Optimal consumption and portfolio for an insider in a market with jumps

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Optimal consumption and portfolio for an insider in a market with jumps

Delphine David, Yeliz Yolcu Okur

To cite this version:

Delphine David, Yeliz Yolcu Okur. Optimal consumption and portfolio for an insider in a market with

jumps. Communications on Stochastic Analysis, 2009, 3 (1), pp.101-117. �hal-00376909�

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INSIDER IN A MARKET WITH JUMPS

DELPHINE DAVID AND YELIZ YOLCU OKUR

Abstract. We examine a stochastic optimal control problem in a financial market driven by a L´evy process with filtrationF= (F_t)_t∈[0,T_]. We assume that in the market there are two kinds of investors with different levels of information: anuninformed agent whose information coincides with the natural filtration of the price processes and aninsiderwho has more information than the uninformed agent. When optimal consumption and investment exist, we identify some necessary conditions and find the optimal strategy by using forward integral techniques. We conclude by giving some examples.

1. Introduction

The consumption-portfolio problem in continuous time market models was first introduced by Merton [22], [23]. He worked with the assumption that stock prices were governed by Markovian dynamics with constant coefficients. This approach is based on stochastic dynamic programming. For a market which consists of only two assets, he formulated the problem of choosing optimal portfolio selection and consumption rules as follows:

max(c,X) E

"

Z T

0

U(c(t), t)dt+g(X(T), T)

# ,

subject to the budget constraint,c(t)≥0,X(t)>0 for allt∈[0, T] withX(0) =x.

HereX(·) represents the wealth process,c(·) is the consumption per unit time,Uis assumed to be a strictly concaveutilityfunction andgis thebequest valuationfunc- tion which is concave in terminal wealthX(T). Recently, many authors have used a martingale representation technique instead of dynamic programming methods:

see Cox and Huang [4], [5], Karatzas, Lehoczky and Shreve [15] and Pliska [26].

In incomplete markets, the theory was studied by He and Pearson [12], Karatzas et al. [16], Karatzas and Zitkovic [18], Kramkov and Schachermayer [20].

In financial markets a trader is assumed to make her decisions with respect to the information revealed by the market events. This information is assumed to be accessible to everyone. However, in reality some agents have superior information

2000Mathematics Subject Classification. Primary 93E20 91B28; Secondary 60H05 60G57.

Key words and phrases. Insider trading; forward integral; Levy process; portfolio optimization.

* This research was partially supported by grants from AMaMeF (Projects No #2062 and

#1850).

1

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about the market. The informed agent possesses information regarding some future movements in stock prices and she has opportunities to gain profits by trading before prices reach equilibrium. Because of this fact, it is more realistic to model stochastic control problems where the information is non-identical. This paper focuses on the characterization of the optimal consumption and portfolio choices of an insider when the market is driven by a L´evy process. We also compare the performance functions of informed and uninformed agents in certain specific cases.

These results can be helpful in detecting informed agents.

Karatzas and Pikovsky [17]’s study on stochastic control problems with the presence of insiders is one of the earliest studies regarding to this approach. In this study, they assume that the informed agents maximize their expected logarithmic utility from terminal wealth and consumption in Brownian motion framework.

See also Amendinger et al. [1], Grorud et al. [11] and Elliott et al. [8]. In a subsequent, Grorud [10] analyzed the optimal portfolio and consumption choices when prices are driven by a Brownian motion and a compound Poisson process.

The first general study of insider trading based on forward integrals (without assuming the enlargement of filtration) was done in Biagini and Øksendal [3].

Thereafter, many authors used Malliavin calculus and forward integration to study the optimal portfolio choices of insiders. See e.g. Biagini and Øksendal [2], Imkeller [13], Kohatsu-Higa and Sulem [19] and Le´on et al. [21] for further discussions of the Brownian motion case. An extension of forward integration in to the case of compensated Poisson random measures was proposed by Di Nunnoet al. [6]. That setting is used in solving the optimal portfolio problem as elaborated by Di Nunno et al. [7] and in solving the optimal consumption rate problem as elaborated by Øksendal [25].

In this paper, we extend the results of Di Nunno et al. [7] and of Øksendal [25] by considering both the optimal portfolio and consumption rate choices of an insider when her portfolio is allowed to anticipate the future. We formulate the associated optimal control problem as follows:

max(c,π)

E

"

Z T

0

e^−δ(t)lnc(t)dt+Ke^−δ(T⁾lnX^(c,π)(T)

# .

δ(t)≥0 is a given bounded deterministic function representing the discount rate, K is a nonnegative constant representing the weight of the expected utility of the terminal wealth in the performance function,T > 0 is a fixed terminal time and X^(c,π)(·) is the wealth process with control parameters (c, π). As we consider the optimal consumption and the optimal portfolio problems together, the solution for the optimal portfolio is more complex than the studies which consider only the optimization with respect to terminal wealth. Moreover, unlike many studies we do not use the initial enlargement of filtration technique. As a result, we prove that if there exist optimal portfolio and consumption, then theF-Brownian motion B(·) is a G= (Gt)t∈[0,T]-semimartingale where Ft ⊂ Gt for allt ∈ [0, T] and we show that it also holds for L´evy processes (see Theorem 4.5).

