BSDE with random terminal time under enlarged filtration and financial applications

Transcription

1 BSDE with random terminal time under enlarged filtration and financial applications - Anne EYRAUD-LOISEL Université Lyon 1, Laboratoire SAF - Manuela ROYER Université Lyon 1, LBBE, CNRS, UMR WP 234 Laboratoire SAF 5 Avenue Tony Garnier Lyon cedex 7

2 BSDE with random terminal time under Enlarged Filtration, and Financial Applications Anne EYRAUD-LOISEL I.S.F.A. Université Lyon 1 5 avenue Tony Garnier 697 Lyon FRANCE anne.eyraud@univ-lyon1.fr Manuela ROYER LBBE UMR CNRS 5558 Université Lyon 1 43 Bd du 11 novembre Villeurbanne Cédex FRANCE royermanuela@yahoo.fr Abstract Markets with asymmetrical information have been already studied, generally in a wealth optimization point of view. We study here a hedging problem for a financial agent possessing an additional information on the market. We extend the results given for hedging strategies under a fixed terminal time to the case of a random terminal time. In particular, we provide tools to understand the behavior of American option pricing by an insider. To achieve this aim, we prove the existence and uniqueness of solution of BSDE with random terminal time under enlarged filtration. Keywords : enlargement of filtration, BSDE, uncertain time horizon, American option hedging, insider trading, asymmetrical information. 1

3 1 Introduction We study here existence and uniqueness of solutions of BSDEs with random terminal time under enlarged filtration, motivated by the following financial problem : we study the financial hedging strategy for an American contingent claim in a market with asymmetrical information. BSDEs were first introduced by E. Pardoux and S. Peng in 199 [26]. Such equations are frequently used, and have a large panel of application fields, especially in mathematical finance. They also appear in several cases such as stochastic control see S. Peng [28], N. El Karoui, S. Peng and M.C. Quenez [9], and X. Zhou and J. Yong [3] or problems linked with PDEs see E. Pardoux [24] and G. Barles, R. Buckdahn and E. Pardoux [2]. BSDEs are interesting in our problem, since these equations naturally appear when describing hedging problems. As we study hedging of contingent claims with random exercise time, we model it with BSDEs with random terminal time. Such equations were introduced by S. Peng 1991 [27], and developped by E. Pardoux 1999 [25] and M. Royer 24 [29] among others, and E. Pardoux 1995 [23] for BSDEs with jumps and random terminal time. To model the asymmetrical information in the market we have chosen initial enlargement of filtration, theory developed by J. Jacod [15], T. Jeulin [18] and M. Yor [31], and often used to model insider trading see A. Grorud and M. Pontier [14], who used this model to construct a statistical test to detect insider traders. A. Eyraud-Loisel stated in [11] existence and uniqueness of solutions of BSDE with a constant terminal time under enlargement of filtration. She stated that an insider trader having an additional information on the market about a terminal time T > T and satisfying a standard hypothesis for the enlargement of filtration H 3, has no other strategy for hedging a European option with terminal exercise time T than if he had not the additional information. For this work, the financial motivation still 2

4 comes from hedging problems for insider traders, but we consider here the hedging of contingent claims with random terminal time, such as American options or Lookback options. For this purpose, we have to consider BSDEs with a random terminal time, under an initial enlargement of the Brownian filtration. We prove that A. Eyraud-Loisel s results stated in [11] remain true when the hedging terminal time is a random stopping time. We take standard existence and uniqueness hypotheses, as in E. Pardoux 1999 [25] who studied BSDEs with random terminal time without enlargement of filtration i.e. under the natural Brownian filtration. We obtain existence and uniqueness results for the BSDE with initial enlargement of filtration under hypothesis H 3 for a bounded random terminal time. We first deal with stopping time a.s. bounded by T < T, and we extend the results for all stopping times a.s. strictly bounded by T. This financially means that an agent who has an initial additional information satisfying hypothesis H 3 will have the same hedging strategy as a non informed agent, for a contingent claim with random terminal time, as it was previously proved for a constant terminal time. This result extends results from [11] to more general claims traded on the market hedging of American or Lookback options for instance, and is consistent with the result for a fixed terminal time. It still differs from the results obtained by A. Grorud and M. Pontier who stated that in a wealth optimization point of view, under the same kind of enlargement of filtration, an insider trader has a different strategy from a non informed trader. 3

