Portfolio optimization with insider s initial information and counterparty risk

Transcription

1 Portfoio optimization with insider s initia information and counterparty risk Caroine HILLAIRET Ying JIAO Juy 11, 212 Abstract We study the gain of an insider having private information which concerns the defaut risk of a counterparty. More precisey, the defaut time τ is modeed as the first time a stochastic process hits a random barrier L. The insider knows this barrier (as it can be the case for exampe for the manager of the counterparty), whereas standard investors ony observe its vaue at the defaut time. A investors aim to maximize the expected utiity from termina weath, on a financia market where the risky asset price is exposed to a sudden oss at the defaut time of the counterparty. In this framework, the insider s information is modeed by using an initia enargement of fitration and τ is a stopping time with respect to this enarged fitration. We prove that the reguator must impose short seing constraints for the insider, in order to excude the vaue process to reach infinity. We then sove the optimization probem and we study the gain of the insider, theoreticay and numericay. In genera, the insider achieves a arger vaue of expected utiity than the standard investor. But in extreme situations for the defaut and oss risks, a standard investor may in average outperform the insider, by taking advantage of an aggressive short seing position which is not aowed for the insider, but at the risk of big osses if the defaut finay occurs after the maturity. Keywords : asymmetric information, enargement of fitrations, counterparty risk, optima investment, duaity, dynamic programming. MSC 21 : 6H3 91B28 91G4 93E2 This research is part of a project of Europace Institute of Finance. We thank Laurent Denis, Nicoe E Karoui, Monique Jeanbanc, Huyên Pham, Abass Sagna, Nizar Touzi and Lioudmia Vostrikova for discussions. CMAP Ecoe Poytechnique, Emai: caroine.hiairet@poytechnique.edu. Financia support by Chair Financia Risks of the Risk Foundation, Chair Derivatives of the Future sponsored by the Fédération Bancaire Française, Chair Finance and Sustainabe Deveopment sponsored by EDF and Cayon. LPMA Université Paris Diderot and Peking University, Emai: jiao@math.univ-paris-diderot.fr. Financia support by Ama Recherche. 1

2 1 Introduction The insider s optima investment is a cassica probem where an investor possessing some extra fow of information aims to maximize the expected utiity on the fina vaue of her portfoio. As the insider has more information, she has access to a arger set of avaiabe trading strategies, eading to a higher expected utiity from termina weath. In the iterature, an interesting question has been studied: what is the cost of the extra information? From an indifference point of view, we search for the vaue at which the investor accepts to buy the information at the initia time, that is, the amount of money she is ready to pay such that this cost is offset by the increase of the maxima expected utiity. This is the approach adopted by Amendinger et a. 1, where the authors study the vaue of an initia information in the setting of a compete defaut free market. The extra information they consider is a termina information distorted by an independent noise, for exampe, a noisy signa of a functiona of the fina vaue of the assets. We adopt a more direct manner: we are interested in the gain of the insider from her investment strategy on the portfoio compared to other investors not having access to the extra information. The originaity of our paper is to study this probem in the context of credit risks: the insider s information concerns the defaut risk of a counterparty firm. During the financia crisis, the counterparty defaut has become an important source of risk we need to take into account. Jiao and Pham 14 have considered an optima investment probem where the risky asset in the portfoio is subjected to the defaut risk of a counterparty firm and its vaue may suffer a sudden oss at the counterparty defaut time τ. This paper is a good benchmark in our study in order to quantify the vaue of the extra information. The accessibe information for a standard investor is described as in the cassica credit risk modeing by Bieecki and Rutkowski 4, using the progressive enargement of a reference defaut-free fitration F = (F t ) t by the defaut τ. To anayze the impact of defaut, the defaut density framework deveoped in E Karoui et a. 5 has been adopted. This current paper concentrates on an insider in comparison with a standard investor. Both agents can invest in the same risk-free asset and risky one and they observe the same market price for each asset. However, the insider possesses more information on the risky asset since it is infuenced by the counterparty defaut on which the insider has additiona knowedge. Due to the extra information, the insider may gain arger profit. The insider s information is modeed by using an initia enargement of fitration as in 1 and in Grorud and Pontier 7. More precisey, in the credit risk context, we mode the defaut time τ as the first time that a stochastic process hits a random barrier L. The insider knows the barrier from the initia time and the other investors ony see its vaue at the defaut time. This extra information is caed the insider s information, or the fu information in Hiairet and Jiao 9. We sha consider the insider s optimization probem in parae with the one studied in 14. The canonica decomposition of processes adapted to the enarged fitration induces to specify the investment strategies on the two sets: before-defaut one {t < τ} and after-defaut one {t τ}, which is a simiar point to 14. However, due to the extra knowedge on the defaut barrier L, the insider s strategy depends on L before the counterparty defaut, which is not the case for the standard investor. If the defaut occurs, the insider s strategy wi depend on the defaut time τ. From the methodoogy point of view, the main difference here is that for the insider, the defaut time is modeed as in the cassica structura approach mode since the random barrier L is known, so that τ becomes a stopping time w.r.t. the reference fitration F. 2

3 Therefore, the defaut density hypothesis, which is crucia in 14, fais to hod for the insider and we can no onger adopt the conditiona density approach in this situation. We appy the theory of initia enargement of fitration, assuming that the conditiona aw of L given F t is equivaent to the aw of L. The corresponding Radon Nikodym derivative process, (p t (.), t ) wi pay a key roe in our methodoogy. The main observation of our study is that, if the short-seing is not reguated, then the insider can obtain unbounded termina weath. This justifies the necessity to consider the optimization probem with portfoio strategies where the short-seing is imited to a given eve. We decompose the optimization probem as an after-defaut probem and a goba before-defaut one, that we sove respectivey by using the dua and the dynamic programming methods. To make comparison with the standard investor in 14, we choose to consider CRRA utiity function. The paper is organized as foows. In section 2, we introduce the mode for the counterparty defaut, and we define and compare the informationa structure of an insider with respect to a standard investor. In Section 3 we present the insider s investment probem and we decompose it into an after defaut and goba before defaut ones, using the Radon Nikodym derivative process. We aso prove the necessity of imposing short seing constraint for the insider to excude the vaue process to reach infinity. In Section 4 we sove the two optimization probems: the after defaut one through duaity methods in a defaut free compete market, and the goba before defaut one through dynamic programming approach. We perform theorica comparison of the vaue process of the after defaut optimization probem, and for the goba before defaut optimization probem, the comparison is done in Section 5 through numerica iustrations. 2 Counterparty defaut mode and information We first introduce the mode for the counterparty defaut, which is a genera and standard mode in the credit risk anaysis. Let us fix a probabiity space (Ω, A, P) equipped with a reference fitration F = (F t ) t satisfying the usua conditions, which represents the defautfree information. Let τ be a positive random time denoting the defaut time of the counterparty, which is not necessariy an F stopping time. The defaut mode We consider the defaut risk of the counterparty in a genera barrier mode. Let (λ t, t ) be a positive F-adapted process representing the defaut intensity process of the counterparty. Denote by Λ t = t λ sds. It is an increasing process. We mode the defaut time as the first passage time of the process Λ to a positive random barrier L, i.e. (2.1) τ = inf{t : Λ t L} where the defaut threshod L is a positive A-measurabe random variabe. In the particuar but widey used case of Cox process mode, L is independent of F and foows the uni-exponentia aw. In the case where L is constant or deterministic, τ is an F-stopping time as in the cassica structura defaut modes. Information of the insider We suppose that, besides the information on the defaut-free market, the insider has compete information on L: this is the case for exampe of the counterparty firm s managers who 3