This paper is organized as follows. In Section 2, we recall some mathematical preliminaries about forward integrals which are relevant to our calculations. In

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Section 3, we introduce the main problem. In Section 4, we characterize the optimal consumption and portfolio choice for the problem introduced in the previous section. In Section 5, we compare the optimal wealth process and the performance function of informed and uninformed agents in certain specific examples using the results in Section 4.

2. Preliminary Notes

In this section, we recall the forward integrals with respect to the Brownian motion and to the compensated Poisson random measure. For further information on the forward integration with respect to the Brownian motion, we refer to Russo and Vallois [27], [28] and [29], Nualart [24], Biagini and Øksendal [2]. For the forward integration with respect to the compensated Poisson random measure we refer to Di Nunnoet al. [6] and [7].

Let (Ω,F, P) be a product of probability space such that (Ω, P) = (ΩB×Ωη, P_B⊗P_η)

on which are respectively defined a standard Brownian motion{B(t)}0≤t≤T and a pure jump L´evy process,{η(t)}0≤t≤T, such that

η(t) = Z t

0

Z

R

zN(dt, dz),˜

where ˜N(dt, dz) = (N −νF)(dt, dz) = N(dt, dz)−νF(dz)dt is a compensated Poisson random measure with L´evy measureνF.

LetBdenote theσ-field of Borel sets.

F_t^B :=σ{B(s), s≤t, t∈[0, T]} ∨ N and

F_t^N^˜ :=σ{N(△) :˜ △ ∈ B(R₀×(0, t)),0≤t≤T} ∨ N

are the augmented filtrations generated byB(·) and ˜N(·,·) respectively. We denote by F_t = F_t^B⊗ F_t^N^˜,0 ≤t ≤ T the augmented filtration generated by B and ˜N.

LetG= (G_t)t∈[0,T] be the filtration such that

Ft⊂ Gt⊂ F ∀t ∈[0, T], whereT >0 is a fixed terminal time.

Letϕ(t, ω) be aG-adapted process. Then Z T

0

ϕ(t, ω)dB(t) (2.1)

makes no sense in the normal settings. Similarly, for aG-adapted processψ(t, z, ω), Z T

0

Z

R⁰

ψ(t, z, ω) ˜N(dt, dz) (2.2) does not make sense either. Therefore, it is natural to use forward integrals to handle this problem and to make the integrals (2.1) and (2.2) well defined.

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Definition 2.1. Let ϕ(·, ω), ω ∈ ΩB be a measurable process. The forward integral ofϕwith respect to Brownian motion is defined by

Z ∞

0

ϕ(t, ω)d⁻B(t) = lim

ε→0

Z ∞

0

ϕ(t, ω)B(t+ε)−B(t)

ε dt

if the limit exists in probability. Then ϕis calledforward integrable with respect to Brownian motion. If the limit exists also inL²(P), we writeϕ∈D^B.

In particular, we recall the following result.

Lemma 2.2. Let ϕ be forward integrable and c`agl`ad (i.e., left continuous with right limits). Then for any partition0 =t0< t1< . . . < tN =T

Z T

0

ϕ(t, ω)d⁻B(t) = lim

|△t|→0

X

j

ϕ(tj)△B(tj), where△B(tj) =B(tj+1)−B(tj)and| △t|= supj=0,...,N−1△t_j.

Remark 2.3. Letϕ∈D^B be a c`agl`ad process. IfB(·) is aG-semimartingale, then RT

0 ϕ(t, ω)dB(t) exists as a semimartingale integral and Z T

0

ϕ(t, ω)d⁻B(t) :=

Z T

0

ϕ(t, ω)dB(t).

Let us now give the corresponding definition of forward integral with respect to the compensated Poisson random measure.

Definition 2.4. Letϕ(·, z, ω),z∈R₀,ω∈Ωηbe a measurable random field. The forward integral ofϕwith respect to the compensated Poisson random measure is defined by

Z ∞

0

Z

R⁰

ϕ(t, z) ˜N(d⁻t, dz) = lim

n→∞

Z ∞

0

Z

Un

ϕ(t, z) ˜N(dt, dz)

if the limit exists in probability. Here, U_n is an increasing sequence of compact sets whereUn ⊆R₀, νF(Un)<∞andS∞

n=1Un =R₀. Then, ϕis calledforward integrablewith respect to the Poisson random measure. If the limit exists inL²(P), we writeϕ∈D^N^˜.