5 2 Model 2.1 Financial Motivation Let W be a standard d-dimensional Brownian motion, and let Ω, F t t T, IP be a filtered probability space, with Ω = C[, T ]; IR d. Let F t t T be the natural filtration of Brownian motion W. We consider a financial market with d risky assets, whose prices are driven by : S i t = S i + t S i s b i s ds + t S i sσ i s, dw s, t T,, i = 1,..., d and where the bond or riskless asset evolves as : S t = 1 + t S s r s ds. Parameters b, σ, r are supposed to be bounded on [, T ], adapted, and take values respectively in IR d, IR d d, IR. Matrix σ t is invertible dt dip a.s. and the Doléans-Dade exponential E σ 1 b r.w is supposed to be integrable. These are the usual conditions to have existence of a risk-neutral probability which implies no-arbitrage. A financial agent has a positive initial wealth Y at time t =, and he wishes to hedge a contingent claim with terminal payoff ξ at uncertain time horizon τ, with maturity T. τ is a F-stopping time. Generally, τ is not given, it may depend on the option owner s strategic decision. It may in particular be the optimal stopping time determined with the help of the notion of the Snell envelope of the price process see N. El Karoui et al. [8], N. El Karoui [7] and I. Karatzas and S. Shreve [22]. But in our study, as the stopping time is not necessarily the optimal stopping time, we do not have to deal with the optimal stopping problem. For example if the agent is an American option s seller with strike K, and pay-off ξ = S τ T K +, then τ is a stopping time, which represents the time when the 4

6 buyer of the option decides to exercise his option. The seller wants to hedge against the risk of this option, and so wants to get ξ at random time τ : Y τ T = ξ. His consumption is here supposed to be zero. His wealth at time t is Y t = n i= θi ts i t. The standard self-financing hypothesis can be written as : dy t = n θtds i t. i i= It means that the consumption is only financed with the profits realized by the portfolio, and not by outside benefits. Then, the wealth of the agent satisfies the following equation : dy t = θ t S t r t dt + n θt i St i b i t dt + i=1 n θts i tσ i t, i dw t. We denote by π i t = θ i ts i t the amount of wealth invested in the i th asset for i = 1,..., d, and we notice that θ t S t = Y t d 1 πi t. We denote also by π t = π i t, i = 1,.., d the portfolio or strategy, and so the total wealth can be written as a solution of the following stochastic differential equation : i=1 dy t = Y t r t dt + π t, b t r t 1 dt + π t, σ t dw t, where 1 is the vector with all coordinates equal to 1. The previous line can also be rewritten by integrating from t τ to T τ : Y T τ Y = T τ T τ T τ Y s r s ds + π s, b s r s 1 ds + π s, σ s dw s a.s. 5

7 so : T τ T τ Y = Y T τ [Y s r s + π s, b s r s 1] ds σ }{{} sπ s, dw }{{} s a.s. fs,y s,z s Z s The wealth equation can be written as a BSDE with random terminal time. A first interest of writing the problem in this way is to model the hedging problem with a unique equation, and another interesting aspect is that such a tool does not use the notion of equivalent martingale measure to solve the hedging problem see N. El Karoui, S. Peng and M.-C. Quenez [9]. We consider two different agents in the market : agent N is a normally informed agent, as presented above, and the other agent, agent I, has got an additional information on the market. He knows, at time t =, a random variable L F T, where T T in general T > T. The filtration representing his information is obtained by enlarging the natural filtration of the Brownian motion. Y t = s>tf s σl. Given that the actualized assets prices are martingales in the initial probability space under a risk-neutral probability, it would be interesting and natural to check if they still have similar properties in the larger space. So we wonder under which condition we have the following useful property : Hypothesis H If M t t T is a given F, IP-martingale or semi-martingale, then M t t T is a Y, IP-semi-martingale. This problem has been introduced and studied by T. Jeulin and M. Yor [2, 19, 21], 6