4 determine the defaut threshod. This fu information is modeed as the initia enargement of the fitration F by L and denoted by G M = (G M t ) t, G M t = F t σ(l). Without oss of generaity, we assume that a the fitrations we dea with in the foowing satisfy the usua conditions. We suppose that a standard investor on the market observes whether the defaut has occurred or not and if so, the defaut time τ, together with the information contained in the fitration F. Mathematicay, this information is represented by the progressive enargement of fitration F by τ, or more precisey, by the fitration G = (G t ) t where G t = F t D t, D t = σ(1 τ s, s t). This is the standard credit risk modeing for an market investor as in 4. The investor s information is incuded in the insider s information fow. We have G t Gt M for any t. In fact, before the defaut τ, i.e., on the set {t < τ}, the insider has additiona information on L, so her information Gt M is in genera stricty arger than G t. After the defaut occurs, both of them observe the defaut event and subsequenty the vaue of L so that they have equa information fow. We reca the canonica decomposition of G M -adapted (respectivey G M -predictabe) processes (see Jeuin 13 Lemma 3.13 and 4.4). Lemma For t, any Gt M -measurabe random variabe can be written in the form Y t = 1 τ>t Yt (L) + 1 τ t Yt 1 (τ) where Yt ( ) and Yt 1 ( ) are F t B(R + )-measurabe. 2. Any G M -adapted process Y admits the decomposition form Y t = 1 τ>t Y t (L) + 1 τ t Y 1 t (τ) where Y ( ) and Y 1 ( ) are F B(R + )-adapted Any G M -predictabe process Y admits the decomposition form Y t = 1 τ t Y t (L)+1 τ<t Y 1 t (τ) where Y ( ) and Y 1 ( ) are P(F) B(R + )-measurabe, P(F) being the predictabe σ-agebra associated with the fitration F. Remark 2.2 To compare with the case of a standard investor, we reca that any G t -measurabe random variabe Z t can be written as Z t = 1 τ>t Zt + 1 τ t Zt 1 (τ) where Zt and Zt 1 ( ) are respectivey F t -measurabe and F t B(R + )-measurabe. 3 Insider s optimization probem 3.1 Portfoio investment strategy and weath process From now on, a finite horizon T is fixed and a investment strategies take pace from time to time T. The insider has access to the same financia market as the standard investor, more precisey, she can invest in two types of financia assets. The first one is a risk-free bond with stricty positive vaues. We choose it as the numéraire and assume, without oss of generaity that the vaue of this bond equa to 1. The other asset is a risky one which is affected by defaut risk of the counterparty firm on which the insider has extra information. The price of this risky asset is observabe by a investors on market at any time t, T. Since it is subject to the counterparty defaut risk, the price process is modeed by a G-adapted 1 Namey for any t, the function Y i t ( ) is F t B(R +)-measurabe. 4

5 process S, which admits the decomposition form (3.1) S t = S t 1 t<τ + S 1 t (τ)1 t τ, t T where S is F-adapted and S 1 ( ) is F B(R + )-adapted. We suppose that the asset suffers a contagious oss at the defaut time of the counterparty, that is, S 1 θ (θ) = S θ (1 γ θ), and that < γ θ < 1 for any θ so that the asset price remains stricty positive. The process γ is F-adapted and represents the proportiona oss at defaut. We consider the trading strategy of the insider, who chooses to adjust the portfoio of assets according to information accessibiity. Therefore, the investment strategy process is characterized by a G M -predictabe process π which represents the proportion of weath invested in the risky asset and is of the form π t = 1 t τ π t (L) + 1 t>τ π 1 t (τ), where π ( ) and π 1 ( ) are P(F) B(R + )-measurabe processes. Starting from an initia weath X R +, the tota weath of the insider s portfoio is then a G M -adapted process given by (3.2) X t = 1 t<τ X t (L) + 1 t τ X 1 t (τ) where the before-defaut weath process satisfies the sef-financing equation (3.3) dx t (L) = X t (L)π t (L) ds t S t, t T and after the defaut τ, the weath process has a change of regime in its dynamics and satisfies (3.4) dxt 1 (τ) = Xt 1 (τ)πt 1 (τ) ds1 t (τ) St 1, t τ, T. (τ) At the defaut time, the weath has a jump in its vaue. Therefore, at time τ, the initia vaue of the after-defaut weath process is (3.5) X 1 τ (τ) = X τ (L) ( 1 π τ (L)γ τ ). We suppose that π τ (L)γ τ < 1, so that the weath remains stricty positive after the jump due to the counterparty defaut. We consider the foowing dynamics for the asset price S on before-defaut set {t < τ} for S and on after-defaut set {t τ} for S 1 : ds t = S t (µ t dt + σ t dw t ), t T ds 1 t (θ) = S 1 t (θ)(µ 1 t (θ)dt + σ 1 t (θ)dw t ), θ t T where the coefficients µ and σ are F-adapted processes, µ 1 (θ) and σ 1 (θ) are F B(R + )-adapted processes, and W is an F-Brownian motion. In addition, we suppose the integrabiity condition T µ t σt 2 dt + T θ µ1 t (θ) 2 dt + σt 1(θ) T 5 σ t 2 dt + T θ σ 1 t (θ) 2 dt <.

