jumps

Proposition A.0.3 Let gbe a function satisfying Assumption4.2.1(i), which is uniformly Lipschitz in (y, z, u). Let Y be a càdlàg g-submartingale (resp. g-supermartingale) in S^∞. Assume further that g is concave (resp. convex) in (z, u) and that

g_t(y, z, u)≥ −M−β|y| −γ

2|z|²− 1

γj_t(−γu).

resp. gt(y, z, u)≤M +β|y|+γ

2|z|²+1 γjt(γu)

Then there exists a predictable non-decreasing (resp. non-increasing) processAnull at0and processes (Z, U)∈H²×J² such that

Y_t=Y_T + Z T

g_s(Y_s, Z_s, U_s)ds− Z T

Z_sdB_s− Z T

U_s(x)eµ(dx, ds)−A_T +A_t, t∈[0, T].

Proof. As usual, we can limit ourselves to theg-supermartingale case. Let us consider the following reflected BSDEJ with lower obstacleY, generatorg and terminal conditionY_T

Ye_t=Y_T + Z T

g_s(Ye_s,Ze_s,Ue_s)ds− Z T

Ze_sdB_s− Z T

Ue_s(x)µ(dx, ds) +e A_T −A_t, t∈[0, T], P−a.s.

Yet≥Yt, t∈[0, T], P−a.s.

Z T 0

Ye_s⁻−Y_s⁻

dA_s = 0, P−a.s. (A.0.1)

The existence and uniqueness of a solution consisting of a predictable non-decreasing process A and a triplet(Y ,e Z,e U)e ∈ S²×H²×J² follow from the results of [58] for instance, since the generator is Lipschitz and the obstacle is càdlàg and bounded.

We will now show that the process Ye must always be equal to the lower obstacle Y, which will provide us the desired decomposition. We proceed by contradiction and assume without loss of generality thatYe₀> Y₀. For any ε >0, we now define the following bounded stopping time

τ_ε := infn

t >0, Ye_t≤Y_t+ε, P−a.s.o

∧T.

By the Skorokhod condition, it is a classical result that the non-decreasing process A never acts before τ_ε. Therefore, we have for anyt∈[0, τ_ε]

Ye_t=Ye_τ_ε+ Z _τ_ε

g_s(Ye_s,Ze_s,Ue_s)ds− Z _τ_ε

Ze_sdB_s− Z _τ_ε

Ue_s(x)µ(dx, ds),e P−a.s. (A.0.2)

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Consider now the BSDEJ on [0, τε] with terminal condition Yτε and generator g (existence and uniqueness of the solution are consequences of the result of [4] or [81])

Ybt=Yτε + Notice also that since Y and g(0,0,0)are bounded, Yb and Ye are also bounded, as a consequence of classical a priori estimates for Lipschitz BSDEJs and RBSDEJs. Then, using the fact that g is convex in (z, u) and that

g_t(y, z, u)≤M+β|y|+γ

2|z|²+ 1

γj_t(γu),

we can proceed exactly as in the Step 2 of the proof of Proposition 4.2.7 to obtain that for any θ∈(0,1) whereC0 is some constant which does not depend onθ.

Lettingθgo to 1, and using the fact that by definition Ye_τ_ε−θY_τ_ε ≤ε, we obtain

But since Y is a g-supermartingale, we have by definition that Yb0 ≤ Y0. Since we assumed that Ye0> Y0, this implies that Ye0 >Yb0. Therefore, we can use Gronwall’s lemma in (A.0.4) to obtain

0<Ye0−Yb0≤C1ε, for some constant C1 >0, independent of ε.

By arbitrariness ofε, this gives us the desired contradiction and ends the proof.

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Technical Results for 2BSDEs with Jumps

B.1 The measures P

^α,ν

Lemma B.1.1 Let τ be an F-stopping time. Letω ∈Ω andS be a bounded F^τ(ω)-stopping time.

There exists an F stopping time S, such thate

S=Se^τ,ω. (B.1.1)

Proof.

(B.1.1) means that

∀ωe∈Ω^τ(ω), S(ω) =e Se^τ,ω(ω) =e S(ωe ⊗_τ(ω)ω).e We set

∀ω₁ ∈Ω, S(ωe ₁) :=S(ω₁^τ(ω)).