8 next by J. Jacod [15], and later by J. Amendinger [1] and A. Grorud and M. Pontier [13]. In the current paper, we will work with the following assumption : Hypothesis H 3 There exists a probability Q equivalent to IP under which F t and σl are independent, t < T. Remark that H 3 implies H see for example J. Amendinger [1] or A. Grorud and M. Pontier [13]. Among the remarkable consequences of this hypothesis, we can notice that, W is a Y, Q-Brownian motion see J. Jacod [15]. Another very important tool that holds under hypothesis H 3 is a martingale representation theorem for Y, Q-martingales with respect to Brownian motion W, stated in J. Jacod and A.N. Shiryaev [16] Theorem III.4.33 p. 189, and used as a key tool in the proof of the main Theorem in A. Eyraud-Loisel [11]. Some of the questions raised in this article that naturally appear when dealing with additional information is : will the informed agent hedge his option as the non informed agent? Will he have a different hedging strategy? The case of European contingent claims was studied by A. Eyraud-Loisel 24 [11], who solved the existence and uniqueness problem of solution of a backward stochastic differential equation with fixed horizon, under an initial enlargement of filtration. Under hypothesis H 3, she proved that the informed agent will have a unique hedging strategy for the European option, which is the same as a non informed agent. The main financial question is : is it the case for American options? This turns out to be mathematically : does the BSDE with random terminal time under enlarged filtration have a unique solution? Is this solution adapted to the small filtration? 7

9 2.2 Mathematical formulation Mathematically speaking, from a more general point of view, we are looking for a solution of the following BSDE with random terminal time : T τ T τ Y = ξ + fs, Y s, Z s ds Z s, dw s, t T, which belongs to the enlarged σ-algebra Y t t T, and where ξ L 2 Y τ is the terminal condition, f : Ω [, T ] IR d IR d d IR d is the driver, Y t is the total wealth of the portfolio at time t, Z t represents the portfolio investments at time t. One of the fundamental results on solutions of BSDEs with random terminal time is an existence and uniqueness theorem given by E. Pardoux [24] which gives the existence and uniqueness of the solution of the BSDE with random terminal time under some Lipschitz hypotheses and also monotonicity conditions on the driver function. P. Briand and Y. Hu [6] improved the result in the one-dimensional case and M. Royer [29] reduced again the hypotheses on monotonicity conditions. 3 BSDE with random terminal time under enlarged filtration 3.1 Stopping time a.s. bounded by T < T Let τ be a stopping time a.s. bounded by T. τ T < T. Suppose f., y, z is F-progressive measurable, and satisfies the following assumptions 8

10 A 1 f is Lipschitz with respect to z Lipschitz constant K, A 2 f is continuous with respect to y and an increasing function ϕ : IR + IR + such that ft, y, ft,, + ϕ y, IP a.s., t, y, A 3 f is monotonous with respect to y : µ IR such that < y y, ft, y, z ft, y, z > µ y y 2, IPa.s., t, y, y, z, A 4 IE IP T ft,, 2 dt <. We denote by Ω, F, P a filtered probability space which stands either for the standard space Ω, F, IP or for the enlarged space Ω, Y, Q. We define, for any time A R +, the following spaces of F-progressively measurable processes : { M 2 P,F, A; IR k = IR k -valued F-adapted process ψ; } A IE P ψ s 2 ds <, } SP,F, 2 A; IR d = {IR d -valued F-adapted process ψ; IEP sup s A ψ s 2 <. We are looking for a solution of the following BSDE with random terminal time : Y = ξ + fs, Y s, Z s ds Z s, dw s, t T. 1 Definition 3.1 A Ω, F, P -solution or a solution on Ω, F, P to equation 1 is a pair of F-progressively measurable processes IR d IR d d -valued Y t, Z t t T such that 9

11 1. Z M 2 P,F, T ; IR d d, 2. On the set {t τ}, we have Y t = ξ and Z t =, 3. t [, T ], we have Y = Y τ + fs, Y s, Z s ds Z s, dw s. We look for a solution successively on the standard space Ω, F, IP and on the enlarged space Ω, Y, Q. Under the Brownian filtration, the previous hypotheses on the driver f guarantee the existence and uniqueness of a Ω, F, IP-solution. see E. Pardoux [24] for a proof. We prove now that under the same hypotheses on the driver f, asking for an additional integration hypothesis under the new probability Q, the BSDE with random terminal time also has a unique solution in the enlarged space Ω, Y, Q. Theorem 3.2 Under the hypotheses A1 to A4 on f, and if IE Q T fs,, 2 ds <, then for all ξ L 2 QY τ, the BSDE Y = ξ + fs, Y s, Z s ds Z s, dw s, t T has a unique Ω, Y, Q-solution. Remark : On the set {t τ}, we have Y t = Y τ = ξ. Proof of Theorem 3.2 Let us first prove the existence of a solution. We fix ξ L 2 QY τ. From existence and uniqueness Theorem in A. Eyraud-Loisel 25 [11], we can define Y t, Z t t T 1

