Non-arbitrage condition and thin random times

Transcription

1 Non-arbitrage condition and thin random times Laboratoire d'analyse & Probabilités, Université d'evry 6th General AMaMeF and Banach Center Conference Warszawa 14 June 2013

2 Problem Question Consider a Default-free Model (Ω, A, F, P, X ) without arbitrage opportunities. Then, incorporate the default event τ to this model. Is the Defaultable Model (Ω, A, F τ, P, X ) arbitrage free? Choulli T., Aksamit A., Deng J., and Jeanblanc M., Non-arbitrage up to Random Horizon and after Honest Times for Semimartingales Models, Working paper, Aksamit A., Choulli T., Jeanblanc M., Thin random times, Working paper, 2013

3 Random times Set a ltered probability space (Ω, A, F, P) and consider a random time τ, i.e. positive A-measurable random variable. The process Z dened as Z t = P(τ > t F t ) is called the Azéma supermartingale associated with τ. The process Z dened as Z t = P(τ t F t ) is second important supermartingale associated with τ. The processes A o and A p denote respectively dual optional and predictable projections of the process A = 1 [τ, ). The Doob Meyer decomposition of Azéma supermartingale Z is Z = µ A p, with µ t = E(A p F t ) and optional decomposition is Z = m A o, with m t = E(A o F t ).

4 Enlargement of ltration Progressively enlarged ltration F τ associated with τ is dened as F τ t = s>t(f s σ(τ s)). Initially enlarged ltration F σ(τ) associated with τ is dened as F σ(τ) t = s>t(f s σ(τ)). For general enlargement of ltration F G we talk about two hypotheses (H) hypothesis: Each F-martingale remains a G-martingale. (H ) hypothesis: Each F-martingale remains a G-semimartingale.

5 Honest times Random time τ is an honest time if for each t 0 there exists F t -measurable random variable τ t such that τ = τ t on (τ < t). If τ is an honest time, then each F-martingale X remains an F τ -semimartingale with semimartingale decomposition given by t τ X t = X t Z s where X is F τ -martingale. t d X, m s τ 1 d X, m s 1 Z s Let τ be a random time. For each F-martingale X, the process X τ is F τ -semimartingale with semimartingale decomposition where X is F τ -martingale. t τ X t = X t Z s d X, m s

6 Non-arbitrage condition NUPBR Let X be an F-semimartingale. We say that X satises No Unbounded Prot with Bounded Risk (NUPBR) if K(X ) := {(H X ) T : H L(X ) and H X 1} is bounded in L 0 (P). Theorem(Takaoka) The F-semimartingale X satises NUPBR if and only if L σ (X ), where L σ (X ) is the set of σ-densities given by L σ (X ) := {L M loc (F) : L > 0 and LX is an σ-martingale}.

7 Non-arbitrage up to Random Horizon and after Honest Time Theorem 1 Let τ be a random time. Then, the following are equivalent: The thin set { Z = 0 & Z > 0} is evanescent. For any process X satisfying NUPBR(F), X τ satises NUPBR(F τ ). 2 Let τ be an honest time satisfying Z τ < 1 a.s.. Then, the following are equivalent: The thin set { Z = 1 & Z < 1} is evanescent. For any process X satisfying NUPBR(F), X X τ satises NUPBR(F τ ).

8 Avoidance of F-stopping times Often in the literature standard assumption on a random time is used: (A) assumption: τ avoids F stopping times, i.e. P(τ = T ) = 0, for any F-stopping time T.

9 Strict and thin random times Denition A random time τ is called 1 a strict random time if [[τ]] [[T ]] = for any F-stopping time T. 2 a thin random time if its graph [[τ]] is contained in a thin set. i.e. if there exists a sequence of F stopping times (T n ) n=1 with disjoint graphs such that [[τ]] [[T n n]]. The sequence (T n ) n is then exhausting sequence of a thin random time. A random time τ is strict and thin random time if and only if τ =.

10 Decomposition of a random time Denition Pair of random times (τ 1, τ 2 ) is the decomposition of a random time τ if 1 τ 1 is a strict random time and τ 2 is a thin random time; 2 τ 1 τ 2 = τ ; 3 τ 1 τ 2 =. Theorem Each random time τ has a decomposition (τ 1, τ 2 ).

11 Decomposition of a random time Dene τ 1 := τ { A o τ =0} and τ 2 := τ { A o τ >0}. We see that the time τ 1 is a strict random time as P(τ 1 = T < ) = E( 0 1 {u=t } 1 { A o u =0}dA o u) = 0. and the time τ 2 is a thin random time as [[τ 2 ]] = [[τ]] { A o > 0} = [[τ]] [[T n ]] [[T n ]]. n n

12 Decomposition of a random time Lemma: Alternative characterisation 1 The random time τ is a thin random time if and only if its dual optional projection is a pure jump process. 2 The random time τ is a strict random time if and only if its dual optional projection is a continuous process.

13 Decomposition of a random time Remark 1 Decomposition of a stopping time, P instead of O. 2 τ = τ 1 τ i 2 τ a 2 τ 1 = τ { A o τ =0} strict part τ i 2 = τ { A o τ >0, A p τ =0} totally inaccessible thin part τ a 2 = τ { A p τ >0} accessible thin part

14 Poisson ltration example Let F X be a ltration of CPP X t = N t k=1 Y k, where N is a Poisson process with parameter η and sequence of jump times (θ n ) n=1 and Y k are i.i.d. positive random variables, independent from N, with cumulative distribution function F. Dene the random time τ = sup{t : µt X t a} with a > 0. Under the condition µ > η E(Y 1 ), the random time τ is nite a.s. Since τ is a last passage time, it is an honest time in the ltration F. Furthermore, since the process µt X t has only negative jumps, one has µτ X τ = a A o = C n 1 1 [Tn, ) with T n = inf{t > T n 1 : µt X t = a} and T 0 = 0.