Using the fact that

for s≥τ(ω), (ω⊗_τ(ω)ω)e ^τ(ω)(s) =ω(τ(ω)) +ω(s)e −ω(τ(ω))−eω(τ(ω)) =ω(s),e we have that Sesatisfies (B.1.1) by construction, indeed

Se^τ,ω(eω) =Sh

ω⊗_τ(ω)ωeτ(ω)i

=S(ω).e

Notice now that for any ω ∈ Ω, Se: Ω → [τ(ω), T] is measurable as a composition of measurable mappings.

Finally, let us prove thatSeis anF-stopping time.

Fort < τ(ω),{Se≤t}={∅} ∈ F_t. Fort≥τ(ω),

{Se≤t}={ω₁ ∈Ω|S(ω^τ(ω)₁ )≤t}

ω₁⊗_τ(ω)ω₁^τ(ω)∈Ω|S(ω₁^τ(ω))≤to

ω₁ ∈Ω|ω₁^τ(ω)∈A^ωo

= Π⁻¹_τ(ω)(A^ω).

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whereA^ω:={ωe ∈Ω^τ(ω)|S(ω)e ≤t} ∈ F_t^τ(ω) and Π_τ(ω) : (Ω,F_t) →

Ω^τ(ω),F_t^τ(ω)

is defined by Π_τ(ω)(ω1) := w^τ(ω)₁ . By definition of the measura-bility ofΠ_τ(ω), we have{Se≤t}= Π⁻¹_τ(ω)(A^ω)∈ F_t. Lemma B.1.2 Let P∈ P_A_˜ and τ be an F^B-stopping time. Then

P^τ,ω ∈ P^τ(ω)_˜

A for P−a.e. ω∈Ω.

Proof.

Step1: Let us first prove that

(Pν)^τ,ω=P^τ(ω)_ν^τ,ω, Pν-a.s. on Ω, (B.1.2) where(Pν)^τ,ωdenotes a probability measure onΩ^τ, constructed from a regular conditional probability distribution (r.c.p.d.) ofPν for the stopping time τ, evaluated atω, andP^τ(ω)_ν^τ,ω is the unique solution of the martingale problem (P¹, τ(ω), T, Id, ν^τ,ω), whereP¹ is such that P¹(B^τ_τ = 0) = 1.

It is enough to show that outside aPν-negligible set, the shifted processes M^τ, J^τ, Q^τ are(Pν)^τ,ω -local martingales, where M, J and Q are defined in Remark 5.2.1. For this, for any ω in Ω, take a bounded F^τ(ω)-stopping time S. By Lemma B.1.1, there exists a bounded F-stopping time S˜ such that S= ˜S^τ,ω. Then, following the definitions in Subsection 6.2.1,

∆B^τ,ω_S (ω) = ∆Be S(ω⊗_τ ω) = ∆(ωe ⊗_τω)(S) = ∆ωe S1{S≤τ}+ ∆ωeS1{S>τ}, and that for S ≥τ

B_S(ω⊗_τω) = (ωe ⊗_τω)(S) =e ω_τ +ωe_S =B_τ(ω) +B_S^τ(eω).

From this we get

M_S^τ,ω(ω) =e M_S(ω⊗_τ ω) =Be _S(ω⊗_τ eω)−X

u≤S

1_|∆B_u_(ω⊗_τ

eω)|>1∆B_u(ω⊗_τ eω) +

Z S 0

x1|x|>1νu(ω⊗_τω)(dx)e du

=B^τ_S(ω) +e Bt(ω)−X

u≤τ

1|∆ω_u|>1∆ωu− X

τ <u≤S

1|∆B_u^τ(ω)|>1e ∆B_u^τ(ω)e +

Z τ 0

x1_|x|>1νu(ω)(dx)du+ Z S

x1_|x|>1ν_u^τ,ω(ω)(dx)e du

=M_S^τ(ω) +e M_τ(ω), ∀ω∈Ω.

And we can now compute E⁽^P^ν⁾

τ,ω[M_S^τ] =E⁽^P^ν⁾

τ,ω

M_S^τ,ω−M_τ(ω)

=E⁽^P^ν⁾

τ,ωh M^τ,ω_˜

S^τ,ω

−Mτ(ω)

=E^Pτ^ν[MS˜](ω)−Mτ(ω) = 0,for Pν-a.e. ω. (B.1.3)

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Since S is an arbitrary bounded stopping time, we have thatM^τ is a (Pν)^τ,ω-local martingale for Pν-a.e. ω.