12 M 2 Q,Y, T ; IR k d M 2 Q,Y, T ; IR k d as the unique Y t -adapted solution of the following BSDE t [, T ], Y t = ξ + T t 1l s τ fs, Y s, Z s ds T t Z s, dw s, which can be rewritten in the following way t [, T ], Y t = ξ + fs, Y s, Z s ds T t Z s, dw s. We fix t [, T ].By definition, Y = ξ + fs, Y s, Z s ds T Z s, dw s. Consequently Y T τ = ξ T τ Z s, dw s, hence T τ Z s, dw s is Y τ -measurable. T This leads to τ Z T s, dw s = IE Q τ Z t s, dw s Y τ =,as Z s, dw s is a Y t, Q-martingale. T Hence IE Q τ Z s 2 ds Z t = Q a.s. on the set {τ < t T }. T 2 = IE Q τ Z s, dw s t =,which means that We deduce Y = ξ + fs, Y s, Z s ds Z s, dw s. Besides, t [, T ], IE Q Y t Y 2 = IE Q t Z s 2 ds. Then it follows that, t [, T ], Y t = Y dt dq a.s., and we have the existence of the solution of our BSDE in the enlarged space. Suppose now that there exists two Ω, Y, Q-solutions to equation 1, denoted by Yt 1, Zt 1 t [,T ] and Yt 2, Zt 2 t [,T ]. Let us recall that t τ, Yt 1 = Yt 2 Z 1 t = Z 2 t =. We set Ŷ = Y 1 Y 2 and Ẑ = Z1 Z 2. = ξ and We first wish to prove that Y 1 = Y 2 Q a.s. 11

13 Let λ IR + and t [, T ]. Applying Itô s formula to e λs Ŷs 2 we get that e λ Ŷ 2 = e λτ Ŷτ 2 2 λ e λs Ŷs 2 ds + 2 e λs Ŷ s Ẑs, dw s e λs Ẑs 2 ds e λs < Ŷs, fs, Y 1 s, Z 1 s fs, Y 2 s, Z 2 s > ds. Hence, by taking the convenient expectation, IE Q e λ Ŷ 2 2µ λ IE Q + 2K IE Q K 2 + 2µ λ IE Q T e λs Ŷs 2 ds e λs Ŷs Ẑs ds IE Q e λs Ŷs 2 ds. e λs Ẑs 2 ds We deduce that for any λ 2µ + K 2, IE Q e λ Ŷ 2, which yields to the uniqueness of Y in M 2 Q,Y, T ; IR d. After replacing Ŷ by in the equation satisfied by Ŷ, Ẑ, we obtain t [, T ], fs, Y 1 s, Z 1 s fs, Y 1 s, Z 2 s ds = Ẑs, dw s. Given that a martingale can be equal to a finite variation process if and only if it is a null process, then IE Q Ẑs 2 ds =, which provides also the uniqueness of Z in M 2 Q,Y, T ; IR k d and ends the proof of Theorem 3.2. Corollary 3.3 Under the same hypotheses as in Theorem 3.2, we get that the unique Ω, Y, Q- 12

14 solution Y t, Z t t T of BSDE 1 satisfies Y t t T S 2 Q,Y, T, IRd. Proof. From Itô s formula and Burckholder-Davis-Gundy s inequality, and by using assumptions A1 to A4, we get that 1 2 IE Q sup Y t 2 t T IE Q ξ 2 + IE Q T fs,, 2 ds µ + K 2 IE Q T Y s 2 ds + 2CBDG 2 T IE Q Z s 2 ds, which is finite thanks to the hypotheses on ξ and f, and because the solution Y, Z of the BSDE 1 is in M 2 Q,Y, T ; IR d M 2 Q,Y, T ; IR d d. 3.2 Stopping time a.s. strictly bounded by T Let now τ be a F-stopping time, a.s. strictly bounded by T : τ < T. We want to solve the following BSDE Y = ξ + fs, Y s, Z s ds Z s, dw s, t T. 2 We will now suppose that f., y, z is F-progressively measurable, and satisfies the following assumptions : A1 f is Lipschitz with respect to z, with Lipschitz constant K, 13