6 So the vaues of the before-defaut and after-defaut weath satisfy the dynamics (3.6) (3.7) dx t (L) = X t (L)π t (L)(µ t dt + σ t dw t ), t T dx 1 t (τ) = X 1 t (τ)π 1 t (τ)(µ 1 t (τ)dt + σ 1 t (τ)dw t ), t τ, T and the jump at defaut of the weath process is given by the equaity (3.5). Finay, we define the admissibe trading strategy famiy A L as the set of pairs (π ( ), π 1 ( )), where π ( ) and π 1 ( ) are P(F) B(R + )-measurabe processes such that ( τ T (3.8) >, T ) πt ()σt 2 dt + πt 1 ( )σt 1 ( ) 2 dt < and πτ ()γ τ < 1, a.s., T where is the F-stopping time defined by := inf{t : Λ t }. Remark 3.1 Let A denote the set of a G M -predictabe processes π such that T π tσ t 2 dt < and π τ γ τ < 1. If (π ( ), π 1 ( )) is an eement in A L, then (π t = π t (L)1 τ t +π 1 t (τ L )1 τ<t, t ) is a processus in the set A. Conversy, given a process π A, there exists a pair (π ( ), π 1 ( )) A L such that π t = π t (L)1 τ t + π 1 t (τ L )1 τ<t for any t, thanks to Lemma The optimization probem The insider has the objective to maximize her expected utiity function on the termina weath of the portfoio. Let U be a utiity function defined on (, + ), stricty increasing, stricty concave and of cass C 1 on (, + ), and satisfying im x + U (x) = + and im x U (x) =. We sha consider the probem (3.9) V = sup π A L EU(X T ) and search for the optima strategy π for the insider. A simiar probem has been studied in 14 for a standard investor with G-predictabe strategy π = (π, π 1 ( )) and G-adapted weath X = (X, X 1 ( )). The admissibe strategy set A consists of pairs π = (π, π 1 ( )) where π and π 1 ( ) are respectivey F-predictabe and P(F) B(R + )-measurabe, and such that θ R +, T T π t σt 2 dt + π 1 t (θ)σt 1 (θ) 2 dt < and π θ γ θ < 1, a.s. θ In this paper, we concentrate on the insider s optimization probem (3.9) and we are interested in the information fow impact on the trading strategies. Intuitivey, the insider shoud have a arger gain of investment due to the extra information. Indeed, if the investor and the insider have the same utiity function and if they can invest in the same financia assets, then the ony difference between them reies on their avaiabe information in the sense that the corresponding fitrations satisfy G G M, which impies the same incusion for the sets of admissibe strategies: A A L. Thus the corresponding vaue functions satisfy (3.1) V := sup EU(X T ) V. π A 6

7 since the supremum in V is taken on a smaer set than the one in V. Remark here that on a given sampe path, it may happen that U(X T ) > U(X T ) if both investor and insider foow their optima strategies π and π respectivey, but one aways have in expectation for optima strategies EU(X π T ) EU(X π T ). as The expectation EU(X T ) can be written, by using the weath decomposition formua (3.2), (3.11) EU(X T ) = E1 T <τ U(X T (L)) + 1 T τ U(X 1 T (τ)). The aim of the above formuation, simiar as in 14, is to reduce the initia optimization probem in an incompete market into two probems : the after-defaut and before-defaut ones. Nevertheess, the approach we adopt here is different since, as mentioned previousy, the random time τ is not a totay inaccessibe random time for the insider and we can no onger use the conditiona density approach to sove the probem. Our approach wi use the theory of initia enargement of fitration (aso caed the strong information modeing in 2) by using the vaue of the random defaut barrier L known to the insider. More precisey, we introduce a famiy of F-stopping times = inf{t : Λ t } for a > which are possibe reaizations of L and we work under an equivaent probabiity measure P L under which L is independent to F T. Thus in our framework, we sha need the Radon-Nikodym derivative process p t (L) which is the density of the historica probabiity measure P with respect to this equivaent probabiity measure P L and it wi pay a simiar roe as the defaut density process in 14. This probabiity density hypothesis is given beow. It is a standard hypothesis in the theory of initia enargement of fitration due to Jacod 11, 12. Assumption 3.1 We assume that L is an A-measurabe random variabe with vaues in, +, which satisfies the assumption : P(L F t )(ω) P(L ), t, T, P a.s.. We denote by P L t (ω, dx) a reguar version of the conditiona aw of L given F t and by P L the aw of L (under the probabiity P). According to 12, there exists a measurabe version of the conditiona density (3.12) p t (x)(ω) = dp t L (ω, x) dp L which is an positive (F, P)-martingae and hence can be written as (3.13) p t (x) = p (x) + t p s (x)ρ s (x)dw s, x, +, t, T for some F-predictabe process (ρ t (x)) t,t. It is proved in 7 that Assumption 3.1 is satisfied if and ony if there exists a probabiity measure equivaent to P and under which F T and σ(l) are independent. Among such equivaent probabiity measures, the probabiity P L defined by the Radon-Nikodym derivative process dp E P L G M dp L t = p t (L) 7

8 is the ony one that is identica to P on F. For exampes of L and expicit computations of corresponding p t (L), interest reader may refer to 1. We reca that we consider the optimization probem (3.9) V = sup π AL EU(X T ). Since the initia σ-fied is non-trivia, it is usefu to consider the conditiona optimization probem (3.14) ess sup π A L EU(X T ) G M, where G M = σ(l). The ink between those two optimization probems (3.9) and (3.14) is given by 1 : if the supremum in (3.14) is attained by some strategy in A L, then the ω-wise optimum is aso a soution to (3.9). Athough the supremum is not necessariy attained in our probem, we wi see in Proposition 4.12 that there exists a sequence of admissibe strategies π n such that EU(X πn T ) GM converges in L1 to (3.14) and we can prove that for the same sequence, EU(X πn T ) converges to V. Thus, we wi first sove the optimization probem (3.14), and then deduce the soution of (3.9) by taking the expectation, as expained in the foowing Proposition: Proposition 3.2 Under the Assumption 3.1, one has EU(X T ) G M = E p T () ( 1 T <τ U(XT ()) + 1 T τ U(XT 1 ( )) ) =L EU(X T ) = E p T () ( 1 T <τ U(XT ()) + 1 T τ U(XT 1 ( )) ) P L (d) R + where for >, := inf{t : Λ t }. Proof: We wi use the change of probabiity to P L in order to reduce to the case where L and F T are independent. Firsty, by (3.11), EU(X T ) G M = E 1 T <τ U(XT (L)) + 1 T τ U(XT 1 (τ)) L = E P L pt () ( 1 T <τ U(XT ()) + 1 T τ U(XT 1 ( )) ) =L = E p T () ( 1 T <τ U(X T ()) + 1 T τ U(X 1 T ( )) ) =L where the ast two equaities foow respectivey from the facts that F and σ(l) are independent under P L and that P L is identica to P on F T. Thus EU(X T ) = E E 1 T <τ U(XT (L)) + 1 T τ U(XT 1 (τ)) L = E p T () ( 1 T <τ U(XT ()) + 1 T τ U(XT 1 ( )) ) P L (d). R + This motivates to introduce, for any >, the set A of pairs π = (π, π 1 ( )), where π and π 1 ( ) are respectivey F-predictabe and P(F) B(R + )-measurabe processes, such that (3.15) τ T π t σ t 2 dt + T T and consider the foowing optimization probem π 1 t ( )σ 1 t ( ) 2 dt < and π γ τ < 1, a.s. (3.16) V () = sup π A E p T () ( 1 T <τ U(X T ) + 1 T τ U(X 1 T ( )) ), where is the F-stopping time inf{t : Λ t }. 8