Notice that since M is a càdlàg local martingale underPν, we can find aPν-negligible setN, such that for any bounded F-stopping times σ1 ≤σ2, for any ω outside of N, we have E^P_σ^ν₁[Mσ2] (ω) = M_σ₁(ω). In particular, the negligible set appearing in the equality (B.1.3) is independent of the choice of S.e

We treat the case of the process J^τ analogously and write J_S^τ,ω(ω) =e M_S^τ,ω(ω)e 2

Then we can compute the expectation E^(P^ν⁾

We have the desired result, and conclude that (B.1.2) holds true. We can now deduce that for any (α, ν)∈A˜

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Step2: We define τ˜ := τ ◦X^α, α˜^τ,ω := α^{τ ,β}^˜ ^α^(ω) and ν˜^τ,ω := ν^˜^{τ ,β}^α^(ω) where βα is a measurable

Step3: We show that E^P

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Then we have

By definition of the conditional expectation, ψ_τ(ω) =E^P

τ,ωh

ψ(ω(t₁), . . . , ω(t_k), ω(t) +B_t^t_k+1, . . . , ω(t) +B_t^t_n)i

, P^α,ν-a.s.,

where t:= τ(ω) ∈ [t_k, t_k+1[, and where the P^α,ν-null set can depend on (t₁, . . . , t_n) and ψ, but we can choose a common null set by standard approximation arguments.

Then by a density argument we obtain E^P

Now following the arguments in the proof of step 3 of LemmaB.1.2, we prove that for any0< t₁ <

· · ·< t_k=t < t_k+1< t_nand any continuous and bounded functionsφ andψ,

Ae. And since all the probability measuresPⁱsatisfy the requirements

of Definition6.2.1, we havePⁿ=P^α,ν ∈ P_H^κ.

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B.2 Other properties

We define the followingF_t-measurable sets E₁ :=n

0 satisfies (B.2.1). Let us prove now that P

0 ∈ P_H^κ(t⁺,P). As in the proof of claim

Now using the same arguments as in the Step3 of the proof of Lemma B.1.2, we obtain that for any t₁ <· · ·< t_k=t < t_k+1<· · ·< t_n and any bounded and continuous functionsφ andψ,

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ThusP

0 coincides with P onF_t, the proof is complete.

B.2.2 L^r-Integrability of exponential martingales

Lemma B.2.2 Let δ >0 andn∈N^∗. Then there exists a constantC_n,δ depending only onδ and n such that

(1 +x)⁻ⁿ−1 +nx≤Cn,δx², for all x∈[−1 +δ,+∞).

Proof. The inequality is clear forx large enough, let’s sayx≥M for some M >0. Then, a simple Taylor expansion shows that this also holds in a neighborhood of0, that is to say forx∈[−ε, ε]for someε >0. Finally, forx∈(−1 +δ,−ε)∪(ε, M), it is clear that we can chooseC large enough such

that the inequality also holds.

Mémin [89] and then Lépingle and Mémin [80] proved some useful multiplicative decompositions of exponential semimartingales. We give here one of these representations that we will use in the proof of Lemma B.2.3.

Proposition B.2.1 (Proposition II.1 of [79]) Let N be a local martingale and let A be a pre-dictable process with finite variation such that ∆A 6= −1. We assume N₀ = A₀ = 0. Then there exists a local martingale N˜ with N˜0 = 0 and such that

E(N +A) =E( ˜N)E(A).

Lemma B.2.3 Letλ >0andM be a local martingale with bounded jumps, such that∆M ≥ −1 +δ, for a fixedδ >0. Let V^−λ be the predictable compensator of

 (ii) There exist a local martingale N˜^−λ such that

E^−λ(M) =E( ˜N^−λ)E(A^−λ).

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Proof. First note that thanks to lemma B.2.2, for λ >0,(1 +x)^−λ−1 +λx≤Cx², and thusW^−λ is integrable. We set

Tn= inf{t≥0 : E(M)_t≤ 1

n} andM_tⁿ=Mt∧T_n.