Remark 2.5. Letϕ∈D^N^˜ be c`agl`ad. IfR· 0

R

R⁰ϕ(t, z, ω) ˜N(dt, dz) is aG-semimartingale, then

Z T

0

Z

R⁰

ϕ(t, z, ω) ˜N(d⁻t, dz) :=

Z T

0

Z

R⁰

ϕ(t, z, ω) ˜N(dt, dz), a.s..

The last result we recall in this section is the Itˆo formula for forward integrals.

We first define the forward process.

Definition 2.6. Aforward process is a measurable stochastic functionX(·), that admits the representation

X(t) =X(0)+

Z t

0

α(s)ds+

Z t

0

β(s)d⁻B(s)+

Z t

0

Z

R⁰

γ(s, z) ˜N(d⁻s, dz), ∀t∈[0, T],

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where Z T

0

(|α(s)|+β(s)²)ds <∞, almost surely,γ(t, z) is continuous inz around zero for a.a. t∈[0, T] such that

Z t

0

Z

R⁰

|γ(s, z)|²νF(ds, dz)<∞, almost surely fort∈[0, T].

Moreover,β(·) andγ(·,·) are forward integrable with respect to Brownian motion and compensated Poisson random measure. A shorthand notation for this is

d⁻X(t) =α(t)dt+β(t)d⁻B(t) + Z

R⁰

γ(t, z) ˜N(d⁻t, dz). (2.3) Theorem 2.7. (Itˆo formula for forward integrals).

LetX(·)be a forward process of the form (2.3) and defineY(t) =f(X(t))for any f ∈C²(R)and all t∈[0, T]. ThenY(·)is also a forward process and

d⁻Y(t) =

f^′(X(t))α(t) +1

2f^′′(X(t))β(t)²+ Z

R⁰

f(X(t⁻) +γ(t, z))

−f(X(t⁻))−f^′(X(t⁻))γ(t, z)

νF(dz)

dt+f^′(X(t))β(t)d⁻B(t) +

Z

R⁰

f(X(t⁻) +γ(t, z))−f(X(t⁻))N˜(d⁻t, dz),

wheref^′ andf^′′ are the first and second derivatives respectively.

Proof. We refer to Russo and Valois [28] for the proof in the Brownian motion case and to Di Nunnoet al. [6] for the pure jump L´evy process case.

3. The main problem

Assume there is a riskless and a risky asset in an arbitrage-free financial market.

The price per unit of the riskless asset is denoted byS0(·) and satisfies the following ordinary differential equation (O.D.E.)

dS0(t) = r(t)S0(t)dt, S0(0) = 1.

The risky asset has a price processS1(·) defined by dS1(t) = S1(t⁻)[µ(t)dt+σ(t)dB(t) +

Z

R⁰

γ(t, z) ˜N(dt, dz)], S1(0) > 0.

Assume that the coefficientsr(t) =r(t, ω), µ(t) =µ(t, ω), σ(t) =σ(t, ω),γ(t, z) = γ(t, z, ω) satisfy the following conditions:

(1) r(·), µ(·), σ(·), γ(·, z) areF-adapted c`agl`ad processes.

(2) γ(t, z)>−1 dt×νF(dz)-a.e.

(3) Z T

0

{|r(t)|+|µ(t)|+σ(t)²+ Z

R⁰

γ(t, z)²νF(dz)}dt <∞ a.s.

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In this paper, we will consider an agent who wants to maximize his expected in- tertemporal utility of consumption and terminal value of wealth when the portfolio is allowed to anticipate the future. Hence, we assume that the informed agent’s portfolio and consumption choices are adapted to the larger filtrationG. Note that G= (G_t)t∈[0,T] can not be chosen freely. For example, if G_t=F_t∨σ(X), where X is any hedgeableFT-measurable random variable, then the informed agent im- mediately obtain the arbitrage opportunity. For further information, see [8], [9], [11] and [17]. In this and the next section, we define the stochastic control problem and characterize the controls under the filtration Gsuch that the additional information for an insider does not blow up the value of the problem.

Letπ(t) be the fraction of the wealth invested in the stock (risky asset) at time tby an insider. Sinceπ(·) is aG-adapted process, it is natural to use the forward integration to make the integrals well defined. The corresponding wealth process X^(c,π)(·) of the insider is given by

d⁻X^(c,π)(t) = X^(c,π)(t⁻)[{r(t) + (µ(t)−r(t))π(t)}dt+σ(t)π(t)d⁻B(t) +π(t)

Z

R⁰

γ(t, z) ˜N(d⁻t, dz)]−c(t)dt with initial valueX^(c,π)(0) =x.