9 3.3 The necessity of imiting short seing for the insider before defaut In this subsection, we show that if the reguators do not impose any constraint on short seing for the insider before the defaut of the counterparty, then the insider can achieve a termina weath that is not bounded in L 1. Proposition 3.3 We suppose that the foowing conditions are satisfied: (1) the process Λ is a.s. stricty increasing on, T, (2) for any in the support of the distribution of the aw of L, one has P(Λ T ) >. Then we have ess sup EX T G M = + a.s. π A L In addition, for any utiity function U such that im U(x) = +, x + ess sup EU(X T ) G M = + a.s. π A L Proof: Let ϕ :, +, + be an increasing function such that ϕ() < for any, +. Let ψ > be a constant. For each, +, we define a strategy π() = (π (), π 1 ( )) A as foows π t () = ψ1 τϕ() <t, π 1 ( ). Note that (π ( ), π 1 ( )) is an admissibe strategy in A L. The vaue at the time of the corresponding weath process X ϕ,ψ () is equa to (1 + γ τ ψ)x ϕ,ψ. By the dynamics of the weath τ process (3.6) and (3.7), on { T }, we have ( ) Moreover, X ϕ,ψ = X (1 + γ τ ψ) exp τ τ ϕ() ( µ t ψ (σ t ) 2 ψ 2) dt τ τ ϕ() σ t ψdw t EX T G M = E p T () ( 1 T <τ XT () + 1 T τ XT 1 ( ) ) =L E 1 T τ p T ()Xτ ϕ,ψ Now fix an increasing sequence (ϕ n ) n 1 of functions such that ϕ n () < for, and that im n + ϕ n () =. By the condition (1), one obtains that τ ϕn() converges a.s. to when n +. The sequence of random variabes T τ ϕn() T σ t ψ dw t, n 1 converges a.s. to. Then by Fatou s emma im inf E1 T τ p T ()X ϕn,ψ n + E1 T τ p T ()X (1 + γ τ ψ) which impies the first assertion since P( T ) > by condition (2). argument and the assumption on U to obtain im ψ + E1 T τ p T ()U(X (1 + γ τ ψ)) =L =.. =L We use the simiar 9

10 Remark 3.4 The strategies mentionned in this proof are not arbitrage strategies because for any fixed function ϕ as in the proof, P(T τ ϕ(l), τ ) > and on this event, the strategy of the insider that consists of betting on the defaut before maturity T turns out to be a wrong bet. Thus, on a non nu probabiity set, the strategy of a standard investor outperforms the one of the insiders, athough the converse inequaity hods on expectation. Proposition 3.3 justifies to consider, instead of A L defined in (3.8), the foowing admissibe trading strategy sets for the insider : for a δ, (3.17) A δ L = {(π ( ), π 1 ( )) A L such that π δ} and to quantify the impact of the ower bound δ. Simiar to (3.15), we define A δ = {(π, π 1 ( )) A such that π δ}. Before considering the main optimization probem (3.18) ess sup EU(X T ) G M π A δ L we first consider an aternative famiy of optimization probems depending on the parameter, +, (3.19) V δ () := sup E p T () ( 1 T <τ U(XT ()) + 1 T τ U(XT 1 ( )) ). π A δ The foowing theorem shows that the optima vaue of the optimization probem (3.18) is actuay equa to V δ (L). Theorem 3.5 With the above notation, we have V δ (L) = ess sup EU(X T ) G M π A δ L a.s. Proof: 3.2 we obtain that Assume that (π ( ), π 1 ( )) is an eement in A δ L, then (π (), π 1 ( )) A δ. By Proposition ess sup EU(X T ) G M V δ (L). π A δ L For the converse inequaity, we sha use measurabe seection theorem. For any ε > and any,, et F ε () be the set of strategies (π, π 1 ( )) A δ which are ε-optima with respect to the probem (3.19), namey such that E p T () ( 1 T <τ U(X T ()) + 1 T τ U(X 1 T ( )) ) { V δ 1/ε, () ε, if V δ () < +, if V δ () = +. By a measurabe seection theorem (cf. Benes 3, Lemma 1), there exists a measurabe (with respect to ) famiy {(π (), π 1 (, ))} R+ such that (π (), π 1 (, )) F ε () for any >. Finay et π ( ) := π ( ), π 1 t (x) := 1 t>x π 1 t (x, Λ x ). 1

11 We have ( π ( ), π 1 ( )) A L and π 1 t ( ) = π 1 t (, ) for any > on { < t}. Therefore, by Proposition 3.2, EU( X T ) G M = E p T () ( 1 T <τ U(XT ()) + 1 T τ U(XT 1 ( )) ) =L { V δ (L) ε, if V δ (L) < +, 1/ε, if V δ (L) = +. Since ε is arbitrary, we obtain EU( X T ) G M V δ (L). 4 Soving the optimization probems In this section, we concentrate on soving the optimization probem (3.19) sup E p T () ( 1 T <τ U(XT ()) + 1 T τ U(XT 1 ( )) ) π A δ for any fixed. We reca that the before-defaut and after-defaut weath processes X and X 1 are governed by two contro processes π and π 1 respectivey, so we need to search for a coupe of optima contros ˆπ = (ˆπ, ˆπ 1 ). In Theorem 4.1 we expain how to decompose the optimization probem into two probems each depending ony on π and on π 1 respectivey. The after-defaut optimization probem wi be soved firsty, using the after-defaut fitration F 1 F 1 := (F τ t) t,t. Remark that the initia σ-fied of the fitration F 1 is not trivia : F 1 = F τ = {A A : A { t} F t, t, T } and is F τ -measurabe. A the F 1 -adapted processes invoved in the after-defaut optimization probem are indexed on the right-upper side by the symbo 1. In particuar, we denote by X 1,x ( ) the soution of the SDE (3.7) defined on the stochastic interva, T starting from the F-stopping time with F τ -measurabe initia vaue x. We define by A 1 the admissibe predictabe strategy set (πt 1 ( ), t, T ) such that T πt 1 ( )σt 1 ( ) 2 dt < a.s.. The goba before defaut optimization probem invoves the soution of the after-defaut optimization probem and wi be soved in a second step, using the stopped fitration F F := (F τ t) t,t. A the F -adapted processes invoved in the after-defaut optimization probem are indexed on the right-upper side by the symbo. The admissibe predictabe strategy set A,δ is (πt (), t, T ) such that T πt ()σt 2 dt <, πt () δ and 1 > πτ ()γ τ, a.s.. 11