Then Mⁿ and E(Mⁿ) are local martingales, E(Mⁿ) ≥ _n¹ and E(Mⁿ)_t = E(M)_t if t < T_n. The assumption∆M >−1shows thatTn tends to infinity whenn tends to infinity. For eachn≥1, we apply Itï¿¹₂’s formula to a C² function fn that coincides withx^−λ on [_n¹,+∞):

E^−λ(Mⁿ)t=1−λ Z t

E^−λ−1(Mⁿ)_s⁻dE(Mⁿ)s+λ(λ+ 1) 2

Z t 0

E^−λ−2(Mⁿ)_s⁻dh(E(Mⁿ))^ci_s

s≤t

hE^−λ(Mⁿ)_s− E^−λ(Mⁿ)_s⁻+λE^−λ−1(Mⁿ)_s⁻∆E^−λ(Mⁿ)_si

=1 + Z t

E(Mⁿ)_s⁻dX_sⁿ, where

X_tⁿ:=−λM_tⁿ+ λ(λ+ 1)

2 h(Mⁿ)^c,(Mⁿ)^ci_t+X

s≤t

(1 + ∆Ms)^−λ−1 +λ∆Ms

i , and then E^−λ(Mⁿ) =E(Xⁿ). Let us define the non-truncated counterpart X ofXⁿ:

X =−λM+ λ(λ+ 1)

2 hM^c, M^ci+W^−λ.

On the interval [0, T_n[, we have Xⁿ =X and E^−λ(M) =E(X), now letting n tends to infinity, we obtain that E^−λ(M) and E(X) coincide on[0,+∞[, which is the point (i) of the Lemma.

We want to use the proposition B.2.1to prove the point (ii), so we need to show that ∆A > −1.

We set

S = inf{t≥0 : ∆A^−λ_t ≤ −1}.

It is a predictable time. Using this, and the fact that M and (W^−λ−V^−λ) are local martingales, we have

∆A^−λ_S =E h

∆A^−λ_S |F_S⁻i

=E[∆XS|F_S⁻] =E h

(1 + ∆MS)^−λ|F_S⁻i , and since {S <+∞} ∈ F_S−,

0≥E h

1_{S<+∞}(1 + ∆A^−λ_S ) i

=E h

1_{S<+∞}(1 + ∆M_S)^−λ i

Then ∆M_S ≤ −1 on {S <+∞}, which means that S = +∞ and ∆A >−1 a.s. The proof is now

complete.

We are finally in a position to state the Lemma onL^r integrability of exponential martingales for a negative exponent r.

Lemma B.2.4 Let λ > 0 and let M be a local martingale with bounded jumps, such that ∆M ≥

−1 +δ, for a fixed δ >0, and hM, Mi_t is bounded dt×P-a.s. Then E^P

hE(M)^−λ_t i

<+∞ dt×P−a.s.

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Proof. Letn≥1 be an integer. We will denoteµ˜M =µM −νM the compensated jump measure of M. Thanks to lemmaB.2.3, we write the decompostion

E(M)⁻ⁿ=E( ˜N⁻ⁿ)E 1

2n(n+ 1)< M^c, M^c>+V⁻ⁿ

where N˜⁻ⁿ is a local martingale and V⁻ⁿ is defined as V^−λ. Using Lemma B.2.2, we have the inequality

V_t⁻ⁿ≤ Z t

Cx²νM(ds, dx) and using the previous representation we obtain

E(M)⁻ⁿ_t ≤ E( ˜N⁻ⁿ)tE 1

2n(n+ 1)hM^c, M^ci+ Z ·

Cx²νM(ds, dx)

≤ E( ˜N⁻ⁿ)texp

2n(n+ 1) +C)hM, Mi_t

≤CE( ˜N⁻ⁿ)t since hM, Mi_t is bounded.

Let us prove now that the jumps ofN˜⁻ⁿ are strictly bigger than −1. We compute

∆ ˜N⁻ⁿ= ∆N⁻ⁿ

1 + ∆A⁻ⁿ whereA⁻ⁿ is defined as in lemma B.2.4

= (1 + ∆M)⁻ⁿ

1 + ∆V⁻ⁿ −1>−1 since −1<∆M ≤B and∆V⁻ⁿ>−1.

This implies that E( ˜N⁻ⁿ)is a positive supermartingale which equals 1 at t= 0. We deduce E

E(M)⁻ⁿ_t

≤CE

hE( ˜N⁻ⁿ)_ti

≤C.

We have the desired integrability for negative integers. We extend the property to any negative real

number by Hölder’s inequality.

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