15 Brownian ltration example: local time approximation Let B be a Brownian motion. For ε > 0, dene a double sequence of stopping times by U 0 = 0, V 0 = 0 U n = inf{t V n 1 : B t = ε}, V n = inf{t U n : B t = 0}. and process D t = max{n : V n t} which is the number of downcrossings of B from level ε to level 0 before time t. Dene a random time τ ε = sup{v n : V n T 1 } with T 1 = inf{t : B t = 1}. A o = εd t T1 + ε and { Ao > 0} = [0, T 1 ] n=0 [[V n]]

16 (H ) hypothesis and decomposition formula Let τ be a thin random time with exhausting sequence (T n ) n. Denote by C n = {τ = T n }, so τ = 1 n Cn T n. Theorem For each thin random time τ, the hypothesis (H ) is satised between F and F τ. Any F martingale X can be decomposed as t τ X t = X t Z s d X, m s + n 1 Cn t {Tn<s} zs n d X, z n s, with z n t = P(C n F t ), where X is F τ local martingale.

17 (H ) hypothesis and decomposition formula Theorem (Jacod) Suppose that F C is an initial enlargement of the ltration F with an atomic σ-eld generated by C = ((C n ) n ). Then, the ltration F C satises (H ) hypothesis and each F martingale X can be decomposed in F C as X t = X t + n t 1 1 Cn d X, z n s 0 z n s with z n t = P(C n F t ), where X is F C local martingale.

18 (H ) hypothesis and decomposition formula F τ predictable process H can be decomposed as H t = 1 {t τ}j t + 1 {τ<t}k t (τ) t > 0 where J is F predictable process and K : R + Ω R + R is P B(R + ) measurable. As τ is thin we can rewrite process H as H t = J t 1 {t τ} + n 1 {Tn<t}K t (T n ) 1 Cn Note that each process 1 {Tn<t}K t (T n ) is F predictable.

19 (H ) hypothesis in general case Corollary Let τ be a random time and (τ 1, τ 2 ) its random time decomposition. Then: 1 The ltration F τ satises (H ) hypothesis if and only if the ltration F τ1 satises (H ) hypothesis. 2 The Azéma supermartingale of τ 2 in ltration F coincides with the Azéma supermartingale of τ 2 in F τ1, i.e. P(τ 2 > t F t ) = P(τ 2 > t F τ1 t ).

20 Thin honest times Let τ be an honest time and (τ 1, τ 2 ) its random time decomposition. Then, times τ 1 and τ 2 are honest times. Theorem For any honest time τ with decomposition (τ 1, τ 2 ) its Azéma supermartingale at τ can be written as Z τ τ 1 {τ< } = 1 {τ=τ1< } + Z τ τ 2 1 {τ=τ2< } where Z τ τ 2 < 1.

21 Thin honest times Remark For a thin honest time τ, the two following decomposition formulas coincide X t = X t + X t = X t + t τ 1 0 Z s t τ 0 1 Z s d X, m s + n t d X, m s 0 1 Cn t {Tn<s} zs n 1 1 {τ<s} 1 Z s d X, m s This is due to two possible representations of predictable process d X, z n s, 1 {τ<t}k(τ) = 1 {τ<t}k(τ t ) from honest time property, 1 {τ<t}k(τ) = n 1 {Tn<t}K(T n ) 1 Cn from thin time property.

22 Local martingale deator in F C and F τ Theorem Assume that all martingales in F are continuous. Let X be an F local martingale. Then: 1 The process L C t := 1 1 n Cn is a local martingale deator in F C for zt n X, i.e. it is strictly positive F C local martingale with L C 0 = 1 and L C > 0 a.s. such that XL C is an F C local martingale. 2 The process L τ t := ( ) 1 1 n Cn is a local martingale z n t 1 z n t Tn deator in F τ for X X τ, i.e. it is strictly positive F τ local martingale with L τ 0 = 1 and Lτ > 0 a.s. such that (X X τ ) L τ is an F τ local martingale.

23 Entropy of the partition Let C = (C n ) n be an F measurable partition of Ω. Then, the quantity H(C) := n P(C n ) log(p(c n )) is an entropy of C. Theorem[Meyer, Yor] Assume that all F local martingales are continuous and H(C) <. Let an F local martingale X be an element of H 2 (F). Then, an F C semimartingale X is an element of H 1 (F C ).

24 Entropy of the partition Meyer P.-A., Sur un Théorème de Jacod Yor M., Entropie d'une Partition et Grossissement Initial d'une Filtration The author of the rst paper posed the question about additional knowledge associated with thin random time: Un problème voisin, mais plus intéressant peut être, consiste à mesurer le bouleversement produit, sur un système probabiliste, non pas en forçant des connaissances à l'instant 0, mais en les forçant progressivement dans le système.

25 Entropy of thin random time In case of progressive enlargement with thin random time τ = n 1 Cn T n we suggest measurement of additional knowledge by H(τ) = n E ( 1 Cn log z n T n ). Remark If τ is an F stopping time then H(τ) = 0. If for any n the set C n is already in F Tn then we do not gain any additional information. H(τ) is invariant under dierent decompositions of τ.

26 Entropy of thin random time To justify this measurement of additional knowledge we give analogous result to the previous one Theorem Assume that all F local martingales are continuous and H(τ) <. Let an F local martingale X be an element of H 2 (F). Then, an F τ semimartingale X is an element of H 1 (F τ ).

27 Thank you for your attention!