12 Theorem 4.1 For any >, we denote by V 1 (x ) the optima vaue of the after-defaut optimization probem (4.1) Vτ 1 (x ) = ess sup Ep T ()U(X 1,x T ( )) F τ, π 1 ( ) A 1 then the goba optima vaue V δ () defined in (3.19) can be written as the soution of a beforedefaut optimization probem as (4.2) V δ () = sup E 1 T <τ p T ()U(XT ()) + 1 T τ V 1 ( X τ ()(1 πτ ()γ τ ) ). π A,δ Proof: Consider firsty an arbitrary admissibe strategy (π, π 1 ( )) A δ for a fixed >. By definition, (πt, t, T ) A,δ and (πt 1 ( ), t, T ) A 1. Taking the conditiona expectation with respect to F τ eads to the foowing inequaities E p T () ( 1 T <τ U(XT ()) + 1 T τ U(XT 1 ( )) ) = E 1 T <τ p T ()U(X T ()) + 1 T τ E ( p T ()U(X 1 T ( )) F τ ) E 1 T <τ p T ()U(X T ()) + 1 T τ V 1 (X ()(1 π ()γ τ )) V δ (). For the converse inequaity, et us assume for the moment that the esssup in the definition (4.1) is achieved for a given ˆπ 1 ( ) (see section 4.1 for the proof). Then for any (π t, t, ) in A,δ, by a measurabe seection theorem, there exists a P(F) B(R + )-measurabe process π 1 ( ) such that π 1 ( ) = ˆπ 1 ( ) on, T and that (π, π 1 ( )) A δ, where we have extended π to an F-predictabe process on R +. Thus V δ () E 1 T <τ p T ()U(X T ()) + 1 T τ V 1 (X ()(1 π ()γ τ )) By taking the supremum over a (π t (), t, ) A,δ, we obtain the desired inequaity. Remark 4.2 The supremum in V δ () can be approached by a sequence of admissibe strategies in A,δ (see Proposition 4.12), which induces a sequence of strategies in A δ L such that the corresponding vaue functions converge to V δ := sup π A δ EU(X T ). Thus V δ = L R + V δ()p L (d). Remark 4.3 The process (p t (), t, T ) is essentia in our approach of initia information and pays a simiar roe to the defaut density process in 14 (α t (θ), t, T ) defined as α t (θ)dθ = P(τ dθ F t ). Thus, to quantify the gain of the insider (as in the forthcoming Proposition 4.5), it is interesting to compare those two processes. In the particuar case where the F t -conditiona aw of L admits a density g t () w.r.t. the Lebesgue measure, the defaut density can be competey deduced (see 6, Proposition 3) in this framework as α t (θ) = λ θ g t (Λ θ ), t θ and α t (θ) = Eα θ (θ) F t for t θ where λ is the process given in Section 2. In the genera case, the aw of L can have atoms, then the defaut density does not exist and the approach in 14 is no onger vaid, whereas the insider s optimization probem can be soved with the process p. (). 12

13 4.1 The after-defaut optimization In this section, we focus on the after defaut optimization probem (4.1) in the fitration F 1 = (F τ t) t,t Vτ 1 (x ) = ess sup Ep T ()U(X 1,x T ( )) F τ π 1 ( ) A 1 where is an F stopping time and the initia weath x is F τ -measurabe. This probem is simiar to a standard optimization probem, we wi extend the resuts in our framework where the initia time is a random time (and is an F-stopping time). We define the process ( τ t µ 1 Z t ( ) = exp u( ) σu(τ 1 ) dw u 1 2 τ t µ 1 u( ) σu(τ 1 ) 2 ) du, t, T. This process is an F 1 -oca martingae (cf. 15, page 2), we assume that the coefficients µ 1 ( ) and σ 1 ( ) satisfy a Novikov criterion (see Theorem 4.4 beow) so that (Z t ( )) t,t is an F 1 - martingae. Theorem 4.4 We assume that for any, the coefficients µ 1 u( ) and σu(τ 1 ) satisfy the Novikov criterion ( 1 τ T E exp µ 1 2 ) u( ) 2 σu(τ 1 ) du <. Then the vaue function process to probem (4.1) is a.s. finite and is given by ( 1 (4.3) ˆV τ (x ) = E p T ()U I ( ŷ τ (x ) Z T ( )) ) F 1 p T () where I = (U ) 1 and the Lagrange mutipier ŷ τ ( ) is the unique F τ B(R + )-measurabe soution of the equation 1 Z t ( ) E Z T ( )I ( ŷ τ (x ) Z T ( )) F 1 = x. p T () The corresponding optima weath is equa to (4.4) ˆX1,x t ( ) = 1 Z t ( ) E Z T ( )I ( ŷ τ (x ) Z T ( )) Ft 1 p T (), t, T. Proof: Note that after the defaut, the market is compete. The process (Z t ( )X 1,x t ( )) t,t is a positive oca F 1 -martingae, and thus a supermartingae, eading to the foowing budget constraint : ( ) (4.5) E Z T ( )X 1,x T ( ) F 1 x. Conversey, the martingae representation theorem on the brownian fitration impies that for any F T B(R + )-measurabe X T ( ), there exists a P(F) B(R + )- measurabe process φ( ) such that τ T X T ( )1 τ <T = E(X T ( )1 τ <T F τ ) + φ u ( )du 13

14 Therefore the after defaut optimization probem is soved by the mean of the Lagrange mutipier ( Vτ 1 (x ) = E p T ()U I ( ŷ τ (x ) Z T ( )) ) F 1 p T () and the optima weath is given by ˆX 1,x t ( ) = 1 Z t ( ) E Z T ( )I ( ŷ τ (x ) Z T ( ) p T () ) F 1 t where I = (U ) 1 and the Lagrange mutipier ŷ τ (x ) is F τ B(R + )-measurabe and satisfies ˆX 1,x ( ) = x. The existence, uniqueness and measurabiity of the Lagrange mutipier ŷ τ (x ) in the case of a non trivia initia σ-fied is proved in Proposition 4.5 of Hiairet 8. We can aready give a first anaytica comparison of the vaue function process of the after defaut optimization probem between the initia information and the progressive information. We reca (see 14, Theorem 4.1) that for the progressive information, this vaue function process at the defaut time θ = is ( V τ 1 (x) = E α T ( )U I ( ȳ τ (x) Z T ( )) ) (4.6) F τ α T ( ) ZT ( ) with y τ such that E Z t ( ) I( ȳ τ (x) Z T ( )) F τ = x α T ( ) which has to be compared to the vaue function (4.3) for an initia information. To do this, we assume that the F t -conditiona aw of L admits a density with respect to the Lebesgue measure, denoted as g t () (see Remark 4.3). Using Assumption 3.1, it is sufficient to assume that the aw of L admits a density with respect to the Lebesgue measure. Proposition 4.5 The vaue function processes of the after defaut optimization probem for the initia information ˆV τ 1 (x) and for the progressive information V τ 1 (x) satisfy V 1 (x) ˆV 1 (x) = g ()λ τ where g (.) denotes the density of the aw of L with respect to the Lebesgue measure and is supposed to be stricty positive. In addition, the ratio α T ( ) p T () is F τ -measurabe and is aso equa to V τ 1 (x) ˆV. τ 1 (x) Proof: On the stochastic interva, T, the density of probabiity Z t ( ) is the same for the initia and the progressive information, but not necessariy the Lagrange mutipiers, athough they satisfy the same type of equation. More precisey, ȳ τ (x) is the unique F τ B(R + )- measurabe soution of ZT ( ) (4.7) E Z t ( ) I( ȳ τ (x) Z T ( )) Fτ = x α T ( ) 14