We assume that the agent has a logarithmic utility function. It is convenient to use such functions because it hasiso-elastic marginal utilitywhich means that an agent has the same relative risk-tolerance as toward the end of his life. Moreover, as X(t) ≥ 0 for all t ∈ [0, T], we can assume that the insider has a relative consumption rateλ(·) defined by

λ(t) := c(t) X^(c,π)(t)

without loss of generality. If X^(c,π)(T) = 0, we put λ(T) = 0. Then the corresponding wealth dynamic of the insider in terms of relative consumption rate λ and portfolioπcan be rewritten by the following forward S.D.E.

d⁻X^(λ,π)(t) = X^(λ,π)(t⁻)[{r(t)−λ(t) + (µ(t)−r(t))π(t)}dt +σ(t)π(t)d⁻B(t) +π(t)

Z

R⁰

γ(t, z) ˜N(d⁻t, dz)], (3.1) for allt∈[0, T], where the initial wealth is X^(λ,π)(0) =x >0.

Problem 3.1. Find the optimal relative consumption rateλ^∗(·) and the optimal portfolio π^∗(·) for an insider subject to his budget constraint, i.e. find the pair (λ^∗, π^∗)∈ Awhich maximizes the performance function given by

J(λ^∗, π^∗) := sup

(λ,π)∈A

E

"

Z T

0

e^−δ(t)ln(λ(t)X^(λ,π)(t))dt+Ke^−δ(T)lnX^(λ,π)(T)

# . Here, δ(t) ≥ 0 is the discount rate for all t ∈ [0, T]. It is a given bounded deterministic function satisfying RT

0 e^−δ(u)du+Ke^−δ(T⁾ 6= 0. T > 0 is a fixed terminal time and X^(λ,π)(·) is the wealth process satisfying the equation (3.1).

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A is the set of all admissible control pairs which will be defined in the following definition.

Definition 3.2. AG-adapted stochastic process pair (λ, π) is calledadmissible if (i)

Z T

0

λ(s)ds <∞a.s..

(ii) π(t),t∈[0, T] is c`agl`ad.

(iii) π(·)σ(·) and π(·)γ(·, z) are forward integrable with respect to Brownian motion and the compensated Poisson random measure, respectively.

(iv) 1 +π(t)γ(t, z)> επ for a.a. (t, z) with respect to dt×νF(dz), for some επ∈(0,1) depending onπ.

(v) Z T

0

|(µ(s)−r(s))π(s)|+σ²(s)π(s)²+ Z

R⁰

π(s)²γ(s, z)²νF(dz)

ds <∞ almost surely.

(vi) E

"

Z T

0

e^−δ(s)|lnλ(s)|ds+Ke^−δ(T)|lnX^(λ,π)(T)|

#

<∞.

4. Characterization of the optimal consumption and investment choice Since we use an iso-elastic utility function, this optimal consumption rate de- pends only on the discount rate. The other parameters in the economy such as interest rates or volatility do not appear. However, the consumption of the agent is depending on these coefficients through the wealth. The next step is to show that the optimal consumption rate found in Øksendal [25] is also optimal for Prob- lem 3.1, i.e., the relative consumption rate can be chosen independently from the optimal portfolio if the terminal wealth is added to the problem when the agent has logarithmic utility function. This result for non-anticipative information was initially established by Samuelson [30].

Theorem 4.1. Defineλˆ as

λ(t) :=ˆ e^−δ(t) RT

t e^−δ(s)ds+K e^−δ(T⁾; t≥0. (4.1) Thenλˆ is an optimal relative consumption rateindependent of the portfolio cho- sen, in the sense that (ˆλ, π)∈ Aand

J(ˆλ, π)≥J(λ, π) for allλandπ such that(λ, π)∈ A.

Proof. The proof can be shown using the additive separability of log-utility func-

tion.

Note that since the optimal relative consumption rate,λ^∗= ˆλdoes not depend on the portfolio choice, the main problem turns to :

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Problem 4.2. Findπ^∗ such that (λ^∗, π^∗)∈ Aand J(λ^∗, π^∗) := ˜J(π^∗)

= sup

(λ^∗,π)∈A

E

"

Z T

0

e^−δ(t)ln(λ^∗(t)X^(λ^∗^,π)(t))dt+Ke^−δ(T⁾lnX^(λ^∗^,π)(T)

# . For all (λ, π)∈ A, let us defineMπ and Yπ(t) as follows:

Mπ(t) :=

Z t

0

µ(s)−r(s)−σ(s)²π(s)− Z

R⁰

π(s)γ(s, z)²

1 +π(s)γ(s, z)νF(dz)

ds +

Z t

0

σ(s)d⁻B(s) + Z t

0

Z

R⁰

γ(s, z)

1 +π(s)γ(s, z)N˜(d⁻s, dz) (4.2) and

Y_π(t) :=

Z t

0

e^−δ(s)M_π(s)ds+M_π(t)Z T t

e^−δ(s)ds+Ke^−δ(T⁾

, (4.3)

for allt∈[0, T]. Moreover, we make the following assumptions:

Assumption 4.3.

(A.1) ∀(λ, π),(λ, β) ∈ A with β bounded, there exists positive τ such that the family{|Mπ+εβ(T)|}0≤ε≤τ is uniformly integrable.