15 A2 f is continuous with respect to y and for the same constant K : ft, y, z ft,, + K y + K z, IP a.s., t, y, A3 f is monotonous with respect to y : µ R such that < y y, ft, y, z ft, y, z > µ y y 2, IP a.s., t, y, y, z, A4 IE IP T ft,, 2 dt <. Definition 3.4 A Ω, F, P -solution or a solution on Ω, F, P to equation 2 is a pair of F-progressively measurable processes IR d IR d d -valued Y t, Z t t T such that 1. Y, Z S 2 P,F, T ; IR d M 2 P,F, T ; IR d d, 2. On the set {t τ}, we have Y t = ξ and Z t =, 3. t [, T ], we have Y = Y τ + fs, Y s, Z s ds Z s, dw s. We look again for a solution successively on the standard space Ω, F, IP and on the enlarged space Ω, Y, Q. Under the Brownian filtration, the previous hypotheses on the driver f guarantee the existence and uniqueness result for ξ L 2 P F τ see E. Pardoux [24], or [25] Theorem 4.1 p. 23. In the same way as in the previous section, if we require an additional integration hypothesis under the new probability Q, the BSDE with random terminal time also has a unique solution in the enlarged space Ω, Y, Q. 14

16 Theorem 3.5 Let us suppose that τ < T a.s. and IE Q T fs,, 2 ds <. We also consider ξ L 2 QY τ. Then, under the hypotheses A1, A2, A3 and A4 on f, 1. the BSDE Y = ξ + fs, Y s, Z s ds Z s, dw s, t T has a Ω, Y, Q-solution Y t, Z t t T, satisfying for any λ 2µ + 2K IE Q sup t τ e λt Y t 2 + C IE Q e λτ ξ 2 + e λt Y t 2 dt + e λt Z t 2 dt e λt ft,, 2 dt Y t, Z t t T is the unique Ω, Y, Q-solution of 2, satisfying 3. Proof of Theorem 3.5 We first prove the existence of the solution. For each n N, we construct a solution {Yt n, Zt n ; t } on the fixed interval [, T 1 n ], of the following BSDE n Yt n = ξ n + fs, Ys n, Zs n ds n Z n s, dw s, t T n, where T n, τ n, ξ n denote respectively T n = T 1 n, τ n = T n τ, and ξ n = ξ 1l τ Tn. This equation has a unique solution in M 2 Q,Y, T n ; IR d M 2 Q,Y, T n ; IR d d, according to the existence and uniqueness result of Theorem 3.2 from previous subsection, for a random terminal time bounded by T 1 n < T. Moreover, {Y n t, Z n t ; t [T n, T ]} is defined by Y n t = ξ n, t > τ n, Z n t =, t > τ n. 15

17 Hence Z n M 2 Q,Y, T ; IR d d, and Y n M 2 Q,Y, T ; IR d, n N. Let us now consider m > n, and define Y t = Y m t Y n t, Z t = Z m t Z n t. We fix λ 2µ+2K We want to prove that Y n n N and Z n n N are Cauchy sequences respectively in Q-probability and in M 2 Q,Y, T ; IR d d. So they will be convergent in the same spaces. Moreover, the limit will satisfy the BSDE 2. We first have for any t T n < T m : m Y t = ξ m ξ n + m fs, Y m s Z m s, dw s + n, Z m s ds Z n s, dw s. n fs, Y n s, Z n s ds Consequently, for any t T n, applying Itô s formula to e λt Y t 2 between t τ and τ n = T n τ, we obtain IE Q e λ Y 2 + IE Q n e λs λ Y s 2 + Z s 2 ds 4 n IE Q e λτn ξ m ξ n 2 + 2IE Q e λs µ Y s 2 + K Y s Z s ds, and using λ 2µ + K 2, we get IE Q e λ Y 2 e λt IE Q ξ m ξ n 2. 5 We also have IE Q ξm ξ n 2 = IE Q ξ 2 1l Tn<τ T m. Because we have supposed τ < T a.s., this term tends to as n goes to infinity according to dominated convergence Theorem. Hence Y goes to in L 2 QY τ as n tends to infinity. 16