15 and ŷ τ (x) is the unique F τ B(R + )-measurabe soution of ZT ( ) (4.8) E Z t ( ) I( ŷ τ (x) Z T ( )) Fτ = x. p T () We reca that g T () is the density of the F T -conditiona aw of L with respect to the Lebesgue measure. According to (3.12) and Remark 4.3, p T () = g T () g () and α T ( ) = λ τ g T (Λ τ ) = λ τ g T (). Thus the ratio α T ( ) p T () = λ τ g () is F τ -measurabe. Furthermore, (4.7) is equivaent to ZT ( ) E Z t ( ) I( ȳ τ (x) p T () α T ( ) Z T ( )) Fτ = x p T () so the unicity of the F τ -measurabe soution of the equation (4.8) impies that the ȳ τ (x) p T () α T (τ) = ŷ τ (x). Therefore, ȳ (x) ŷ τ (x) = λ g () and U ( I ( ȳ τ (x) Z T ( ))) ( α T ( ) = U I (ŷτ (x) Z T ( ))) p T (). We concude the proof by using again that the ratio α T ( ) p T () = λ τ g () is F τ -measurabe. 1 Remark The constant g () is a threshod to compare with λ. For a given scenario, if λ τ is smaer than 1 g (), which is often the case in practice, then the absoute vaue of the vaue function for the initia information is greater than the one for the progressive information. In this case, the insider puts a higher weight p T () on the after defautoptimization probem (4.1), compared to a standard investor whose weight is α T ( ). This means that p t () conveys more information than α t ( ). Moreover, the smaer g () is, the greater is the gain of an insider. The interpretation is as foows: sma vaue of g () means that P(L d) is sma, thus the insider, who knows the rea vaue of L, has very reevant information. 2. Concerning the optima weath ˆX 1 ( ) in (4.4), we can prove, using the same argument as in the proof of Proposition 4.5 that starting from a same weath x at the defaut time, the optima weath process of the after-defaut optimization probem is the same for the initia and the progressive information, i.e. ˆX1,x ( ) = X 1,x ( ). This resut is not surprising, since after the defaut, the two information fows coincide. But naturay the input weath of the after-defaut optimization probem wi not be the same for the two information fows since they are not the same before. We wi quantify numericay the gain of an insider compared to an standard investor for the goba optimization probem. We now consider, as in 14, Constant Reative Risk Aversion (CRRA) utiity functions U(x) = xp p, < p < 1, x > and I(x) = x 1 p 1. Direct computations from the previous theorem yied the optima weath ˆX 1,x t ( ) = x Z t ( ) E E p T () p T () ( ) p ZT ( ) p 1 p T () ( ZT ( ) p T () ) p p 1 Ft 1, t, T F τ 15

16 and the optima vaue function (4.9) ˆV 1 τ (x ) = xp p ( E p T () ( ) p ZT ( ) p 1 p T () F ) 1 p =: xp p K ony de- ( ( ) p where the F τ -measurabe random variabe K τ = E p T () ZT ( ) p 1 p T () pends on the stopping time and on market parameters. ) 1 p F 4.2 The before-defaut optimization We now consider the optimization probem (4.2) with CRRA utiity functions. Using (4.9), we have to sove : (4.1) V δ () = sup E 1 T <τ p T ()U(X T ()) + 1 T τ K τ U ( X τ ()(1 π τ ()γ τ ) ) π A,δ where the F τ -measurabe random variabe K τ does not depend on the contro process π A,δ. We wi use a dynamic programming approach. Reca that F = (F τ t) t,t is the stopped fitration at the defaut time. Since 1 t<τ is F τ t-measurabe, we have E 1 T <τ p T ()U(XT ()) + 1 T τ K τ U ( Xτ ()(1 πτ ()γ τ ) ) F τ t = 1 t τ K τ U ( Xτ ()(1 πτ ()γ τ ) ) + E 1 T <τ p T ()U(XT ()) + 1 t<τ T K τ U ( Xτ ()(1 πτ ()γ τ ) ) F τ t For any ν A,δ, we introduce the famiy of F -adapted processes (4.11) X t (ν) := ess sup E π A,δ (t,ν) 1 T <τ p T ()U(X T ()) + 1 t<τ T K τ U ( X ()(1 π ()γ τ ) ) F τ t where A,δ (t, ν) is the set of contros coinciding with ν unti time t: for any t, T, ν A,δ, A (t, ν) = {π A,δ : π. t = ν. t }, X ν, denotes the weath process derived from the contro ν A,δ. We have V δ () = X (ν) for any ν A,δ. In the foowing resut, we show that the short seing constraint pays a crucia roe in the optimization. In fact, it is the optima strategy at the defaut time. Proposition 4.7 For any π A,δ, there exists a sequence of strategies (πn A,δ ) n N such that πn,τ = δ and im X (π n + n) X (π ). 16

17 Proof: Let (τ n ) n N where τ n < be a increasing sequence of F stopping times that converge to. Starting from a strategy π A,δ, we define another strategy πn = 1,τnπ 1 τn,τ δ that remains in A,δ. We denotes as X () and Xn() the corresponding weath before defaut, and as X (π ) and X (πn) the corresponding vaue function for those strategies of the before defaut goba optimization probem. On the one hand, by dominated convergence theorem, it is easy to check that On the other hand, on the event {T } X n, ()(1 π n, ()γ τ ) X ()(1 π ()γ τ ) π A,δ im E 1 T <τ p T () U(X n + T ()) U(Xn,T ()) =. = (1 + δγ ( ) τ (1 πτ ()γ τ ) exp ( (πs + δ)µ s + 1 τ ) τ n 2 (σ s) 2 ((πs) 2 δ 2 ))ds (πs + δ)σsdw s τ n impies that π () δ and (1+δγ τ ) (1 π ()γ τ ) 1 (because γ > ). τ n impies that the exponentia term tends to 1 a.s. Thus im n + X n, ()(1 π n, ()γ τ ) X ()(1 π ()γ τ ) 1 and im E 1 T τ K τ U(X n + n, ()(1 πn,τ ()γ τ )) E 1 T τ K τ U(Xτ ()(1 πτ ()γ τ )). Consequenty, im X (π n + n) X (π ). We now characterize the optima strategy process. From the dynamic programming principe, the foowing resut hods: Lemma 4.8 For any ν A,δ, the process ξ ν t := X t (µ) + 1 t τ K τ U(X ν, ()(1 ν τ ()γ τ )), t T is an F -supermartingae. Furthermore, the optima strategy π is characterized by the martingae property : (ξ π t ) t T is an F -martingae. Proof: Let s, t be two times such that s t T. E X t (µ) + 1 t τ K τ U(Xτ ν, ()(1 ν τ ()γ τ )) F τ s = E X t (µ) + 1 s<τ tk τ U(Xτ ν, ()(1 ν τ ()γ τ )) F τ s + 1 s τ K τ U(Xτ ν, ()(1 ν τ ()γ τ )) 17