(A.2) For all t ∈ [0, T] the process pair (λ, π) where π(s) := χ_(t,t+h](s)β0(ω), with h > 0 and β0(ω) being a bounded G_t-measurable random variable, belongs toA.

The following theorem plays a crucial role to obtain the necessary conditions for the optimal consumption and investment choice.

Theorem 4.4. Suppose (λ^∗, π^∗) ∈ A is optimal for Problem 3.1. Then Y_π∗(·) defined in (4.3)is aG-martingale.

Proof. Suppose that (λ^∗, π^∗)∈ A is optimal for the insider. We can chooseβ(·) such that (λ^∗, β)∈ A. Then (λ^∗, π^∗(·) +yβ(·))∈ A, for ally small enough. Since J(λ^∗, π^∗+yβ) also denoted as ˜J(π^∗+yβ) is maximal at π^∗, we have

d

dyJ˜(π^∗+yβ)|y=0= 0, which implies

E

"

Z T

0

e^−δ(s) Z s

0

β(u){µ(u)−r(u)−π^∗(u)σ(u)²

− Z

R⁰

(γ(u, z)− γ(u, z)

1 +π^∗(u)γ(u, z))νF(dz)}du+ Z s

0

β(u)σ(u)d⁻B(u) +

Z s

0

Z

R⁰

β(u)γ(u, z)

1 +π^∗(u)γ(u, z)N˜(d⁻u, dz)

ds

(4.4)

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+Ke^−δ(T) Z T

0

β(u)σ(u)d⁻B(u) + Z T

0

Z

R⁰

β(u)γ(u, z)

1 +π^∗(u)γ(u, z)N˜(d⁻u, dz) +

Z T

0

β(u){µ(u)−r(u)−π^∗(u)σ(u)²

− Z

R⁰

(γ(u, z)− γ(u, z)

1 +π^∗(u)γ(u, z))νF(dz)}du

= 0. (4.5)

Let us fixt∈[0, T) andh >0 such thatt+h≤T. We can chooseβ of the form

β(s) :=χ(t,t+h](s)β0 (4.6)

where β0 is a bounded Gt-measurable random variable. Rewriting equation (4.5) by using the equation (4.6), we obtain :

E

"

β0

Z t+h

t

e^−δ(s) Z s

t

{µ(u)−r(u)−π^∗(u)σ(u)²− Z

R⁰

π^∗(u)γ(u, z)²

1 +π^∗(u)γ(u, z)νF(dz)}du +

Z s

t

σ(u)d⁻B(u) + Z s

t

Z

R⁰

γ(u, z)

1 +π^∗(u)γ(u, z)N˜(d⁻u, dz)

ds +β0

Z T

t+h

e^−δ(s) Z t+h

t

σ(u)d⁻B(u) + Z t+h

t

Z

R⁰

γ(u, z)

1 +π^∗(u)γ(u, z)N˜(d⁻u, dz) +

Z t+h

t

nµ(u)−r(u)−π^∗(u)σ(u)²− Z

R⁰

π^∗(u)γ(u, z)²

1 +π^∗(u)γ(u, z)νF(dz)o du

! ds

#

+E

"

Ke^−δ(T)β0

Z t+h

t

µ(u)−r(u)−π^∗(u)σ(u)²− Z

R⁰

π^∗(u)γ(u, z)²

1+π^∗(u)γ(u, z)νF(dz)

du

+ Z t+h

t

σ(u)d⁻B(u) + Z t+h

t

Z

R⁰

γ(u, z)

1 +γ(u, z)π^∗(u)N˜(d⁻u, dz)

!#

= 0.

Let us defineM_π(·) as in equation (4.2), then the above equation turns to : E

"

β0

Z t+h

t

e^−δ(s)Mπ^∗(s)ds+Mπ^∗(t+h) Z T

t+h

!

−Mπ^∗(t) Z T

t

!!#

= 0.

Let us define

N_π∗(t) :=

Z t

0

e^−δ(s)M_π∗(s)ds and

Pπ^∗(t) :=Mπ^∗(t) Z T

t

! . Hence, we have

Eh β0

Nπ^∗(t+h)−Nπ^∗(t) +Pπ^∗(t+h)−Pπ^∗(t)| Gt

i= 0.

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By using the equation (4.3), we get

E[β0(Yπ^∗(t+h)−Y_π∗(t))] = 0.

Since this holds for all boundedG_t-measurableβ0, we have : E[Yπ^∗(t+h)|Gt] =Yπ^∗(t).

HenceYπ^∗ is a G-martingale.

The initial enlargement of filtration technique is a common methodology used in stochastic control problems with anticipative information. The main assumption in this method is that the conditional distribution of the random time is absolutely continuous to a measure. It implies that everyF-martingale is a semimartingale with respect to the enlarged filtration. In this paper, we do not assume this strict condition and indeed we have this as a result by using Theorem 4.4 and forward integral techniques. In the following theorem, we show that if there exists admissible optimal portfolio and consumption choices, thenF-Brownian motion is aG-semimartingale.