18 Next, consider the case of t [T n, T m ]. Then Yt n = ξ n, and m Y t = ξ m ξ n + fs, Ys m, Zs m ds m Z m s, dw s. We can also apply Itô s formula on e λt Y t 2 between t τ and τ m. Remarking that Z n s = s T n, and so Z m s = Z s for s t, we obtain : m e λ Y 2 + = e λτm ξ m ξ n m e λs λ Y s 2 + Z s 2 ds m e λs Y s, Z s dw s e λτm ξ m ξ n m m e λs Y s fs, ξ n, ds 2 e λs < Y s, fs, Ys m, Z s > ds e λs µ Y s 2 + K Y s Z s ds m e λs Y s, Z s dw s. 6 We take the Q-expectation of the previous inequality, the martingale term has expectation. Hence, using standard inequalities, we deduce IE Q e λ Y 2 IE Q e λτm ξ m ξ n 2 + 2µ + K λ IE Q m + IE Q m e λs fs, ξ n, 2 ds e λt IE Q ξm ξ n 2 + IE Q m τ n e λs fs, ξ n, 2 ds e λs Y s 2 ds, 7 because λ 2µ + K Besides, from assertion A2, fs, ξ n, 2 2 ft,, 2 + 2K ξ n 2 2 ft,, 2 + 2K ξ 2. 17

19 Consequently, hypotheses of the theorem provide the Q-integrability of e λs fs, ξ n, 2 ds. Moreover, given that τ m τ n tends to zero Q a.s. when n goes to infinity, the right term of 7 tends to zero as n tends to infinity. This proves that Y t also tends to zero in L 2 QY τ. We obtain that Z t also converges to in M 2 Q,Y, T ; IR d d by taking the limit as n tends to infinity in inequality 6. Finally, we consider t [T m, T ]. As said previously, Y m t Y n t = ξ m ξ n converges Q a.s. to zero as n tends to the infinity, so in this case also, Y t goes to in Q-probability. Hence Y n t n is a Cauchy sequence in Q-probability. We now need to state the following a-priori estimate, which provides an upper bound of the expected wealth process Y under probability Q, whenever ξ L 2 QY τ. Lemma 3.6 Y n, Z n satisfies the following inequality, for any λ 2µ + 2K , IE Q e λ Y n 2 e λt E Q ξ 2 + IE Q e λs fs,, 2 ds, and also IE Q sup t τ e λt Yt n 2 + C IE Q e λτ ξ 2 + e λs Ys n 2 ds + e λs fs,, 2 ds. e λs Zs n 2 ds Proof of Lemma 3.6 Let Y n n N be the sequence previously defined in the proof of existence in Theorem 3.5. We first deal with the first inequality. We apply Itô s formula to e λs Y n s 2 between t τ and τ n, and as previously, we use standard inequalities to obtain the majoration. We have, t T n, n e λ Y n 2 + = e λτn ξ n e λs λ Ys n 2 + Zs n 2 ds n e λs < Y n s, fs, Y n s, Z n s > ds 2 n e λs Y n s, Z n s dw s 18

20 n e λτn ξ n n e λs µ Ys n 2 + K Ys n Zs n n ds 2 e λs Ys n, Zs n dw s e λs Y n s fs,, ds. 8 So, IE Q e λ Y n 2 IE Q e λτn ξ n 2 + IE Q n e λs fs,, 2 ds. 9 For the second inequality, let us first apply Burkholder-Davis-Gundy s inequality to inequality 8. We obtain, η, ε > : n IE Q [1 η sup e λt Yt n 2 + e λs λ 2µ εk 2 1 Ys n ] t τ ε Zn s 2 ds IE Q e λτn ξ n 2 n n + IE Q e λs fs,, 2 ds + 2CBDGIE 2 Q e λs Zs n ds. Taking ε = 2 and η = 1/2 provides the following : IE Q sup t τ n n e λt Yt n 2 + e λs Ys n 2 ds + e λs Zs n 2 ds 2IE Q e λτn ξ n 2 n + IE Q e λs fs,, 2 ds + 2CBDG 2 n 1 e λs Zs n ds. In order to get an upper bound on the last term, we use again Itô s formula on e λs Y n s 2 and take the expectation under Q. We obtain : IE Q n e λs Zs n 2 ds 2 IE Q e λτn ξ 2 n + IE Q e λs fs,, 2 ds. We replace it in inequality 1, and obtain finally : IE Q sup t τ n n e λt Yt n 2 + e λs Ys n 2 ds + e λs Zs n 2 ds IE Q e λτn ξ n 2 n + IE Q e λs fs,, 2 ds, C 2 BDG 19