18 We make expicit the conditiona expectation : E X t (µ) + 1 s<τ tk τ U(Xτ ν, ()(1 ν τ ()γ τ )) F τ s = E ess sup (4.12) π A,δ (t,ν) ess sup π A,δ (s,ν) E 1 T <τ p T ()U(XT ()) + 1 t<τ T K τ U(Xτ ()(1 πτ ()γ τ )) F τ t + 1 s<τ tk τ U(Xτ ν, ()(1 ν τ ()γ τ )) F τ s E 1 T <τ p T ()U(XT ()) + 1 s<τ T K τ U(Xτ ()(1 πτ ()γ τ )) F τ s the ast inequaity foowing from the fact that in the ast esssup, the optima contro is taken from the date s t. Thus E X t (µ) + 1 t τ K τ U(Xτ ν, ()(1 ν τ ()γ τ )) F τ s X s (µ) + 1 s τ K τ U(Xτ ν, ()(1 ν τ ()γ τ )) and (ξt ν ) t T is an F -supermartingae. It is an F -martingae if and ony if the inequaity (4.12) is an equaity for a t, T, meaning that ν is the optima contro on, t, for a t T. This characterizes the optima strategy. Remark that the F -adapted process (4.13) Y t := X t(µ) U(X ν, t ()) ( X = ess sup E 1 T <τ p T () T () ) p ( X τ () ) p(1 + X ν, 1t<τ T K τ π t () X ν, τ t () ()γ ) F τ t π A,δ (s,ν) t T does not depend on ν A,δ, and is constant after. We wi give a characterization of the process Y in terms of a backward stochastic differentia equation (BSDE) and of the optima strategy. Before this, we give a characterization of F -martingae. Lemma 4.9 Let (M t ) t,t be an F -martingae. Then there exists an F-predictabe process φ in L 2 oc (W ) such that M t = M + t φ s1 s τ dw s, t, T. Proof: We first prove that (M t ) t,t is aso an F-martingae. Indeed, for s t T M s = E(M t F τ s) = E(E(M t F τ ) F s ) = E(M t F s ) because M t is F τ -measurabe. Thus, by representation theorem for the F-martingae, and since (M t ) t,t is stopped at time, there exists φ an F-predictabe process such that M t = M + t φ s1 s τ dw s. We are now ready to characterize the optima strategy. Remark that Y t = on, (and zero after ) thus Y L + (F ) where Xt(µ) U(X ν, t ()) is positive L + (F ) := {Ỹ : F -adapted processes s.t. Ỹ t > for t, and Ỹt = for t, }. 18

19 Theorem 4.1 The process Y defined in (4.13) is the smaest soution in L + (F ) to the BSDE (4.14) (1 + δγ τ ) p T τ T τ Y t = 1 T <τ p T () + 1 t<τ T K τ + f(θ, Y θ, φ θ )dθ φ θ dw θ, t, T, p t t for some φ L 2 oc (W ), and where (4.15) f(s, Y s, φ s ) = p ess sup ν A,δ,s.t. ν τ = δ (µ s Y s + σsφ s )ν s 1 p 2 Y s ν s σs 2 The optima vaue function is attained as the imit of a sequence of admissibe strategies (ˆπ t ) t,t τ converging to the supremum in (4.15). Remark 4.11 As in Theorem 4.2 in 14, the optima strategy before defaut is characterized through the optimization of the driver of a BSDE. However, the main difference reies in the fact that in our case, the driver has a jump at the defaut time. Nevertheess, since the jump occurs (if it occurs) ony at the termina date of the BSDE, standard theory on BSDE sti appy. Proof: By Lemma 4.8, for any ν A,δ ξt ν = U(X ν, t ())Y t + 1 t τ K τ U(X ν, ()(1 ν τ ()γ τ ) t T is an F -supermartingae. In particuar, by taking ν =, we see that the processes (Y t + K τ 1 t τ ) t T, and thus (Y t ) t T are F -supermartingaes. By the Doob-Meyer decomposition and Lemma 4.9, there exists φ L 2 oc (W ), and a finite variation increasing F -predictabe process A such that: (4.16) dy t = φ t dw t da t, t, T. From Itô s formua, we deduce that the finite variation process in the decomposition of the F -supermartingae ξ ν, ν A,δ, is given by A ν with da ν t = (X ν, t ()) p{ 1 p da t (µ t Y t + σt φ t 1 t τ )ν t dt 1 p 2 Y t ν t σt 2 (1 ν t γ t ) p dt K t d1 t τ }. p A ν is nondecreasing and the martingae property of ξˆπ and impies that da t = p (µ t Y t + σt φ t 1 t τ ))ˆπ t dt 1 p 2 Y t ˆπ t σt 2 (1 ˆπ t γ t ) p dt + K t d1 t τ p A t = A + p ess sup ν A,δ t ( (µ s Y s + σsφ s )ν s 1 p 2 Y s ν s σs 2) (1 ν τ γ τ ) p ds + 1 t τ K τ. p Maximizing at eads to ν τ = δ (see Proposition 4.7) and A t = A + p ess sup ν A,δ s.t. ν τ = δ t ( µ s Y s + σsφ s )ν s 1 p 2 Y s ν s σs 2) (1 + δγ τ ) p ds + 1 t τ K τ. p 19