Theorem 4.5. Suppose γ(t, z)6= 0 and σ(t)6= 0 for a.a. (t, z, ω). Suppose that there exist optimal relative consumption rate and optimal portfolio (λ^∗, π^∗) ∈ A for Problem 3.1. Then

(i) B(·)is a(G, P)-semimartingale. Therefore, there exists an adapted finite variation process α(·)such that the processB(·)ˆ defined as

B(t) =ˆ B(t)− Z t

0

α(s)ds; ∀t∈[0, T] is aG-Brownian motion.

(ii) The process Z ·

0

Z

R⁰

γ(s, z)

1 +π(s)γ(s, z)N˜(d⁻s, dz) is aG-semimartingale.

(iii) The process

Z ·

0

Z

R⁰

γ(s, z) ˜N(ds, dz) is aG-semimartingale.

(iv) The optimal portfolio πsatisfies the following equation:

Z t

0

Z

R⁰

γ(s, z) 1 +π(s)γ(s, z)

Z T

s

e^−δ(u)du+Ke^−δ(T⁾

!

(νG−νF)(ds, dz) +

Z t

0

µ(s)−r(s)−σ(s)²π(s) +σ(s)α(s)− Z

R⁰

π(s)γ(s, z)²

× Z T

s

!

ds= 0, ∀t∈[0, T].

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(v) The optimal relative consumption rate is given by λ^∗(t) = e^−δ(t)

RT

t e^−δ(s)ds+K e^−δ(T); ∀t∈[0, T].

Proof. Let (λ, π) ∈ A be an optimal control choice for the Problem 3.1. By applying the Fubini’s Theorem toY_π(t) in the equation (4.3), we can write:

Yπ(t) = Z t

0

Z

R⁰

γ(s, z) 1 +π(s)γ(s, z)

Z T

s

!

N(d˜ ⁻s, dz)

+ Z t

0

σ(s) Z T

s

!

d⁻B(s) +

Z t

0

R⁰

π(s)γ(s, z)²

× Z T

s

! ds.

Note that since F_t⊂ G_t for allt ∈[0, T], the Poisson random measureN(dt, dz) has a unique compensator with respect to the enlarged filtrationG, sayνG(dt, dz) for all (t, z)∈[0, T]×R₀. Using the orthogonal decomposition into a continuous partY_π^c(t) and a discontinuous partY_π^d(t), we get as follows

Y_π^c(t) = Z t

0

σ(s) Z T

s

e^−δ(u)du+Ke^−δ(T)

!

d⁻B(s)

− Z t

0

σ(s)α(s) Z T

s

! ds

Y_π^d(t) = Z t

0

Z

R⁰

γ(s, z) 1 +π(s)γ(s, z)

Z T

s

!

N(d˜ ⁻s, dz)

+ Z t

0

Z

R⁰

θ(s, z)(νF−νG)(ds, dz),

whereα(s) andθ(s,·) areG_s-measurable processes for alls∈[0, T] such that Z t

0

Z

R⁰

θ(s, z)(νF−νG)(ds, dz)− Z t

0

σ(s)α(s) Z T

s

! ds

= Z t

0

R⁰

π(s)γ(s, z)²

× Z T

s

! ds.

(i) For the continuous partY_π^c(t), we use the fact that Z t

0

1 σ(s)

RT

0 e^−δ(u)du+Ke^−δ(T⁾dY_π^c(s) =B(t)− Z t

0

α(s)ds, t∈[0, T]

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is aG-martingale. Then we obtain directly thatB(·) is aG-semimartingale.

(ii) SinceY_π(·) is aG-martingale, we can easily show that Γ(·) defined as Γ(t) :=

Z t

0

Z

R⁰

γ(s, z) 1 +π(s)γ(s, z)

Z T

s

N˜(d⁻s, dz) +

Z t

0

σ(s)Z T s

d⁻B(s), t∈[0, T] is aG-semimartingale. Then,

Z ·

0

Z T

s

e^−δ(u)du+Ke^−δ(T)−1

dΓ(s)

= Z ·

0

Z

R⁰

γ(s, z)

1 +π(s)γ(s, z)N˜(d⁻s, dz) + Z ·

0

σ(s)d⁻B(s) is also aG-semimartingale. Finally, using (i) we conclude that

Z ·

0

Z

R⁰

γ(s, z)

1 +π(s)γ(s, z)N˜(d⁻s, dz) is aG-semimartingale.