21 which completes the proof of Lemma 3.6. Let us go back to the proof of Theorem 3.5. Lemma 3.6 establishes the expected inequalities only for Y n, Z n. We should obtain identical results for the solution by passing to the limit. For this purpose, we take the conditional expectation of equation 8. We obtain : T e λ Y n 2 e λt IE Q ξ 2 + fs,, 2 ds Y. This proves that Y n 2 is dominated by a Q-integrable process independent of n, and the same is true for sup t τ e λt Y n t 2. This allows us to dominate also Y 2, and to apply dominated convergence Theorem. We conclude that Y n is a Cauchy sequence in M 2 Q,Y, T ; IR d. Taking the limit in equation 4, we deduce that the same holds for Z t, and Z n t n is also a Cauchy sequence in M 2 Q,Y, T ; IR d d. This allows us to define Y, Z as the limit of the sequence Y n, Z n n. Next, taking the limit in equation 9 as n tends to infinity, we obtain finally IE Q e λ Y 2 e λt E Q ξ 2 + IE Q e λs fs,, 2 ds, which provides the first inequality. And taking the limit as n goes to infinity in equation 11 leads to : IE Q sup t τ e λt Y t 2 + e λs Y s 2 ds + e λs Z s 2 ds IE Q e λτ ξ 2 + IE Q e λs fs,, 2 ds C 2 BDG This has just allowed us to define the process Y, Z as the limit of the sequence Y n, Z n n. This process satisfies equation 2 so it is solution of the BSDE f, ξ, τ, and satisfies equation 3 : existence is proved. 2

22 We have now to state the uniqueness. Let Y, Z and Y, Z be two solutions, which satisfy equations 2 and 3. Let Ȳ, Z = Y Y, Z Z. It follows from Itô s formula applied between t τ and τ T n = τ n, and the assumptions A1, A2, A3 and A4 that for all λ R and for t T n, n e λ Ȳ 2 + e λτn Ȳτ n n e λs λ Ȳs 2 + Z s 2 ds n e λs µ Ȳs 2 + K Ȳs Z s ds e λs Ȳs, Z s dw s. 12 Combining the above inequality with 2K Ȳs Z s Z s 2 +K 2 Ȳs 2, we deduce n n e λ Ȳ 2 e λτn Ȳτ n 2 + 2µ + K 2 λ e λs Ȳs 2 ds 2 e λs Ȳs, Z s dw s. Then we take expectation under Q of the previous inequality. As W s s<t is a Y, Q-Brownian motion, so that the last term is a Y, Q-martingale on [, T n ], null at T n, so with expectation. Hence E Q e λ Ȳ 2 E Q e λτn Ȳτ n 2 t [, T n ]. e λτn Ȳτ n 2 is dominated by e λt 2 sup t T Y t sup t T Y t 2, which is Q- integrable by definition of a solution. By dominated convergence Theorem, since τ n goes to τ, and T n goes to T as n goes to infinity, we deduce that E Q e λ Ȳ 2 E Q e λτ Ȳτ 2. 21

23 And as Ȳτ =, since Y τ = Y τ = ξ, this proves that t τ, Ȳt =, Q a.s. As this is also true for t τ, as Y t = Y t = ξ by Definition 3.4, we deduce Ȳt = t [, T ] Y T = Y T = ξ as τ < T a.s. so this also holds for T. By replacing Ȳt by in equation 12, we also obtain Z t = t [, T ], Q a.s. Uniqueness is then proved, and Theorem 3.5 is stated. 3.3 BSDE with jumps For simplicity reasons in the proofs, we have chosen to develop these Theorems only for BSDEs driven by a Brownian motion. Nevertheless, these results can easily be extended to the case of BSDEs driven by both a Brownian motion and a Poisson measure, as the same results have been proved by A. Eyraud-Loisel [11] in the case of a fixed terminal time. The proofs are similar to the previous ones. 4 Financial Interpretation 4.1 Hedging by an informed agent We supposed at the beginning a no-arbitrage market, with in particular σ invertible. This allows us to deduce from the solution Y s, Z s of the BSDE, the strategy π s of the portfolio in the risky assets. The main consequence of our result Theorem 3.5 is that the informed agent in our model will have a unique hedging strategy for any contingent claim which satisfies the hypotheses, in particular for which the exercise time is a bounded stopping time maturity strictly bounded by T, the time at which the private information is revealed. This is very often the case for most financial contingent claims traded in the market, as it is presented in the following paragraph for American and Lookback options. This extends the work of A. Eyraud-Loisel 25 [11]. 22