20 Furthermore, Y T = 1 T <τ p T () and (Y t ) t T is constant after, thus (Y, φ) soves the BSDE (4.14). Note that Y is not a continuous process, it may jump at time. We now prove that Y is smaest soution in the L + (F ) to the BSDE (4.14). Let Ỹ L+ (F ) be another soution, and we define the famiy of nonnegative F -adapted processes ξ ν (Ỹ ), ν A,δ, as ξt ν (Ỹ ) = U(Xν, t ())Ỹt + 1 t τ K τ U(Xτ ν, ()(1 ν τ ()γ τ )), t, T. By simiar cacuations as above, dξt ν (Ỹ ) = d M t ν dãν t, where Ãν is a nondecreasing F -adapted process, and M ν is a F - oca martingae. By Fatou s emma, this impies that the process ξ ν (Ỹ ) is a F -supermartingae, for any ν A,δ. Since ỸT = 1 T <τ p T (), we deduce that for a ν A,δ E U(X ν, T )1 T <τ p T () + 1 t τ K τ U(Xτ ν, ()(1 ν τ ()γ τ ) Ft U(X ν, t )Ỹt, t, T. Since p >, U(X ν, t ) is positive. By dividing the above inequaities by U(X ν, t ), we deduce by definition of Y (see (4.13)), and arbitrariness of ν A,δ, that Y t Ỹt, t T. This shows that Y is the smaest soution to the BSDE (4.14). For optimizing (4.13) via the BSDE (4.14), a naive approach wi consist in optimizing π at time, eading to an π = δ, and then optimizing for s < the driver f (s, Ys, φ s) = ess sup p ( µ sys + σsφ s)ν s 1 p 2 Y s ν s σs 2 where Y is soution to the BSDE Y t ν s δ = 1 T <τ p T ()+1 t<τ T K τ (1 + δγ τ ) p p T τ T τ + f (θ, Y θ, φ θ )dθ φ θ dw θ, t, T, t t eading to the optima portfoio ˆπ s. Thus, the natura candidate to be the optima strategy before defaut is (4.17) π np := 1,τ ˆπ δ1 τ, but unfortunatey π np is not a predictabe process. Nevertheess, we wi prove the existence of a sequence of predictabe strategies in A,δ such that the corresponding vaue function tends to the vaue function reative to this non predictabe strategy. To do this, for any strategy π A,δ, we reca the corresponding vaue function of the before defaut goba optimization probem (4.18) X (π ) = E 1 T <τ p T ()U(XT π ()) + 1 T τ K τ U(Xτ π ()(1 πτ ()γ τ )). Note that (4.18) can aso be defined for a strategy π that is predictabe ony on, (and not necessary on, ). Using Proposition 4.7, we have the foowing resut: Proposition 4.12 The strategies (πn = 1,τnˆπ 1 τn,τ δ) where ˆπ is the optima process for the driver of the BSDE Yt (1 + δγ τ ) p T τ T τ = 1 T <τ p T ()+1 t<τ T K τ + f (θ, Y θ, φ θ )dθ φ θ p dw θ, t, T, t t 2

21 are in A,δ and satisfy im X (π n + n) = V () = E (µ f (s, y, φ) = p sup s y + σsφ)ν 1 p ν δ 2 y νσ s 2 1 T <τ p T ()U(XT ˆπ ()) + 1 T τ K τ U(Xτ ˆπ ()(1 + δγ τ )). Proof: Let (τ n ) n N be an increasing sequence of F-predictabe stopping times that converge to. For any integer n 1, the strategy πn := 1,τnˆπ 1 τn,τ δ is in A,δ and πn converges to the non-predictabe optima strategy π np defined in (4.17) when n. Moreover, for any n N, X (πn) X (π np ) and by Proposition 4.7 X (π np ) im X (π n + n) V (ˆπ ). But the proof of Proposition 4.7 sti hods if we change the vaue at time of the portfoio π, thus the converse inequaity X (π np ) im n + X (πn) hods and E 1 T <τ p T ()U(XT ˆπ ()) + 1 T τ K τ U(Xτ ˆπ ()(1 + δγ τ )) = X (π np ) = X (1,τ ˆπ δ1 τ ) = im n + X (1,τnˆπ 1 τn, δ) = im n + X (π n) 5 Numerica iustrations We now iustrate our previous resuts by expicit modes and we aim to compare the optimization resuts for an insider and a standard investor. We reca that a investors start with an initia weath X. For the purpose of comparison, we choose a simiar mode with the one studied in 14. More precisey, we et the parameters µ, σ, γ to be constant, and µ 1 (θ), σ 1 (θ) are deterministic functions of θ given by (5.1) µ 1 (θ) = µ θ T, σ1 (θ) = σ (2 θ ), θ, T, T which means that the ratio of the after-defaut and before-defaut for the return rate of the asset is smaer than 1 and for the voatiity is arger than 1. Moreover, these ratios increases or decreases ineary with the defaut time respectivey: the after-defaut rate of return drops to zero, when the defaut time occurs near the initia date, and converges to the before-defaut rate of return, when the defaut time occurs near the finite investment horizon. For the voatiity, this ratio converges to the doube (resp. initia) vaue of the before-defaut voatiity, when the defaut time goes to the initia (resp. termina horizon) time. Moreover, in order to satisfy the hypothesis in the simuation part of 14, we have to assume that the defaut barrier L has no atoms (to ensure the density hypothesis, see Remark 4.3) and that L is independent of the fitration F (so that the defaut density is a deterministic function). In this case, p T (L) = 1. 21

22 Consider the CRRA utiity U(x) = xp p, < p < 1, the after-defaut vaue function is given from (4.9) by Vτ 1 (x) = K τ U(x) where K τ = ) ( (EZ T ( ) p 1 p p 1 1 = exp 2 p 1 p ( µ 1 ( ) ) 2(τ σ 1 T )) ( ) Furthermore, the soution of the before-defaut optimization probem is given by V () = Y U(X ) where Y is the soution of the BSDE (4.14) when etting φ =, i.e. (1 + δγ) p (5.2) Y t = 1 T <τ + 1 T τ K τ + p where f(t, y) = p ess sup ν A,δ,ν τ = δ τ T t f(θ, Y θ )dθ {µ ν t 1 p 2 (ν tσ ) 2 }y. We notice that in the case where the defaut time occurs after the maturity T, the optima strategy coincides with the cassica Merton strategy with constraint π δ, 1 γ. In the case where occurs before T, the process Y is stopped at, with the termina vaue depending on the quantity K τ, and the short-seing strategy δ at. We use an iterative agorithm to sove the equation (5.2). The foowing resuts are based on the mode parameters described beow: µ =.3, σ =.2, T = 1, the risk aversion parameter p =.8. For the standard investor, we use the deterministic mode as in 14 etting P(τ > t) = e λt, λ > so that the density function is α(θ) = λe λθ for a θ. Figure 1 compares the optima vaue function for insider, investor and Merton strategy. We fix the short-seing constraint δ =.5, the oss given defaut γ =.2 and the defaut intensity λ =.3. This corresponds to a reativey high risk of defaut. At the defaut time, the vaue function suffers a bruta oss for a the three strategies. The insider outperforms the other two strategies before and after the defaut occurs. Before the defaut, the vaue function for the standard investor is smaer than the Merton one because the atter does not consider at a the potentia defaut risk. However, when the defaut occurs, the investor outperforms the Merton strategy since the defaut risk is taken into account from the beginning. In Figure 2, we fix a smaer short-seing constraint δ =.1 and we study the impact of the oss given defaut γ. We observe that the vaue functions for both insider and investor are increasing with respect to the oss vaue γ. It is interesting to emphasize this phenomenon as a consequence of the short-seing where both insider and the investor wi profit the defaut event and obtain a arger vaue function. More precisey, in the eft-hand figure, we consider a reativey ow defaut risk (with the defaut intensity λ =.1) and we observe that the gain of the insider with respect to the investor remains stabe in time and aso for different vaues of γ. Whereas in the right-hand figure with a higher defaut risk (λ =.3), the impact of γ is more important for the investor: before the defaut the vaue function of the investor increases more significanty as γ increases, and there is no onger a drop at the defaut time. This can be expained by the fact that the investor has no imit for the short-seing strategy. 22