(iii) By (ii) and Remark 2.5, we know that Z t

0

Z

R⁰

γ(s, z)

1 +π(s)γ(s, z)(νF−νG)(ds, dz), t∈[0, T]

is of finite variation. Using Hypothesis (iv) of Definition 3.2, it follows that

Z t

0

Z

R⁰

γ(s, z)(νF−νG)(ds, dz), t∈[0, T] is of finite variation. Note that theG-martingale

Z ·

0

Z

R⁰

γ(s, z)(N−νG)(ds, dz) can be written as :

Z t

0

Z

R⁰

γ(s, z)(N−νG)(ds, dz)

= Z t

0

Z

R⁰

γ(s, z) ˜N(ds, dz) + Z t

0

Z

R⁰

γ(s, z)(νF−νG)(ds, dz), t∈[0, T] and sinceR

0

R

R⁰γ(s, z)(νF−νG)(ds, dz) is of finite variation then Z ·

0

Z

R⁰

γ(s, z) ˜N(ds, dz) is aG-semimartingale.

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(iv) Let us rewriteY_π(t) as Yπ(t) =

Z t

0

Z

R⁰

γ(s, z) 1 +π(s)γ(s, z)

Z T

s

(N−νG)(ds, dz) +

Z t

0

Z

R⁰

γ(s, z) 1 +π(s)γ(s, z)

Z T

s

Z t

0

R⁰

π(s)γ(s, z)²

×Z T s

e^−δ(u)du+Ke^−δ(T⁾ ds +

Z t

0

σ(s)Z T s

e^−δ(u)du+Ke^−δ(T) dB(s).ˆ

Hence by the martingale representation theorem, we conclude that the finite variation part is zero, i.e.,

Z t

0

Z

R⁰

γ(s, z) 1 +π(s)γ(s, z)

Z T

s

Z t

0

R⁰

π(s)γ(s, z)²

×Z T s

e^−δ(u)du+Ke^−δ(T) ds= 0.

Finally, as a Corollary, let us present the results for an uninformed agent:

Corollary 4.6. Suppose Ft = Gt, for all t ∈[0, T]. Then the optimal portfolio π(·)solves the following equation:

µ(t)−r(t)−σ(t)²π(t)− Z

R⁰

π(t)γ(t, z)²

1 +π(t)γ(t, z)νF(dz) = 0, ∀t∈[0, T] and the optimal relative consumption rate λ(t)is given by

λ(t) = e^−δ(t) RT

t e^−δ(u)du+K e^−δ(T); ∀t∈[0, T].

Proof. These results can be directly derived from Theorem 4.5.

5. Examples

In this section, we give two examples to illustrate our results. These examples were already treated in other papers for the optimal portfolio choices. The aim of this section is to show that our approach is coherent and give the same results as the ones obtained using enlargement of filtration theory.

Example 1: The Brownian motion case.

Suppose thatγ(t, z) = 0 andσ(t)6= 0 for almost all (t, z). We denote byπ_i^∗(t) andπ^∗_h(t) the optimal portfolios for the insider and the uninformed (honest) agent,

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respectively. By Theorem 4.5, the optimal portfolio π_i^∗(t) satisfies the following equation

Z t

0

Z T

s

!

µ(s)−r(s)−σ(s)²π^∗_i(s) +σ(s)α(s) ds= 0, for allt∈[0, T]. Then, we obtain an explicit solution forπ_i^∗(t):

π_i^∗(t) =µ(t)−r(t) σ(t)² +α(t)

σ(t), ∀t∈[0, T]

and the optimal relative consumption rate for the insiderλ^∗_i(t) given by : λ^∗_i(t) = e^−δ(t)

RT

t e^−δ(s)ds+Ke^−δ(T)·

For the uninformed agent, by Corollary 4.6,π^∗_h(t) andλ^∗_h(t) are given by : π^∗_h(t) =µ(t)−r(t)

σ(t)² , λ^∗_h(t) = e^−δ(t) RT

t e^−δ(s)ds+K e^−δ(T⁾ =λ^∗_i(t)·

By (i) in Theorem 4.5,B(t) is a Gt-semimartingale then X

“cλ∗

i,πi^∗

”

i (t) =X

“cλ∗

h,πh^∗

”

h (t) exp

1 2

Z t

0

α(s)²ds+ Z t

0

α(s)dB(s)ˆ

(5.1)

where X

“cλ∗

i,π^∗i

”

i (t) and X

“cλ∗

h,π^∗h

”

h (t) are the optimal wealth processes for the insider and the uninformed agent respectively.

Hence,

Ji cλ^∗i, π_i^∗

=Jh cλ^∗h, π_h^∗ +1

2E

"

Z T

0

e^−δ(t) Z t

0

α(s)²ds dt

#

+1 2KE

"

e^−δ(T⁾ Z T

0

α(s)²ds

# .

(5.2)

Remark 5.1. Note that although the optimal relative consumption rates are the same, the optimal consumption rates are not the same among the informed and uninformed agent by equation (5.1).

Proposition 5.2. LetGt=F_t^B∨σ(B(T0)), T0> T for allt∈[0, T]. Assume that there exist optimal control choices for the Problem 3.1. Let δ(t) and γ(t, z) are equal to zero for all(t, z)∈[0, T]×R₀. Then the additional performance function of an insider is equal to

1 2

(T0−T) ln(T0−T) +T

+K 2 ln

T0

T0−T

.