24 The results mean in particular that information L F T satisfying H 3 does not provide any additional hedging strategy to the informed agent. In fact, for a ξ F τ, both agents will have a unique admissible hedging strategy, respectively F and Y- adapted. But as the BSDE is the same for both agents, the F-adapted solution of the non informed agent is also a solution of the enlarged BSDE because it is also adapted to the enlarged filtration, so it is the unique solution of this BSDE. In other words, the strategy of the informed agent is adapted to the small filtration F. Then both agents have the same hedging strategy. Remark : Let us notice that the fact that obtaining this result only strictly until T is not very surprising. The results can not be generalized until T. This is mathematically due to the hypotheses of enlargement of filtration, which can hold only strictly until T, but financially, it is quite easy to find some examples of contingent claims with terminal time T, where an information on time T provides a different hedging strategy. Let us take for instance a digital option, 1l ST K and suppose that the information is L = S T or even 1l ST [a,b] then, the insider trader will hedge this option by doing nothing if S T > K and invest in the non risky asset otherwise, which is different from what would do an ordinary agent. 4.2 Examples Most of the contingent claims traded in the market do not have only a finite exercise time as European options, but rather have an exercise period as for American and Lookback options for instance, and the terminal time of the hedging problem therefore is a random stopping time. This is the reason why we tried to generalized the previous results to the case of random terminal time for the BSDE with enlarged 23

25 filtration. We will see here after that our results can definitely be applied to the financial framework described in the introduction, because in the particular case of American and Lookback options, the driver f satisfies the required hypotheses of Theorems 3.2 and 3.5. Let us precise that Theorem 3.5 gives a mathematical generalization of Theorem 3.2, but does not give more financial applications American options If we set the financial problem of hedging an American option with maturity T < T, when the exercise time τ may occur at any time before T, the obtained BSDE is a BSDE with random terminal time. In an enlarged filtration, we can solve the BSDE as stated in the previous results Theorem 3.2 as τ T < T. The generator is in this case the following : fs, y, z = r s y + b s r s σ s z. 13 And the payoff has the form ξ = S τ K +. In order to satisfy the hypotheses required in Theorem 3.2, we have to check some b s r s properties on coefficients : A1 needs ess sup to be finite ; A2 s σ s b s r s is verified with setting K = ess sup r s ess sup ; for A3 we set s s σ s µ = ess sup r s ; and A4 is satisfied since ft,, =. s As a conclusion, ξ L 2 b s r s F τ, ess sup r s and ess sup finite are sufficient conditions to ensure existence and uniqueness of an hedging portfolio for s s σ s an American option with or without an additional information satisfying hypothesis H 3. 24

26 Remark : In this financial application, ft,, = so we can write in this case see inequality 3 for instance : IE Q sup t τ e λt Y t 2 + IE Q IE Q e λt Y t 2 e λt E Q ξ 2 e λt Y t 2 dt + e λt Z t 2 dt C IE Q e λτ ξ 2, which provides in particular an upper bound on the expected wealth process and the expected maximum wealth process Lookback options Another example is Lookback option : an option with maturity T < T which pays at exercise the supremum of the price process among the last 2 months. So ξ has mathematically the form : ξ = sup t [τ 2,τ] S t. Applying Itô and Burkholder-Davis-Gundy on the SDE of the price process provides that ξ L 2 F τ. Besides, the generator f is the same as in the equation 13. So with the same conditions on the parameters, we can apply Theorem 3.2 or 3.5 to the case of Lookback options also. 5 Conclusion The results obtained here are consistent with the results obtained for a fixed terminal time see A. Eyraud-Loisel 25 [11]. A further mathematical approach of such problems of BSDE with enlarged filtration and random terminal time will be to look at BSDE with uncertain horizon : the terminal time is not a random time any more, but a general random variable. It will be interesting to look at existence and uniqueness of solutions of such equations A. Eyraud-Loisel and M. Royer 27 25

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