Proof. If we restrict the enlarged filtration to be G_t = F_t^B ∨σ(B(T0)), T0 > T, then

α(t) = B(T0)−B(t) T0−t ·

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and by the equation (11), the performance function of the informed agent can be written in terms of the performance function of the uninformed one as follows :

J_i c_λ^∗_i, π_i^∗

= J_h c_λ^∗_h, π^∗_h +1

2 Z T

0

Z t

0

1

T0−sds+K 2

Z T

0

1 T0−sds

= Jh cλ^∗h, π^∗_h +1

2

(T0−T) ln(T0−T) +T

+K 2 ln

T0

T0−T

.

Example 2: The mixed case.

Suppose that γ(t, z) = z and σ(t) 6= 0 for almost all (t, z). We consider the enlarged filtration G_t^′ =Ft∨σ(B(T0), η(T0)), T0 > T and take the following assumptions:

(1) The informed agent has access to the filtrationGtsuch thatFt⊆ Gt⊆ G_t^′, t∈[0, T].

(2) The L´evy measure νF is given byνF(ds, dz) =ρδ1(dz)dswhere δ1(dz) is the unit point mass at 1.

(3) η(t) is defined as η(t) = Q(t)−ρt with Q being a Poisson process of intensityρ.

Using the results of Di Nunnoet al. [7] Section 5, we obtain the following optimal portfolioπ_i^∗(t):

π_i^∗(t) =π_h^∗(t) + ζ(t) σ(t) with

π^∗_h(t) =µ(t)−r(t) σ(t)² − ρ

σ(t)², ζ(t) = 1

2σ(t)

h−µ(t) +r(t) +ρ+σ(t)α(t)−σ(t)² +

q

(µ(t)−r(t)−ρ+σ(t)α(t) +σ(t)²)²+ 4σ(t)²θ(t)i , α(t) =E[B(T0)−B(s)|G_s]⁻

T0−s , θ(t) =E[Q(T0)−Q(s)|Gs]⁻

T0−s ,

where the notationE[...]⁻ denotes the left limit ins.

Moreover we have the optimal consumption rates λ^∗_i(t) and λ^∗_h(t) for the informed and uninformed agents respectively:

λ^∗_i(t) = e^−δ(t) RT

t e^−δ(s)ds+Ke^−δ(T⁾ =λ^∗_h(t).

Substituting these equalities into the wealth process equation and by Theorem 4.5, we can express the optimal wealth process of the informed agent in terms of the

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optimal wealth process of the uninformed agent : X

“ cλ∗i,π^∗i

”

i (t) =X

“ cλ∗

h,πh^∗

”

h (t) exp

Z t

0

−1

2ζ(s)²+ζ(s)α(s)

ds +

Z t

0

Z

R⁰

ln

1 + zζ(s)σ(s)

σ(s)²+ (µ(s)−r(s)−ρ)z

νG(ds, dz) + Z t

0

ζ(s)dB(s)ˆ +

Z t

0

Z

R⁰

ln

1 + zζ(s)σ(s)

σ(s)²+ (µ(s)−r(s)−ρ)z

(N−νG)(ds, dz)

. Hence,

J_i c_λ^∗_i,π^∗_i

=J_h c_λ^∗_h, π^∗_h +E

"

Z T

0

e^−δ(t) Z t

0

−1

2ζ(s)²+ζ(s)α(s)

ds dt

#

+E

"

Z T

0

e^−δ(t) Z t

0

Z

R⁰

ln

1 + zζ(s)σ(s)

σ(s)²+ (µ(s)−r(s)−ρ)z

νG(ds, dz)dt

#

+KE

"

e^−δ(T⁾ Z T

0

−1

2ζ(s)²+ζ(s)α(s)

ds

#

+KE

"

e^−δ(T⁾ Z T

0

Z

R⁰

ln

1 + zζ(s)σ(s)

σ(s)²+ (µ(s)−r(s)−ρ)z

νG(ds, dz)

# . Acknowledgment. We are grateful to our supervisors B. Øksendal, N. Pri- vault, G. Di Nunno for the useful comments and suggestions. We also thank to M. Pontier and G. W. Weber for their fruitful remarks and suggestions. We wish to thank theAdvanced Mathematical Methods for Finance (AMaMeF)programme of theEuropean Science Foundation (ESF) for the financial supports during this work.

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Delphine David: Laboratoire d’Analyse et Probabilit´es, Universit´e d’Evry-Val- d’Essonne, France

E-mail address:delphine.david@univ-evry.fr URL:http://perso.univ-lr.fr/ddavid/Home.html

Yeliz Yolcu Okur: Centre of Mathematics for Applications (CMA), University of Oslo, Norway

E-mail address:yelizy@math.uio.no